j. ksiam vol.13, no.1, 1 11, 2009 a structured modeling ...€¦ · cases of peptic ulcer disease...

J. KSIAM Vol.13, No.1, 1–11, 2009

A STRUCTURED MODELING APPROACH OF PEPTIC ULCERS ANDH. PYLORI INFECTION

TAE SUG DO1†, GLENN LEDDER2, AND YOUNG S LEE3

1DEPARTMENT OF MATHEMATICS EDUCATION, KWANDONG UNIVERSITY, SOUTH KOREA

E-mail address: [email protected]

2DEPARTMENT OF MATHEMATICS, UNIVERSITY OF NEBRASKA-LINCOLN, NE, USAE-mail address: [email protected]

3DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, N. MANCHESTER, IN, USAE-mail address: [email protected]

ABSTRACT. Current therapies against ulcers caused by H. pylori infection consist of antibi-otics, an acid reducer, and some clinical trials underway to develop a H. pylori vaccine. Wedevelop a structured model with age-dependent mortality of peptic ulcers and H. pylori infec-tion. Our main goal is to analyze our structured model mathematically and to compare it toour previously unstructured model to examine the disease transmission dynamics in terms ofannual prevalence and annual incidence of the disease.

1. INTRODUCTION

A peptic ulcer is a sore on the lining of the stomach or duodenum, which is the beginning ofthe small intestine. Helicobacter pylori (H. pylori) is an emerging, spiral-shaped pathogen. Theconnection between peptic ulcers and H. pylori bacterium was discovered in 1983. it is nowbelieved that most ulcers are caused by H. pylori infection or the use of common nonsteroidalanti-inflammatory drugs instead of stress, acid, and spicy foods.

Approximately two-thirds of the world’s population is infected with H. pylori [1]. In theUnited States, 20% of people under 40 years old and 60% of people over 60 years old areinfected [2]. United States military personnel on foreign deployments have been found to havean increased risk of acquiring H. pylori infection, with an annual infection rate of between1.9% and 7.3% [3] since it is widespread in the developing countries as high as 80% to 90%[4]. About 10% of Americans suffer from peptic ulcers during their lifetime.

It is not known why H. pylori does not cause ulcers in every infected person. Most likely,infection depends on characteristics of the infected person, the type of H. pylori, and otherfactors yet to be discovered. About 15%-20% of subjects with H. pylori colonization will

Received by the editors December 16 2008; Accepted February 26 2009.2000 Mathematics Subject Classification. 37N25.Key words and phrases. Structured Model, Peptic Ulcers, H. Pylori.† Corresponding author.

1

2 DO, LEDDER, AND LEE

develop ulcers in their lifetimes [5][6]. In the United States, there are 500,000 to 850,000 newcases of peptic ulcer disease each year [1]. The infection is found in 90% to 95% of patientswith duodenal ulcers and 75% to 80% of patients with stomach ulcers [5][6]. Current therapiesagainst ulcers caused by H. pylori infection consists of antibiotics and an acid reducer. A testfor H. pylori infection and two or three weeks of medication cure 90% of the disease and therecurrence of ulcer after successful eradication of H. pylori is rare, compared with up to 80%when the infection persists [7]. The efficacy at the level of general practitioners is lower dueto poor compliance and increasing antibiotics resistance [8][9]. Also, many doctors are stilltreating ulcers without antibiotics. The fact that the prevalence of peptic ulcers in the UnitedStates is 4.5 millions with less than 850,000 new cases annually shows that changing medicalbelief and practice takes time.

We assume that 10% of the population is not susceptible to H. pylori based on [6]. Al-though H. pylori can be transmitted from person to person by a fecal-oral or oral-oral routeand the infection is decreasing due to sanitary awareness, prevention is difficult. In some en-demic developing countries, reinfection after eradication is serious. Recent studies have shownpromise with attenuated live vaccines used in combination with H. pylori antigens [10][11] andrecently it is shown that Mongolian gerbil can be successfully vaccinated against the infection[12]. Clinical trials are underway with the goal of producing an inexpensive, safe, and effectivevaccine in the near future.

Our previous work [13] developed an unstructured model of peptic ulcers and H. pylori withvaccination to discuss the disease transmission dynamics through stability and sensitivity anal-ysis. Rupnow et al [15][16] had quantified the population dynamics by modeling three differentage groups using partial differential equations. These and the cohort theory on peptic ulcersdeveloped by [14] motivated us to develop a structured model with age-dependent mortality.Our main goal is to analyze our structured model mathematically and then compare it to ourpreviously unstructured model.

We organize our paper as follows: Section 2 introduces two models, one new structured andone previously unstructured, and reduces the structured model to an integral equation, Section 3contains our mathematical analysis of the structured model, Section 4 estimates our parametersfor numerical analysis, and the last section concludes with the comparison of the two models.

2. TWO EPIDEMIOLOGICAL MODELS OF H. PYLORI

We consider two different models, one unstructured and one structured, for the dynamics ofH. pylori infection and related peptic disease. Both models include four classes: susceptible(S), infective and asymptomatic (H), infective and symptomatic (U ), and removed (R). Theunstructured model allows for investigation of dynamics that can occur if a vaccine is developedfor H. pylori; hence, it has an additional class (V ) which we omit from the structured populationmodel. It will also be frequently useful to consider a combined infective class I = H + U andthe total population N .

Individuals move among the classes by a number of different processes. A natural deathprocess removes individuals from all classes at a rate µ per capita. It is widely held that 10% to

STRUCTURED MODEL OF PEPTIC ULCERS 3

20% of people are naturally resistant to H. pylori infection; hence, the birth process adds peopleinto the removed class, as well as the susceptible and vaccinated classes, at rates (1 − n)bN ,n(1 − p)bN , and npbN respectively. Both models have a transmission process by whichsusceptibles become asymptomatic infectives at rate β′SI and a progression process by whichasymptomatics become symptomatic at rate αH . In the otherwise more complicated structuredmodel, we assume a curative process by which symptomatics become removed at rate ρU . Inthe unstructured model, we assume that symptomatics move into the vaccinated class whencured; we also include a loss of immunity process by which vaccinated individuals move intothe susceptible class at rate γ′V . See the schematic diagrams of Figures 1 and 2.

V S H U R

? ? ? ? ?

? ? ?- - - -γ′V β′IS αH

npbN n(1− p)bN (1− n)bN

µV µS µH µU µR

ρU

FIGURE 1. A schematic diagram of the unstructured model

S H U R

? ? ? ?

? ?- - -β′IS αH ρU

nbN (1− n)bN

µS µH µU µR

FIGURE 2. A schematic diagram of the structured model

2.1. The Unstructured Model. Letting r = b − µ, the unstructured model consists of theequations

dN

dt= rN, (1)

dS

dt= n(1− p)(r + µ)N + γ′V − β′S

I

N− µS, (2)

dI

dt= β′S

I

N− ρU − µI, (3)


dU

dt= αH − ρU − µU, (4)

dR

dt= (1− n)(r + µ)N − µR, (5)

H = I − U, V = N − S − I −R. (6)

2.2. The Structured Model. As an alternative H. pylori model, we consider a structuredmodel that accounts for age-dependent mortality. For simplicity, we assume that the parame-ters α, β, and ρ are age-independent, that immunity is permanent (γ = 0), and that there isno vaccination (p = 0). Thus, the patients cured of ulcers move into the removed class (seethe schematic in Figure 2). Let Sa, Ha, Ua, Ra, and Na be the age-density functions for thesusceptible, asymptomatic, symptomatic, removed, and total populations, where a measuresthe age from birth, not the age from entry into a particular class. Thus, the total population ofall ages and the total infective population of all ages are

N(t) =∫ ∞

0Na(a, t) da, I(t) =

∫ ∞

0[Ha(a, t) + Ua(a, t)] da, (7)

and the totals for the other groups are computed similarly. Because the infection process isage-independent, the rate at which susceptibles of age a are infected is proportional to the totalfraction of infectives. Thus, we have the equations

∂Na

∂t+

∂Na

∂a= −µ(a)Na, Na(0, t) = bN(t), (8)

∂Sa

∂t+

∂Sa

∂a= −β′Sa

I

N− µ(a)Sa, Sa(0, t) = nbN(t), (9)

∂Ha

∂t+

∂Ha

∂a= β′Sa

I

N− αHa − µ(a)Ha, Ha(0, t) = 0, (10)

∂Ua

∂t+

∂Ua

∂a= αHa − ρUa − µ(a)Ua, Ua(0, t) = 0, (11)

Ra = Na − Sa −Ha − Ua. (12)Suppose now that the population is constant. Equation 8 then becomes

dNa

da+ µ(a)Na = 0, Na(0) = bN,

with solutionNa = bNe−

∫ a0 µ(a) da.

Given that the total population is N , we obtain1b

=∫ ∞

0e−M(a) da, M(a) =

∫ a

0µ(a) da. (13)

The survival probability to age a is

Na(a)Na(0)

= e−M(a).


Thus, equation (13( expresses the life expectancy 1/b in terms of survival probability.We complete the steady-state model by normalizing all of the populations, as with the struc-

tured model. With β = nβ′, Sa = nNsa, I = nNi, and so on, we obtain the modeldsa

da+ [βi + µ(a)] sa = 0, sa(0) = b, (14)

dha

da+ [α + µ(a)]ha = βisa, ha(0) = 0, (15)

dua

da+ [ρ + µ(a)]ua = αha, ua(0) = 0, (16)

h =∫ ∞

0ha(a) da, u =

∫ ∞

0ua(a) da, i = h + u. (17)

2.3. Reduction of the structured model to an integral equation. The solution of Equation(14) is

sa = be−M(a)e−xa, x = βi.

We can now solve Equations (15)–(16) with the following computational lemma.

Lemma 2.1. Ifdza

da+ [c + µ(a)]za = Ae−M(a)e−ka and za(0) = 0, then

za =A

k − ce−M(a)

(e−ca − e−ka

),

∫ ∞

0za(a) da = A

L(c)− L(k)k − c

,

where

L(y) =∫ ∞

0e−M(a)e−ya da. (18)

From Lemma 2.1, we obtain the desired solutions for h and u as

ha =bβi

x− αe−M(a)

(e−αa − e−xa

), h =

bβi

x− α[L(α)− L(x)],

u =αbβi

x− α

(L(ρ)− L(α)

α− ρ− L(ρ)− L(x)

x− ρ

).

Substituting these results into i = h + u, we obtain an integral equation for x:

(ρ + α− x)L(x)− x2

bβ+

[ρ + α

bβ+

ρL(α)− αL(ρ)ρ− α

]x =

ρ2L(α)− α2L(ρ)ρ− α

+αρ

bβ. (19)

If there is an endemic equilibrium, it can be found by solving Equation (19), after which wehave

i =x

β, h = bx

L(α)− L(x)x− α

, u = i− h. (20)

The number of new ulcer cases per year and the total prevalence of ulcers, both relative to thetotal population, are

C =αH

N= αnh, P =

U

N= nu. (21)


3. ANALYSIS OF THE STRUCTURED MODEL

We now examine the qualitative behavior of the model of Equations (18)–(20) by focusingon the issue of existence of the endemic disease equilibrium (EDE), which is determined fromEquation (19). Consider first the limiting case β′ → ∞, which corresponds to β → ∞ whilekeeping x/β constant. In this limit, Equation (19) reduces to

−x2

bβ+

ρL(α)− αL(ρ)ρ− α

x = 0,

from which we obtain a solution for x and the corresponding equilibrium result

i? ∼ bρL(α)− αL(ρ)

ρ− α, β′ →∞.

In this special case, we have i? → 1 as ρ → 0 and i? → bL(α) < 1 as ρ → ∞. Thus,the combination of infinite transmission coefficient and no treatment leads to infection of theentire population, while the size of the infective class is positive but bounded as treatmentbecomes standard and instantaneous. These results are consistent with the model assumptionsand the results of the unstructured model; the combination of infinite transmission coefficientand treatment speed leads to an EDE with 0 < i? < 1 owing to the lag between infection anddevelopment of symptoms.

It also follows immediately from (19) that there is no EDE in the limit β′ → 0. As welower β′ from a value sufficient for EDE, we gradually reduce i? to 0, at which point the EDEdisappears. Thus, the critical case is when x = 0 in Equation (19). The criterion for eliminationof the EDE is therefore

1bnβ′

>ρ2[L(0)− L(α)]− α2[L(0)− L(ρ)]

αρ(ρ− α)≡ F (ρ; α), (22)

where the notation for F indicates that we are thinking of the treatment rate ρ as variable whilethe ulcer-development rate α is fixed. The remainder of the analysis of the structured model willconsist in determining the properties of the function F to see if the EDE elimination criterion(22) indicates properties similar to the corresponding criterion nβ′ < bφ of the unstructuredmodel. To that end, we begin by computing F (ρ) for the case µ(a) = b, which makes thestructured model equivalent to the unstructured model. Indeed, with µ = b, we have L(y) =(b + y)−1, and F reduces to

F (ρ) =ρ + α + b

b(α + b)(ρ + b)=

1b2φ

,

making the criteria for the two models identical.Additional properties of F are summarized in Theorem 1.

Theorem 3.1. F is positive, continuous, and decreasing for ρ ≥ 0, with F (0) the average ageof the population and F (∞) = α−1[L(0)− L(α)]. The range of F decreases as α → 0, withlimα→0[F (0)− F (∞)] = 0.

Two important corollaries follow immediately from Theorem 1:


(1) Existence of the EDE is possible only if nβ′ is greater than the ratio of life expectancy1/b to the average age

∫∞0 ae−M(a) da.

(2) bnβ′ > α/[L(0)−L(α)] is sufficient to guarantee existence of the EDE, regardless ofρ.

Proof. Note first that L is decreasing, with

L′(y) = −∫ ∞

0ae−M(a)−ya da,

and L(∞) = 0. From the definition of F and the properties of L, we then have

limρ→∞F (ρ) =

L(0)− L(α)α

> 0,

limρ→0

F (ρ) = limρ→0

L(0)− L(ρ)ρ

= −L′(0) =∫ ∞

0ae−M(a) da,

limα→0

[F (0)− F (∞)] = limρ→0

L(0)− L(ρ)ρ

− limα→0

L(0)− L(α)α

= 0,

limρ→α

F (ρ) = limε→0

(α + ε)2[L(0)− L(α)]− α2[L(0)− L(α + ε)]εα2

=2α

[L(0)−L(α)]+L′(α).

¤

It remains to show that F ′ < 0. We begin with two computational lemmas.

Lemma 3.2. xye−x + e−yx ≥ (1 + x)e−x ∀x, y ≥ 0.

Proof. The statement is immediately true for x = 0. Let x > 0 be given and define f(y) =xye−x + e−yx. Then f ′(y) = x(e−x − e−yx), so f ′ < 0 for y < 1 and f ′ > 0 for y > 1.Hence, f(y) ≥ f(1) = (1 + x)e−x. ¤

Lemma 3.3. y[(y − 1)x + y − 2]e−x + e−yx ≤ (y − 1)2 ∀x, y ≥ 0.

Proof. The statement is immediately true for y = 0. Let y > 0 be given and define f(x) =y[(y−1)x+y−2]e−x+e−yx. Then f(0) = (y−1)2 and f ′(x) = y[(1+x)e−x−xye−x−e−yx];by Lemma 3.2, f ′ ≤ 0. Hence, f(x) ≤ f(0) = (y − 1)2. ¤

Now let ρ and α be chosen such that F ′ > 0. Then

(ρ2 − αρ)[2ρL(0) + α2L′(ρ)− 2ρL(α)] > (2ρ− α)[(ρ2 − α2)L(0) + α2L(ρ)− ρ2L(α)].

After some algebraic simplification, this inequality reduces to

(αρ− α2)ρL′(ρ) + (α2 − 2αρ)L(ρ) + ρ2L(α) > (α− ρ)2L(0).

From the definition of L, we have∫ ∞

0[(α2 − αρ)ρae−ρa + (α2 − 2αρ)e−ρa + ρ2e−αa]e−M(a) da >

∫ ∞

0(α− ρ)2e−M(a) da.


Now let x = ρa and y = α/ρ. With these substitutions, we have the inequality∫ ∞

0

(y[(y − 1)x + y − 2]e−x + e−yx

)e−M(a) da >

∫ ∞

0(y − 1)2e−M(a) da.

However, this inequality contradicts the result of Lemma 3.3; hence, there are no values of ρand α for which F ′ > 0.

4. PARAMETER ESTIMATION

We can estimate the values of the parameters from the following data:

life expectancy µ−1 77 yrU.S. population N 300MH. pylori prevalence I 90Mulcer prevalence U 15Mnew cases per year αH 0.6Msusceptibility n 0.9

TABLE 1. Primary data for H pylori

Assuming the data for I , U , and αH represent equilibrium values, we have

µ ≈ 0.013, i? =I?

nN≈ 1

3, q =

U?

I?=≈ 1

6, α =

αH?

I? − U?≈ 0.008, (23)

ρ =α

q− α− µ ≈ 0.027, ν =

ρq

µ≈ 0.346, β =

1(1 + ν)−1 − i?

≈ 2.44. (24)

The estimate for α is consistent with that of Rupnow et al.

5. COMPARISON OF THE TWO MODELS

Figure 3 shows a comparison of the structured and unstructured models. In each case, welet β0 be the value estimated from the data. Figures 3a and 3b show the effect of changes intreatment rate on ulcer prevalence and incidence rate. In both cases, the upper curves, whichwere calculated with β = β0, are comparable, indicating that the effect of ρ is the same forboth models. Specifically, increasing the treatment rate up to about 0.2 has a significant impacton peptic disease, but further increases in treatment rate make little difference. This is easilyexplained by the unstructured model, where the parameter q, representing the fraction of infec-tives who are symptomatic, is easily calculated. Treatment of ulcer sufferers with antibioticsdecreases the number of symptomatics, but cannot decrease the number of asymptomatics. Theeffect on the infection rate is noticeable only until q is very small. With ρ = 0.2, only 3% ofinfectives are symptomatic. Even the rather low value of ρ estimated from the current data islarge enough for treatment to have already had most of its demographic impact.

The lower curves in Figures 3a and 3b show that the effect of changes in β are qualitativelysimilar, but quantitatively different, for the two models. A 20% decrease in transmission makes


0 0.1 0.2 0.3 0.40

2

4

6

8

10

ρ

ulce

r pr

eval

ence

(%

)

unstructuredstructured

0 0.1 0.2 0.3 0.40

5

10

15

20

25

ρan

nual

inci

denc

e pe

r 10

000

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

ρ

β c/β0

0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

β/β0

annu

al in

cide

nce

per

1000

0

FIGURE 3. Comparison of the two models: a) ulcer prevalence with β0 and.8β0, with β0 = 1.5 for the unstructured model and β0 = 2.52 for the struc-tured model, b) annual incidence with β0 and .8β0, c) critical β values forelimination of the endemic equilibrium, d) annual incidence with ρ = 0.2

a much larger difference when the age structure of the population is accounted for in the model.This effect is peculiar to diseases such as ulcers, in which all of the rates in the model are com-parable in magnitude to the mortality rate. When the course of the disease is rapid compared tonon-disease-related mortality, the effect of neglecting age structure is unimportant. The stan-dard assumption of age-independent mortality yields a very unrealistic age structure. A simplecalculation shows that if µ = 0.013 is used in the structured model, the average of the popula-tion is identical to the expected lifespan. The population has a mean age of 77 years because


the age distribution includes a noticeable number of individuals who are more than 120 yearsold. This unrealistic structure only matters if the disease process is comparably slow, as in thecase of peptic disease.

Figure 3c indicates the effect of changes in treatment on the critical degree of infectivitynecessary to eliminate the endemic equilibrium. Once again, treatment makes little differencebeyond about ρ = 0.2. The unstructured model presents an inaccurately bleak picture ofthe possibility of eliminating the endemic state through further reduction in the transmissionrate. Based on the structured model, decreasing β by just over 25% would be sufficient toeliminate endemic peptic disease in the United States. Of course this is assuming no furtherincrease in lifespan, which would allow infectives more time to transmit the infection. Figure3d elaborates on the same theme by indicating the effect of β on the endemic ulcer incidencerate, given ρ = 0.2. The models agree at β = βc because the transmission coefficients werechosen to achieve the current incidence rate. Decreases in β reduce ulcer incidence markedlyin the model that accounts for age structure, and much less so in the model that does not.

One could make a more sophisticated structured model by making parameters other than µage-dependent. However, there are two reasons why it is probably not advantageous to do so.First, the other parameters are unlikely to be as strongly age-dependent as is the mortality rate.Second, the age-dependence of these parameters is not well known, while the age-dependenceof mortality has the Makeham-Gompertz theoretical model with parameters that are relativelywell-known, as compared with the crude estimate that 90 million U.S. residents are infectedwith H pylori, for example. A further complication is that parameters having to do with in-fectivity and development of symptoms may depend as much on the amount of time sincebecoming infected as on the age of the patient. Even the relatively crude unstructured modelused here displays the right qualitative behavior, so it is arguable that one could make betterquantitative predictions than our structured model by using a model that is more precise butrelies on extremely crude estimates of parameters. Our unstructured model is adequate forstudies for which parameter values likely to be sensitive to population age structure are notchanged much from current estimates. For studies that do make significant changes to suchparameter values, our structured model gives sufficiently different results to warrant its use inspite of the mathematical complications.

REFERENCES

[1] Centers for Disease Control and Prevention www.cdc.gov/ulcer/md.htm[2] National Digestive Disease Information Clearinghouse

http://digestive.niddk.nih.gov/ddiseases/pubs/hpylori[3] S. Becker, R. Smalligan, J, H. Kleanthous, T. Tibbitts, T. Manath and K. Hyamset, Risk of H. pylori infection

among long-term residents in developing countries, Am. J. Trop. Med. Hyg., 60 (1999)[4] L.G. Robinson, F.L. Black, F.K. Lee, A.O. Sousa, M. Owens, D. Danielson, A.J. Nahmias, B.D. Gold, Heli-

cobacter pylori prevalence among indigenous peoples of South America, J. Infect Dis, 186 (2002)[5] Veterans and Agent Orange: Update 2004 (2005), Board on Health Promotion and Disease Prevention, The

National Academic Press[6] E. Kuipers, Helicobacter pylori and the risk and management of associated diseases: gastritis, ulcer disease,

atrophic gastritis and gastric cancer. Aliment Pharmacol Ther, 11 (1997)


[7] K. Knigge, The role of H. pylori in gastrointestinal disease, Postgraduate Medicine, 110 (2001)[8] Proceedings of the fourth Global Vaccine Research Forum, Department of immunization, Vaccines and Bio-

logical World Health Organization, Geneva, Switzerland[9] P. Rugglero, S. Peppoloni, D. Berti, R. Pappuoli, G. Del Gludice, Expert Opinion on Investigational Drugs,

11 (2002)[10] D. Burmann, P.R. Jungblut, T.F. Meyer, Helicobacter pylori vaccine development based on combined subpro-

teome analysis, Proteomics 4 (2004)[11] F. Hardin, R. Wright, Helicobacter pylori: Review and Update, Hospital Physicians, May (2002)[12] AHT Jeremy, Y. Du, M.F. Dixon, P.A. Robinson, J.E. Crabtree, Protection against Helicobacter pylori infec-

tion in the Mongolian gerbil after prophylactic vaccination, Microbes and Infection, 8 (2006)[13] Tae S. Do, Glenn Ledder, Young Lee, The Dynamics of Peptic Ulcers and H. Pylori with Vaccination, Inter-

national Journal of Pure and Applied Mathematics, 43 (2008)[14] M. Susser, Z. Stein, Civilization and peptic ulcer, Lancet, 20 (1962)[15] M. Rupnow, R. Shachter, D. Owens, J. Parsonnet, A Dynamic Tranmission Model for Prediction Trends in

Helicobacter pylori and Associated Diseases in the United States, Emerging Infections Diseases, 6 (2000)[16] M. Rupnow, R. Shachter, D. Owens, J. Parsonnet, Quantifying the population impact of a prophylactic Heli-

cobacter pylori vaccine, Vaccine 20 (2002)[17] P. Ven den Driessche and J. Watmough, Reproduction numbers and sub-threshold Endemic Equilia for Com-

partmental models of disease transmission, 180 (2002)[18] National Center for Health Statistics http://www.cdc.gov/nchs/fastats/lifexpec.htm

Received by the editors December 29 2008; Accepted February 20 2009. 2000 Mathematics Subject Classification. 68W10, 68W40. Key words and phrase : Block Lanczos Method, Parallel Performance Enhancement, Eigenvalue Problem,

Mutifrontal algorithm, IPSAP. † Corresponding author.

13

J. KSIAM Vol.13, No.1, 13-20, 2009

PERFORMANCE ENHANCEMENT OF PARALLEL MULTIFRONTAL SOLVER ON BLOCK LANCZOS METHOD

Wanil BYUN1 AND Seung Jo KIM1,2†

1SCHOOL OF MECHANICAL AND AEROSPACE ENG, SEOUL NATIONAL UNIV, SOUTH KOREA E-MAIL address: [email protected] 2FLIGHT VEHICLE RESEARCH CENTER, SEOUL NATIONAL UNIV, SOUTH KOREA E-MAIL address: [email protected]

ABSTRACT. The IPSAP which is a finite element analysis program has been developed for high parallel performance computing. This program consists of various analysis modules – stress, vibration and thermal analysis module, etc. The M orthogonal block Lanczos algorithm with shift-invert transformation is used for solving eigenvalue problems in the vibration module. And the multifrontal algorithm which is one of the most efficient direct linear equation solvers is applied to factorization and triangular system solving phases in this block Lanczos iteration routine. In this study, the performance enhancement procedures of the IPSAP are composed of the following stages: 1) communication volume minimization of the factorization phase by modifying parallel matrix subroutines. 2) idling time minimization in triangular system solving phase by partial inverse of the frontal matrix and the LCM (least common multiple) concept.

1. INTRODUCTION

Structural vibration analysis considered in IPSAP is to obtain eigenvalues and eigenvectors of the eigen problem of symmetric positive semi-definite stiffness and mass matrix which are obtained by the finite element method. The eigenvalue problem of structural vibration is of the form:

K λM (1)

Since the stiffness and the mass matrix are symmetric and semi-definite, the well-known

Lanczos iteration is applicable to Equation (1). Exactly speaking, the shifted invert transformation as in Equation (2) should be used in presence of the mass matrix M.

LT LT K σM LLT (2)

The shifted eigenvalue α is associated with original one by the relation α λ σ.

Wanil BYUN and Seung Jo KIM

14

The matrix L is the lower triangular matrix, where M=LLT. The existence of the inverse of K σM requires a linear equation solver during Lanczos iterations. The presence of LT implies that the Lanczos basis should be M orthogonal. It is notable that the restarted Lanczos iteration [1] has been proposed as an alternative for the shift-invert transform.

2. ALGORITHM In the shift-invert Lanczos method, a linear equation solver is needed to solve accurately

a linear equation with the coefficient matrix K σM. One of the reliable means is a direct method [2]. Nowadays, due to the bottleneck in scalability and memory requirement of a parallel direct solver, iterative linear equation solvers or reduction methods are getting interest [3-4]. However, since the linear equation solver should be more accurate than the desired accuracy of eigenvalues, iterative solvers may be less powerful even with its great parallel scalability and memory efficiency.

In the IPSAP solver, we implemented a parallel block Lanczos eigensolver using M

orthogonal iteration equipped with a direct linear equation solver (Algorithm 2.1). Algorithm 2.1 M orthogonal block Lanczos algorithm with shift-invert transformation

Let V be a set of initial vectors with VTMV I

j 0 while(required eigenvalue > converged eigenvalue)

U MV K σM)W U , solve for W W W V BT C VTMW W W VC W V B , QR factorize for V

compute eigenvalue of T , j j 1 re-orthogonalize V against V , i 0 j 1

end

step 1 step 2 step 3 step 4 step 5 step 6 step 7 step 8

Step 2 in Algorithm 2.1 is a linear equation solving procedure which takes most of the

time consumed in structural vibration analysis using Lanczos iteration. The parallel multifrontal solver was chosen as a linear equation solver. The algorithm of the solver can be regarded as a FEM-oriented version of a conventional multifrontal solver. This solver does not require a globally assembled stiffness matrix while the element concept of the finite element mesh is utilized as graph information. Therefore, the assembly of element matrices takes place automatically during the factorization phase. The implemented parallel factorization in the multifrontal solver is the well-known Cholesky factorization


15

(Algorithm 2.2). It is noteworthy that the factorization needs to be conducted only once before the Lanczos loop if the shift is not varied.

Algorithm 2.2 Parallel sparse Cholesky factorization

for i 0, , N 1

K K SYMK K

factorize K , K , update K j i while rem j, 2 1 extend_add K with another domain branch in tree factorize K , K , update K j j 1 /2 end

end parallel procedure n 1 while n N extend_add K with another processor branch in tree

factorize K , K , update K n 2n end

In a matrix form, the Cholesky factorization factorize K , K and the update of a dense matrix update K can be written in three steps.

K L LT (3)

L K K (4)

K K K TK (5)

Equation (3)-(5) are based on the assumption that the lower part of the symmetric frontal matrix is active and that the sub-matrix K is allocated according to the memory structure in Figure 1. The feature of the frontal matrix structure in Figure 1 is that the sub-matrix K is firstly assigned at the initial position of the allocated memory. Such a structure of the frontal matrix makes the extend_add K ) operation be easily implemented because K remains always at the head of the allocated memory. If the extend_add operation is


16

implemented appropriately, the major communication overhead of the proposed algorithm is caused by parallel dense matrix operation such as factorize K , K and update K .

FIGURE 1. Memory structure of frontal matrix K

During the phase of solving triangular systems, a forward elimination followed by a

backward substitution is a general procedure. They use the same processor mapping and frontal matrix distribution of the factorization phase. If the RHS (right-hand side) is indicated by W, the forward elimination is generally composed of two steps as follows.

L W W (6)

W W K TW (7) The backward substitution is also completed by the following two steps.

W W K W (8)LT W W (9)

Here, W is the solution we are looking for. There are many communication overheads occurring from panel or block broadcast and summation in parallel operation of Equation (3)-(9). The main objectives of performance enhancement in these routines are to reduce the total communication volume and to minimize the idling time for transferring panel or block matrices.

3. PARALLEL PERFORMANCE ENHANCEMENT AND NUMERICAL TEST

Performance tuning of the multifrontal solver is conducted from three points of views. The first one is to reduce communication occurring in the Cholesky factorization. It is

notable that the naming convention of subroutines used in the eigensolver is based on those of BLAS (Basic Linear Algebra Subprograms)[5] and LAPACK (Linear Algebra Package) [6]. As shown in Figure 2, the symmetric frontal matrix is composed of three matrix entities. Therefore, three routines are required to factorize the frontal matrix. In parallel matrix operations based on panel communication which are adopted in the present research, each routine performs panel or block communications which are column(or row) broadcasting or reducing type. Considering the detailed communication pattern, it is apparent that some parts of broadcasting or reducing are duplicated. For example, column broadcasting of panels of K in POTRF is also present in TRSM (Figure 3). If combining


17

three routines into one is possible, duplicated communications will be avoidable. In Figure 3, they show a communication pattern of one condensation subroutine by combining duplicated communications among three subroutines.

FIGURE 2. Frontal matrix and factorization routines

FIGURE 3. Communication pattern of each subroutine

Table 1 lists some numerical test comparison without condensation and with

condensation. The condensation subroutine which combines duplicated communications between nodes and nodes takes less time compared with non-condensation one.

TABLE 1. Numerical test comparison without condensation and with condensation

process map=4x8, block size=100 with 32 nodes (32 CPUs) matrix dimension without condensation with condensation difference

16000x16000 209.064 sec 201.258 sec -7.806 sec process map=8x8, block size=100 with 64 nodes (64 CPUs)

matrix dimension without condensation with condensation difference 16000x16000 129.261 sec 122.929 sec -6.332 sec 32000x32000 1018.96 sec 816.922 sec -202.038 sec


18

The second performance tuning can be conducted by topology control in panel or block

communication. We implemented two kinds of topology which are ‘increasing ring’ and ‘split ring’. The condensation subroutine is composed of five topology options. Therefore, there is 32 topology sets. When the topology set is ‘issii’, the best performance in our parallel environment is achieved (Table 2). It is notable that the best topology set depends on own network environment.

TABLE 2. Topology comparison of the condensation subroutine

Topology Sets – 32 cases

sssss ssssi sssis sssii ssiss ssisi ssiis ssiii

sisss sissi sisis sisii siiss siisi siiis siiii

issss isssi issis issii isiss isisi isiis Isiii

iisss iissi iisis iisii iiiss iiisi iiiis iiiii

Matrix dimension = 16000x16000, process map = 8x8, block size = 100 with 64 nodes (64 CPUs)

115.432 111.501 116.717 111.242 120.286 113.605 120.819 112.83

120.961 111.241 119.621 113.135 124.059 114.537 122.407 114.573

115.825 110.642 117.704 109.211 120.259 112.433 120.02 111.096

120.185 111.083 121.129 111.52 123.041 113.296 124.083 113.459

Matrix dimension = 32000x32000, process map = 8x8, block size = 100 with 64 nodes (64 CPUs)

607.318 589.898 600.976 586.587 627.982 612.927 624.301 609.839

611.379 593.751 608.472 594.00 638.387 616.156 635.957 614.28

600.697 588.547 598.778 585.828 624.406 608.557 622.545 606.83

611.518 591.975 610.131 590.734 637.907 615.17 633.893 611.836

The last one is to tune the triangular system solving routines. Operations involved in

solving the triangular system are generally composed of TRSM and GEMM with F and F . Since TRSM routine has data-dependent algorithmic flow, there may be idle processors between panel broadcasting. One idea to resolve this bottleneck is converting TRSM into TRMM by computing inverse of L by using TRTRI routine. Although TRMM requires the same number of floating point operations and communication volume as TRSM does, the LCM (least common multiple) concept can apparently reduce the idling time. The LCM concept which is originally proposed in the research of GEMM [7] can also be applied to any case of data-independent communication pattern. In order to apply such a concept to (6) and (9), the factorization phase should be modified so that the operation can be conducted only with matrix multiplication as follows.


19

L W W (10)L TW W (11)

In this approach, performance negotiation between computing time for inverse of L

and gains by using TRSM should be taken into account.

TABLE 3. Performance results of eigenvalue problem (32 CPUs)

Case 1 Case 2 Case 3 Difference

(Case 1 ③ - Case 3 ③)

Hex8 1260 x 1260 x 1 (10M DOF)

10 eigenvalues (9 loop)

① 239.119 sec ② 266.858 sec ③ 517.371 sec

① 227.354 sec ② 267.217 sec ③ 505.865 sec

① 239.676 sec ② 225.588 sec ③ 476.549 sec

- 40.822 sec( ↓ 7.89% )

Hex8 1260 x 1260 x 1 (10M DOF)


① 239.066 sec ② 648.929 sec ③ 899.302 sec

① 225.603 sec ② 642.754 sec ③ 879.649 sec

① 237.000 sec ② 521.791 sec ③ 770.086 sec

- 129.217 sec( ↓ 14.4% )

TABLE 4. Performance results of eigenvalue problem (64 CPUs)

Case 1 Case 2 Case 3 Difference

(Case 1 ③ - Case 3 ③)

Hex8 2000 x 2000 x 1 (24M DOF)


① 484.178 sec ② 797.078 sec ③ 1296.588 sec

① 424.165 sec ② 791.127 sec ③ 1230.586 sec

① 483.794 sec ② 694.068 sec ③ 1193.168 sec

- 103.420 sec( ↓ 7.98% )

Hex8 1260 x 1260 x 1 (10M DOF)


① 143.696 sec ② 1170.720 sec ③ 1320.015 sec

① 135.516 sec ② 1166.260 sec ③ 1307.391 sec

① 152.044 sec ② 909.117 sec ③ 1066.752 sec

- 253.263 sec( ↓ 19.2% )

Hex8 1260 x 1260 x 1 (10M DOF)


① 147.117 sec ② 4378.680 sec ③ 4531.387 sec

① 139.744 sec ② 4374.520 sec ③ 4519.864 sec

① 144.483 sec ② 3032.570 sec ③ 3182.611 sec

- 1348.776 sec( ↓ 29.8% )

Hex8 1260 x 1260 x 1 (10M DOF)


① 143.202 sec ② 8414.730 sec ③ 8563.521 sec

① 136.644 sec ② 8408.190 sec ③ 8550.448 sec

① 143.147 sec ② 6215.300 sec ③ 6364.019 sec

- 2199.502 sec( ↓ 25.7 % )

The eigenvalue analysis of simple finite element models was performed considering the

three performance enhancement issues. Table 3 and 4 list the performance results in 32 and 64 CPUs parallel computing environment. Case 1 is for normal elimination-substitution algorithm with optimized network topology, Case 2 is for a condensation based on Case 1 and Case 3 is for partial inverting algorithm with LCM concept based on case 2. Then, ① lists the Cholesky factorization time, ② lists Triangular solution time and ③ lists MFS


20

elapsed time. The results of the Cholesky factorization time show performance enhancement compares with Case 1 and Case 2. And the Cholesky factorization time is similar between Case 1 and Case 3 because of condensation in spite of adding inverting routine TRTRI. In a view point of triangular system solving time, Case 3 has a good performance enhancement if the iteration loop is more and more.

4. CONCLUSION

Parallel performance tuning of the multifrontal algorithm in block Lanczos method was conducted from three points of views in this research. Using condensation to reduce communication volume results in better performance during Cholesky factorization phase. In a viewpoint of topology, a network topology optimization also needs to have a better performance. The best topology set is dependent on network structure and device type. In this research, about 8~12% enhancement can be obtained by topology control and condensation in case of parallel computing environment (64 CPUs). At last, the factorization time increases when applying the LCM concept because inverse subroutine (TRTRI) is added. However, if the number of Lanczos iteration is large, a partial inverting algorithm is more efficient and shows good performance. About 8~25% enhancement in this research can be obtained by using LCM concept due to the data-independent communication. Applying the LCM concept inside an iteration loop shows that it is significant for reducing the computing time in parallel matrix operations.

ACKNOWLEDGMENTS

This research has been partially supported by the ‘Rapid Design of Satellite Structures by Multi-disciplinary Design Optimization’ project from Aerospace Research Institute(KARI) and NRL program administered via the Institute of Advanced Aerospace Technology at Seoul National University.

REFERENCES

[1] D. Calvetti, L. Reichel and D. Sorensen, “An Implicit Restarted Lanczos Method for Large Symmetric Eigenvalue Problems,” Electronic Transaction on Numerical Analysis, Vol. 2, pp. 1-21, 1994

[2] K. Wu and H. Simon, “An Evaluation of the Parallel Shift-and-Invert Lanczos Method,” Proceedings of International Conference on Parallel and Distributed Processing Techniques and Applications, pp. 2913-2919, Las Vegas USA, June 1999

[3] R. Morgan and D. Scott, “Preconditioning the Lanczos Algorithm for Sparse Symmetric Eigenvalue Problems,” SIAM Journal on Scientific Computing, Vol. 14, pp. 585-593, 1993

[4] Y.T. Feng and D.R.J. Owen, “Conjugate Gradient Methods for Solving the Smallest Eigenpair of Large Symmetric Eigenvalue Problems,” International Journal for Numerical Methods in Engineering, Vol. 39, pp. 2209-2229, 1996

[5] http://www.netlib.org/blas [6] http://www.netlib.org/lapack [7] Choi, J., “A Fast Scalable Universal Matrix Multiplication Algorithm on Distributed-Memory Concurrent

Computers”, Proceedings of the IPPS, pp. 310-314, 1997

J. KSIAM Vol.13, No.1, 21–29, 2009

INSTABILITY IN A PREDATOR-PREY MODEL WITH DIFFUSION

SHABAN ALY1

DEPT OF MATHEMATICS, FACULTY OF SCIENCE, KING KHALID UNIV, ABHA 61413, SAUDI ARABIA


ABSTRACT. This paper treats the conditions for the existence and stability properties of sta-tionary solutions of a predator-prey interaction with self and cross-diffusion. We show that at acertain critical value a diffusion driven instability occurs, i.e. the stationary solution stays sta-ble with respect to the kinetic system (the system without diffusion) but becomes unstable withrespect to the system with diffusion and that Turing instability takes place. We note that thecross-diffusion increase or decrease a Turing space ( the space which the emergence of spatialpatterns is holding) compared to the Turing space with self-diffusion, i.e. the cross-diffusionresponse is an important factor that should not be ignored when pattern emerges

1. INTRODUCTION

The theory of spatial pattern formation via Turing instability (see [11])-wherein an equilib-rium of a nonlinear system is asymptotically stable in the absence of diffusion but unstable inthe presence of diffusion - plays an important role in ecology, embryology and elsewhere inbiology and chemistry (see [4], [5], [8], [9]). Since the relation between the organisms and thespace seems to be essential to stability of an ecological system, the effect of diffusion on thepossibility of species coexistence in an ecological community has been an important subject inpopulation biology (see [7], [10], [11]).

In recent years there has been considerable interest to investigate the stability behavior ofa system of interacting populations by taking into account the effect of self as well as cross-diffusion. The term self-diffusion implies the per capita diffusion rate of each species is influ-enced only by its own density, i.e. there is no response to the density of the other one. Cross-diffusion implies the per capita diffusion rate of each species is influenced not only by its ownbut also by the other ones density. The value of the cross-diffusion coefficient may be positive,negative or zero. The term positive cross-diffusion coefficient denotes the movement of thespecies in the direction of lower concentration of another species and negative cross-diffusioncoefficient denotes that one species tends to diffuse in the direction of higher concentrationof another species. The dynamics of interacting population with self and cross-diffusion areinvestigated by several researchers (see [2], [3], [4], [6]) and others.

Received by the editors January 16 2009; Accepted February 20 2009.2000 Mathematics Subject Classification. 35K57, 92B25, 93D20.Key words and phrases. Predator-prey, self and cross-diffusion, stability, Turing instability.1Permanent Address: Dept of Mathematics, Faculty of Science, Al-Azhar Univ, Assiut 71511, Egypt.

21

22 SHABAN ALY1

In this paper we are going to study the effect of the self and cross-diffusion on the stabilityof the equilibria in a reaction diffusion predator-prey model of Cavani-Farkas type (see [2],[4] ) and we explore under which parameter values Turing instability can occur giving rise tonon-uniform stationary solutions, this model described by the following equations:

∂N

∂t= εN(1− N

K)− βNP

β + N+ d11

∂2N

∂x2+ d12

∂2P

∂x2, x ∈ [0, l], t > 0, (1)

∂P

∂t= −P (

γ + δP

1 + P) +

βNP

β + N+ d21

∂2N

∂x2+ d22

∂2P

∂x2, x ∈ [0, l], t > 0,

where N(t), P (t) represent the population density of prey and predator at time t, respectively.The prey grows with intrinsic growth rate ε up to carrying capacity K in the absence of pre-dation. Here γ > 0 and δ > 0 are the minimal mortality and the limiting mortality of thepredator, respectively (the natural assumption is 0 < γ ≤ δ), dik > 0 are the diffusion coef-ficients, i, k = 1, 2. The meaning of the half saturation constant is that at N = β the specificgrowth rate βN

β+N (called also a Holling type functional response) of the predator is equal to halfits maximum β (the conversion rate is taken to be equal to the half saturation constant for sakeof simplicity). The advantage of this model over the more often used models is that here thepredator mortality is neither a constant nor an unbounded function, still, it is increasing withthe predator abundance.

Assuming that prey and predator are diffusing according to Fick′s law in the interval x ∈[0, l]. We are interested in solutions N : R+ × [0, l] −→ R+, P : R+ × [0, l] −→ R+ withno-flux boundary conditions

Nx(t, 0) = Nx(t, l) = Px(t, 0) = Px(t, l) = 0, (2)

and initial conditionsN(x, 0) ≥ 0, P (x, 0) ≥ 0, x ∈ [0, l]. (3)

Properly posed initial and boundary conditions yield well posed problem concerning this sys-tem of PDE′s in forward time if

D =(

d11 d12

d21 d22

), (4)

has eigenvalues with positive real parts, i.e.,

d11d22 > d12d21 (5)

which means that ′′self-diffusion is stronger than cross diffusion′′, i.e. the flow of the respectivedensities in the spatial domain depends strongly on their own density than on the others.

Let us set F = (F1, F2), U = (N, P ) where

F1(N,P ) = εN(1− N

K)− βNP

β + N, F2(N, P ) = −P (

γ + δP

1 + P) +

βNP

β + N, (6)

INSTABILITY IN A PREDATOR-PREY MODEL WITH DIFFUSION 23

the system takes the form

Ut = F (U) + D∂2U

∂x2, (7)

while the boundary conditions remain

Ux(t, 0) = Ux(t, l) = 0. (8)

Clearly, a spatially constant solution U(t) = (N(t), P (t)) of (7) satisfies the boundary condi-tions (8) and the kinetic system

Ut = F (U). (9)

2. THE MODEL WITHOUT DIFFUSION

In this section, we will study system (1)-(2) without diffusion, i.e.,.

N = εN(1− N

K)− βNP

β + N, (10)

.P = −P (

γ + δP

1 + P) +

βNP

β + N.

The following conditions are reasonable and natural:

γ < β ≤ δ, (11)

β < K, (12)

γ <βK

β + K. (13)

Condition (11) ensures that the predator mortality is increasing with density, and that the preda-tor null-cline has a reasonable concave down shape; (12) ensures that for the prey an Allee-effect zone exists where the increase of prey density is favourable to its growth rate; (13) isneeded to have a positive equilibrium point of system (10). System (10) is made up by twoidentical uncoupled systems. Under these conditions the system has at least one equilibriumwith positive coordinates. This is the point of intersection of the prey null-cline

P = H1(N) =ε

βK(K −N)(β + N) (14)

and the predator null-cline

P = H2(N) =(β − γ)N − βγ

(δ − β)N + βδ. (15)

Thus, denoting the coordinates of a positive equilibrium by (N, P ), these coordinates satisfyP = H1(N) = H2(N).

Note that if K > β, we have an interval u1 ∈ (0, K−β2 ), where the Allee-effect holds, i.e.,

the increase of the prey quantity is beneficial to its growth rate.In particular, we will focus our attention to the existence of equilibria and their local stability.

This information will be crucial in the next section where we study the effect of the diffusionparameters on the stability of the steady states.

24 SHABAN ALY1

The Jacobian matrix of the system (10) linearized at (N,P ) is

A =(

Θ1 −Θ2

Θ3 −Θ4

), (16)

where

Θ1 =εN(K − β − 2N)

K(β + N), Θ2 =

βN

β + N, Θ3 =

β2P

(β + N)2, Θ4 =

(δ − γ)P(1 + P )2

. (17)

The characteristic equation is given by

λ2 − traceAλ + det A = 0, (18)

wheretraceA = Θ1 −Θ4, detA = Θ2Θ3 −Θ1Θ4 (19)

If (N,P ) lies outside the Allee-effect zone then Θ1 < 0. Since, obviously, Θ2, Θ3, Θ4 > 0in this case traceA < 0 and detA > 0, i.e. all eigenvalues of the matrix A, have negative realparts.

The equilibrium point (N, P ) lies in the Allee-effect zone if

H1(k − β

2) < H2(

k − β

2). (20)

Recall that (N,P ) lies in the Allee-effect zone and is locally asymptotically stable if Reλ < 0,which is equivalent to have traceA < 0 and detA > 0. For this, we will assume that

Θ1 < Θ4 and Θ2Θ3 > Θ1Θ4. (21)

3. TURING INSTABILITY

Definition: We say that the equilibrium (N,P ) is Turing unstable if it is an asymptoticallystable equilibrium of the kinetic system (10) but is unstable with respect to solutions of (1)-(2)(see [11]).

We linearize system (1) at the point (N,P ). Introducing the new coordinates V = (V1, V2) =(N −N, P − P ), the linearized system assumes the form

Vt = AV + D∂2V

∂x2, (22)

while the boundary conditions remain

Vx(t, 0) = Vx(t, l) = 0. (23)

We solve the linear boundary value problem by Fourier’s method. Solutions are assumed in theform V (t, x) = y(t)ψ(x). The functions y : [0,∞) → R2, ψ : [0, l] → R are to satisfy

.y = (A− λD)y, (24)

where dot denotes differentiation with respect to time, and

ψ′′

= −λψ, ψ′(0) = ψ

′(l) = 0, (25)


where prime denotes differentiation with respect to the spatial variable x.The eigenvalues of the boundary value problem (25) are

λj = (jπ/l)2, j = 0, 1, 2, 3, ... (26)

with corresponding eigenfunctions

ψj(x) = cos(jπx/l). (27)

Clearly, 0 = λ0 < λ1 < λ2 < .... These eigenvalues are to be substituted into (24). Denotingtwo independent solutions of (27) taken with λ = λj by y1j , y2j , the solution of the boundaryvalue problem (22)-(23) is obtained in the form

V (t, x) =∞∑

j=0

(a1jy1j(t) + a2jy2j(t)) cos(jπx/l) (28)

where aij(i = 1, 2; j = 0, 1, 2, ...) are to be determined according to the initial conditionV (0, x). If e.g. y1j(0) = (1, 0), y2j(0) = (0, 1) for j = 0, 1, 2, ... then

[a10

a20

]=

1l

∫ l

0V (0, x)dx,

[a1k

a2k

]=

2l

∫ l

0V (0, x) cos(jπx/l)dx (k = 0, 1, 2, ...).

The following notations will be used:

B(λ) = A− λD, Bj = B(λj) = A− λjD. (29)

According to Casten, Holland (see [1]) if for all j both eigenvalues of Bj have negative realparts then the equilibrium (N, P ) of (7)-(8) is asymptotically stable; if at least one eigenvalueof at least one matrix Bj has positive real part then (N,P ) of (7)-(8) is unstable.

traceBj = Θ1 −Θ4 − λj(d11 + d22), (30)

detBj = detD λ2j + [Θ4d11 + Θ2d12 −Θ1d22 −Θ3d21]λj . + detA. (31)

We note that traceBj < 0, thus from Routh-Hurwitz criteria, the stability of the equilibrium(N, P ) depends on the sign of det Bj . Since the coefficient matrix of the ODE in (24) is stableif and only if detBj > 0 for each j = 0, 1, 2, ..., therefore using the Theorem 1 in [1], onecan easily see that in order to have Turing instability the quadratic polynomial (31) must benonpositive for some j = 0, 1, 2, ... . Dropping the index this condition assumes the form

ρ(λ) := detD λ2 −Πλ + det A ≤ 0, (32):= λ(detD λ−Π) + detA ≤ 0 for some λ > 0

whereΠ = Θ1d22 −Θ4d11 + Θ3d21 −Θ2d12.

The polynomial ρ(λ) has two positive roots 0 < λ ≤ λ and (32) holds for λ if and only if

Π > 0 and Π2 ≥ 4 det(AD)

26 SHABAN ALY1

since λ := Π−√

Π2−4 det(AD)

2 det D , λ := Π+√

Π2−4 det(AD)

2 det D . This with (5) imply that the polynomialρ(λ) is nonpositive for some λ > 0 between their positive roots 0 < λ ≤ λ.

We are going to establish the conditions of a Turing instability for the following cases un-der the assumption that traceBj < 0 and the predator diffusion coefficient d22 > 0 as thebifurcation parameter.

Case 1: d12 = d21 = 0 (self-diffusion).

This shows that each species moves along its own concentration gradient.Theorem: Suppose that traceBj < 0 and detA > 0.(i) If

d11 ≥ Θ1/λ1 (33)

then the zero solution of the linear problem (22)-(23) is asymptotically stable for all d22 > 0.(ii) If

Θ1/λ1 > d11 ≥ Θ1/λ2 (34)

then at

d22 := d∗22crit =detA + λ1d11Θ4

λ1(Θ1 − λ1d11)(35)

the zero solution of the linear problem (22)-(23) undergoes a Turing instability.Proof. (i) From (31) we have

detBj = detA + λjd11Θ4 − λjd22(Θ1 − λjd11).

Since λj , j = 0, 1, 2, , ...from a monotone increasing sequence (33) implies detBj > 0 forall j = 0, 1, 2, , ....From (30) we see that traceBj < 0 for all j = 0, 1, 2, , ...hence the aerosolution of (22)-(23) is asymptotically stable.

(ii) If d11 satisfies (34) and d22 is chosen according to (35) then det B1 = 0. Clearly, for0 < d22 < d∗22crit we have detB1 > 0, and for d∗22crit < d22 we have detB1 < 0. In all thesecases det Bj > 0, j 6= 1. Thus, taking into account what has been quoted after formula (29),for 0 < d22 < d∗22crit the zero solution is asymptotically stable, for d∗22crit < d22 it is unstable.

Case 2: d12 = 0, d21 6= 0.In this case under conditions (33) and (34) the matrix Bj have negative real parts and the

equilibrium point (N,P ) is Turing unstable if

d22 > d22crit :=detA + λ1(d11Θ4 − d21Θ3)

λ1(Θ1 − λ1d11). (36)

If d21 < 0, this implies that the predator species tends to diffuse in the direction of higherconcentration of the prey species, and the prey species moves along its own concentrationgradient.

We note that d22crit > d∗22crit . This implies that decreasing the Turing space compared tothe first case, so that the emergence of spatial patterns is holding in small regions of parameterspace. This situation is a usual phenomenon in nature.


If d21 > 0, this implies that the predator species tends to diffuse in the direction of lowerconcentration of the prey species, and the prey species moves along its own concentrationgradient. Such a case arise in nature where the predator prefers to avoid group defense by alarge number of prey and chooses to catch its prey from a smaller group unable to sufficientlyresist.

We note that d22crit < d∗22crit . This implies that increasing the Turing space compared tothe first case, so that the emergence of spatial patterns is holding in large regions of parameterspace.

Case 3: d12 6= 0, d21 = 0.In this case under conditions (33) and (34) the matrix Bj have negative real parts and the


d22 > d22crit :=detA + λ1(d11Θ4 + d12Θ2)

λ1(Θ1 − λ1d11). (37)

If d12 < 0, this implies that the prey species moves towards the higher concentration of thepredator species, and the predator species moves along its own concentration gradient. Thissituation can be compared in nature where the predator attracts the prey towards itself as apredation technique and the suicidal tendencies among the prey exist.

Also here we note that d22crit < d∗22crit . This implies that increasing the Turing spacecompared to the first case, so that the emergence of spatial patterns is holding in large regionsof parameter space.

If d12 > 0, this implies that the prey species moves in the direction of lower concentration ofthe predator species, and the predator species moves along its own concentration gradient. Thissituation can be compared in nature where the prey moves towards the lower concentration ofthe predator in search of new food.

We note that d22crit > d∗22crit . This implies that decreasing the Turing space compared tothe first case, so that the emergence of spatial patterns is holding in small regions of parameterspace. This situation is a usual phenomenon in nature.

Case 4: d12 6= 0, d21 6= 0.In this case under conditions (33) and (34) the matrix Bj have negative real parts and the


d22 > d22crit :=detA + λ1d11Θ4 + λ1(−d12d21λ1 + d12Θ2 − d21Θ3)

λ1(Θ1 − λ1d11). (38)

1) If d12 < 0, d21 < 0, this implies that the prey species tends to diffuse in the directionof higher concentration of the predator species and the predator species tends to diffuse in thedirection of higher concentration of the prey species. Such situations are common in naturewhen prey represent the investment capital and the predator represent labour force.

We note that Turing space increase (resp. decrease) if λ1 > d12Θ2−d21Θ3d12d21

i.e. d22crit <

d∗22crit ( resp. < d12Θ2−d21Θ3d12d21

i.e. d22crit > d∗22crit), so that the emergence of spatial patterns isholding in large (resp. small) regions of parameter space.

28 SHABAN ALY1

2) If d12 > 0, d21 > 0, this implies that the prey species moves in the direction of lowerconcentration of the predator species and the predator species tends to diffuse in the directionof lower concentration of the prey species. Such a case arise in nature where the prey movestowards the lower concentration of the predator in search of new food. and the predator prefersto avoid group defense by a large number of prey and chooses to catch its prey from a smallergroup unable to sufficiently resist.


i.e. d22crit <

d∗22crit ( resp. < d12Θ2−d21Θ3d12d21

i.e .d22crit > d∗22crit), so that the emergence of spatial patternsis holding in large (r×esp. small) regions of parameter space.

3) If d12 > 0, d21 < 0, this implies that the prey species tends to diffuse in the directionof lower concentration of the predator species, and the predator species tends to diffuse in thedirection of higher concentration of the prey species. Such situations are common in nature forthe survival of the prey predator species.

We note that d22crit > d∗22crit . This implies that decreasing the Turing space compared tothe first case, so that the emergence of spatial patterns is holding in small regions of parameterspace.

4) If d12 < 0, d21 > 0, his implies that the prey species tends to diffuse in the directionof higher concentration of the predator species and the predator species tends to diffuse in thedirection of lower concentration of the prey species.


i.e. d22crit <

d∗22crit (resp. < d12Θ2−d21Θ3d12d21

i.e .d22crit > d∗22crit), so that the emergence of spatial patternsis holding in large (resp. small) regions of parameter space.

4. CONCLUSIONS

In this paper we have considered a Cavani-Farkas type predator-prey interacting model withself as well as cross-diffusion and investigated the stability conditions in different environmen-tal consequences. We show that at a certain critical value a diffusion driven instability occurs,i.e. the stationary solution stays stable with respect to the kinetic system (the system withoutdiffusion) but becomes unstable with respect to the system with diffusion and that Turing in-stability takes place. We note that the cross-diffusion increase or decrease a Turing space (thespace which the emergence of spatial patterns is holding) compared to the Turing space withself-diffusion, i.e. the cross-diffusion response is an important factor that should not be ignoredwhen pattern emerges.

ACNOWLEDGEMENTS

I would like to express my sincere thanks to Prof. Dongwoo Sheen (SNU University) for hisdetailed criticism that helped me to improve this paper.


REFERENCES

[1] Casten R. G. , Holland C. F., Stability properties of solutions to systems of reaction-diffusion equations,SIAAM J. Appl. Math., 33 (1977) 353-364.

[2] Cavani M. Farkas M. Bifurcation in a predator-prey model with memory and diffusion II: Turing bifurcation.Acta Math. Hungar. 63 (1994) 375-393.

[3] Chattopadhyay J., Sarkar, A.K., Tapaswi, P.K., Effect of cross-diffusion on a diffusive prey-predator system-anonlinear analysis. J. Biol. Sys. 4(1996) 159-169.

[4] Farkas M. Dynamical Models in Biology, Academic Press, 2001.[5] Farkas M. Two ways of modeling cross diffusion, Nonlinear Analysis, TMA., 30 (1997) 1225-1233.[6] Freedman, H.I., Shukla, J.B., The effect of a predator resource on a diffusive predator-prey system. Nat. Res.

Model. 3 (3) (1989) 359-383.[7] Murray J. D. Mathematical Biology, Berlin, Springer-Verlag, 1989.[8] Okubo A. Diffusion and Ecological Problems: Models, Springer-Verlag, Berlin, 1980.[9] Okubo, A., Levin S. A. Diffusion and Ecological Problems: Modern Perspectives, Second Edition (Springer,

Berlin), 2000.[10] Takeuchi Y. Global Dynamical Properties of Lotka-Volterra system, World Scientific, 1996.[11] Turing A. M. The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London B237, (1953) 37-72,

reprinted: Bull. Math. Biol. 52 (1990) 153-197.

J. KSIAM Vol.13, No.1, 31–40, 2009

TRANSFORMATION OF DIMENSIONLESS HEAT DIFFUSION EQUATION FORTHE SOLUTION OF DYNAMIC DOMAIN IN PHASE CHANGE PROBLEMS

MUHAMMAD ASHRAF1, R. AVILA2, AND S. S. RAZA3

1DEPT OF MATH, COMSATS INST OF INFORMATION TECHNOLOGY, ISLAMABAD, PAKISTAN


2THERMAL FLUID LAB, FAC OF MECH ENG, NATL AUTONOMOUS UNIV OF MEXICO (UNAM), MEXICO

3GLOBAL CHANGE IMPACT STUDY CENTER, ISLAMABAD, PAKISTAN

ABSTRACT. In the present work transformation of dimensionless heat diffusion equation forthe solution of moving boundary problems have been formulated. The formulation is basedon 1-D, 2-D and 3-D, unsteady heat diffusion equations. These equations are first turned intodimensionless form by using dimensionless quantities and their transformation was formulatedin liquid and solid phases. The salient feature of this work is that during the transformationof dimensionless heat diffusion equation there arises a convective term v which is responsiblefor the motion of interface in liquid as well as solid phase. In the transformed heat equation, acorrection factor β also arises naturally which gives the correct transformed flux at interface.

1. INTRODUCTION

Transient heat transfer problems, involving melting or solidification, generally referred toas phase-change or moving body problems are important in many engineering applications.The solution of such problem is difficult because the interface between the two phases is mov-ing as latent heat is absorbed or released at the interface [1]. For the numerical solution ofsuch problems dimensionless form of partial differential equation is required. The problemsinvolving moving boundaries are very complicated at the interface, so transformation of dimen-sionless partial differential equation is a helpful technique for their solution in liquid and soliddomain [4]. During the process of melting or solidification, interface moves in between liquidand solid part of the domain. A transformation technique must therefore be used to insurethat the final projection from the old to new domain is exact. The basis of above mentionedtransformation are the classical heat equations T1(x, y, z, t) and T2(x, y, z, t) [1].

Received by the editors January 26 2009; Accepted February 26 2009.2000 Mathematics Subject Classification. 35Kxx, 35K05.Key words and phrases. Heat Equation, Diffusion Transformation.

31

32 MUHAMMAD ASHRAF, R. AVILA, AND S. S. RAZA

FIGURE 1. Domain definition for a two dimensional moving boundary problem.

ρCP

∂T1

∂t= K1∇

2T1 in D1 (1)

ρCP

∂T2

∂t= K2∇

2T2 in D2 (2)

ρLVn = K2∇T2 · n2 − K1∇T1 · n1 on∑

(t). (3)

Here ρ is the density, CP is the specific heat, and K is the thermal conductivity, L is thelatent heat of fusion and vn is the interface velocity in the direction n1 normal to the surface∑

(t) . The subscripts 1 and 2 refer to the liquid and solid phases as from Figure 1, respectively.The initial conditions on the temperature and the interface position are given by

T1(x, y, z, t = 0) (4)

T2(x, y, z, t = 0) (5)

∑

(t = 0) =o∑

. (6)

And the boundary conditions are assumed to be of the form:

a1∇T1 · n1 = c1 on Γ1 (7)

a2∇T2 · n2 = c2 on Γ2 (8)

T1 = T2 = Tm,s. (9)

Here ai, bi, and ci are constants and Tm,s is the melting or solidification temperature of thematerial. Γ1 and Γ2 are the liquid and solid boundaries of the domain.

TRANSFORMATION OF DIMENSIONLESS HEAT DIFFUSION EQUATION ... 33

2. DIMENSIONLESS FORM OF THE HEAT EQUATIONS

The dimensionless form of the 1-D, 2-D, and 3-D heat diffusion equations by using dimen-sionless quantities is given as under:

2.1. Dimensional form of 1-D Heat Equation. The dimensionless form non dimensionalheat diffusion equation is as follows:

∂Θ1

∂τ=

∂2Θ1

∂ξ2, 0 < ξ < η (10)

∂Θ2

∂τ=

∂2Θ2

∂ξ2, η < ξ < 1 (11)

γdη

dτ= −

∂Θ1

∂ξ+ K

∂Θ2

∂ξξ = η. (12)

2.2. Dimensional form of 2-D Heat Equation. The dimensionless form two dimensionalheat diffusion equation is as follows:

∂Θ1

∂τ=

∂2Θ1

∂ξ21

+∂2Θ1

∂ξ22

, 0 < ξ < η, 0 < ξ2 < η (13)

∂Θ2

∂τ= K

(

∂2Θ2

∂ξ21

+∂2Θ2

∂ξ22

)

, 0 < ξ < η, 0 < ξ2 < η (14)

γdη

dτ= −

(

∂Θ1

∂ξ1

+∂Θ1

∂ξ2

)

+ K

(

∂Θ2

∂ξ1

+∂Θ2

∂ξ2

)

, ξ1 = η, ξ2 = η. (15)

2.3. Dimensional form of 3-D Heat Equation. The dimensionless form three dimensionalheat diffusion equation is as follows:

∂Θ1

∂τ=

∂2Θ1

∂ξ21

+∂2Θ1

∂ξ22

+∂2Θ1

∂ξ23

, 0 < ξ1 < η, 0 < ξ2 < η, 0 < ξ3 < η (16)

∂Θ2

∂τ= K

(

∂2Θ2

∂ξ21

+∂2Θ2

∂ξ22

+∂2Θ2

∂ξ23

)

, 0 < ξ1 < η, 0 < ξ2 < η, 0 < ξ3 < η (17)

γdη

dτ= −

(

∂Θ1

∂ξ1

+∂Θ1

∂ξ2

+∂Θ1

∂ξ3

)

+ K

(

∂Θ2

∂ξ1

+∂Θ2

∂ξ2

+∂Θ2

∂ξ3

)

, ξ1 = η, ξ2 = η, ξ3 = η.

(18)Dimensionless quantities are:

ξ =x

l, τ =

α1t

l2, Θ1 =

(

T1 − Tb

4T

)

Θm,s =

(

Tm,s − Tb

4T

)

, γ =L

CP4T

η =s

l, α =

k

ρCP

, 4T = Ta − Tb, ξ1 =x1

l1, ξ2 =

x2

l2, ξ3 =

x3

l3


FIGURE 2. The dynamic regions d1 < ξ < η(τ) and η(τ) < ξ < d2 aretransformed to fixed domains through an interface-local transformation tech-nique.

Θ2 =

(

T2 − Tb

4T

)

.

3. TRANSFORMATION OF DIMENSIONLESS HEAT DIFFUSION EQUATIONS

The transformation of dimensionless heat diffusion equation is given as under:

3.1. Transformation of 1-D dimensionless Heat Diffusion Equation. The dimensionlessform of the heat diffusion equation for the liquid part of the domain is as under:

∂Θ1

∂τ=

∂2Θ1

∂ξ2, 0 < ξ < η. (19)

The dimensionless heat equations are defined on dynamic domains described by the motionof the interface boundary

∑

(t) . In time dependent moving mesh technique, at every time stepthe heat equation in each phase (4-8) are first solved on a fixed domain corresponding to the oldposition of the phase interface. The new interface position is then calculated on the basis of (4c-8c), the geometry and domain of the problem are updated. A transformation technique musttherefore be used to insure that the final projection from the old to new domain is exact [4]. Wenow present the particular interface local transformation technique that we have employed; asthe mapping procedure is identical in both phases. We consider only the liquid phase and dropthe subscript 1, [2]

∂Θ

∂τ=

∂2Θ1

∂ξ2, 0 < ξ < η(τ). (20)

Let ξ be the coordinate on the time dependent domain and ξ be the coordinate on the asso-ciated fixed (old) domain. Our transformation is then given by:

ξ = ξ, 0 < ξ < d1 (21)


FIGURE 3. (3a) The dynamic regions d1 < ξ1, ξ2, ξ3 < η(τ) and η(τ) <

ξ1, ξ2, ξ3 < d2 are transformed to fixed domains through an interface-localtransformation technique in 3D. (3b) Interpretation of the moving interface, η

is the position of the interface.

ξ =

(

η(τo) − d1

η(τ) − d1

)

(ξ − d1) + d1, d1 < ξ < η(τ) (22)

which maps ξ ∈ [0, η(τ)] to ξ ∈ [0, η(τo)], as shown in Figure 2. The transformation involvesonly that part of the dynamic domain d1 < ξ < η(τ), with d1 chosen near the interface. Whereη(τo) and η(τ) corresponds to the interface position at the old time step and at new time steprespectively [2,7,8].

If we see that the left hand side of the equation (20) can be written as:

∂Θ

∂τ=

∂Θ

∂ξ

dξ

dτ+

∂Θ

∂τ(23)

where ∂Θ

∂τis the temporal term in dynamic domain, and by using (22) we have

dξ

dτ= −

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ − d1). (24)

Thus equation (23) can be written as:

∂Θ

∂τ= −

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ − d1)

∂Θ

∂ξ+

∂Θ

∂τ. (25)


We can also write∂2Θ

∂ξ2=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ2. (26)

By substituting (25) and (26) in (20), we have the equation of the form

∂Θ

∂τ−

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ − d1)

∂Θ

∂ξ=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ2. (27)

The equation (27) can be arranged as follows:

1

β

∂Θ

∂τ+ v

∂Θ

∂ξ= β

∂2Θ

∂ξ2(28)

v = −1

β

dτ(η)

dτ

β =ξ − d1

τ(η) − d1

.

The motivation behind the transformation is identity Θ(ξ, τ) = Θ(ξ, τ) that is, any solutionof convection-diffusion 0 < ξ < η(τo) is identical to the solution of the heat diffusion on thenew domain 0 < ξ < η(τ). This implies that the solution on the the old fixed domain canbe projected directly on the new domain .Where the additional convective term v reflects thatthe domain is fixed near the phase interface. The correction factor β in the transformed heatequation are distributed so that the equivalent variational statement naturally gives the correcttransformed flux at the interface.

3.2. Transformation of 2-D dimensionless Heat Diffusion Equation. The 2-D, dimension-less form of the heat diffusion equation for the liquid part of the domain is as under:

∂Θ

∂τ=

∂2Θ1

∂ξ21

+∂2Θ1

∂ξ22

, 0 < ξ1 < η(τ), 0 < ξ2 < η(τ). (29)

Let ξ1,and ξ2 be the coordinate on the time dependent domain ξ1 and ξ2 be the coordinateson the associated fixed old domain, the transformation is thus:

ξ1 = ξ1, 0 < ξ1 < d1 (30)

ξ2 = ξ2, 0 < ξ2 < d1 (31)and

ξ1 =

(

η(τo) − d1

η(τ) − d1

)

(ξ1 − d1) + d1, d1 < ξ1 < η(τ) (32)

ξ2 =

(

η(τo) − d1

η(τ) − d1

)

(ξ2 − d1) + d1, d1 < ξ2 < η(τ) (33)

andξ1 ∈ [0, η(τ)] , ξ2 ∈ [0, η(τ)]


toξ1 ∈ [0, η(τo)] , ξ2 ∈ [0, η(τo)] .

The left hand side of the equation (29) can be written as

∂Θ

∂τ=

∂Θ

∂ξ1

dξ1

dτ+

∂Θ

∂ξ2

dξ2

dτ+

∂Θ

∂τ. (34)

By using (32) and (33) the equation (34) can be written as

∂Θ

∂τ= −

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ1 − d1)

∂Θ

∂ξ1

−(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ2 − d1)

∂Θ

∂ξ2

+∂Θ

∂τ. (35)

And the terms on the right hand side of the equation (29) can be written as

∂2Θ

∂ξ21

=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ21

(36)

∂2Θ

∂ξ22

=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ22

. (37)

By using equation (35), (36), and (37) the equation (29) is of the form

∂Θ

∂τ−

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ1 − d1)

∂Θ

∂ξ1

−(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ2 − d1)

∂Θ

∂ξ2

=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ21

+∂2Θ

∂ξ22

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ22

. (38)

The above equation (38) can be arranged as follows

1

β

∂Θ

∂τ+ v1

∂Θ

∂ξ 1

+ v2

∂Θ

∂ξ 2

= β

(

∂2Θ

∂ξ21

+∂2Θ

∂ξ22

)

. (39)

The important terms in equation (39), have the values.

v1 = −1

β

dτ(η)

dτ

′

v2 = −1

β

dτ(η)

dτ

β =ξ1 − d1

τ(η) − d1

, β =ξ2 − d1

τ(η) − d1

where v1 and v2 are convective terms in two dimensions and are responsible for the movementof interface in liquid phase,and is the same for the solid phase, and the correction factor β in thetransformed heat equation are distributed so that the equivalent variational statement naturallygives the correct transformed flux at the interface in two dimensions problems.


3.3. Transformation of 3-D dimensionless Heat Diffusion Equation. The 3-D, dimension-less form of the heat diffusion equation for the liquid part of the domain is as under:

∂Θ

∂τ=

∂2Θ1

∂ξ21

+∂2Θ1

∂ξ22

+∂2Θ1

∂ξ23

, 0 < ξ1 < η(τ), 0 < ξ2 < η(τ). (40)

Let ξ1, ξ2, and ξ3 be the coordinate on the time dependent domain ξ1, ξ2, and ξ3 be thecoordinates on the associated fixed old domain as from Figure 3, the transformation is thus:

ξ1 = ξ1, 0 < ξ1 < d1 (41)

ξ2 = ξ2, 0 < ξ2 < d1 (42)ξ3 = ξ3, 0 < ξ3 < d1 (43)

ξ1 =

(

η(τo) − d1

η(τ) − d1

)

(ξ1 − d1) + d1, d1 < ξ1 < η(τ) (44)

ξ2 =

(

η(τo) − d1

η(τ) − d1

)

(ξ2 − d1) + d1, d1 < ξ2 < η(τ) (45)

ξ3 =

(

η(τo) − d1

η(τ) − d1

)

(ξ3 − d1) + d1, d1 < ξ3 < η(τ) (46)

whereξ1 ∈ [0, η(τ)] , ξ2 ∈ [0, η(τ)] , ξ3 ∈ [0, η(τ)]

toξ1 ∈ [0, η(τo)] , ξ2 ∈ [0, η(τo)] , ξ3 ∈ [0, η(τo)] .

The left hand side of the equation (40) can be written as

∂Θ

∂τ=

∂Θ

∂ξ1

dξ1

dτ+

∂Θ

∂ξ2

dξ2

dτ+

∂Θ

∂ξ3

dξ3

dτ+

∂Θ

∂τ. (47)

By using equations (41-46) the above equation (47) can be written as

∂Θ

∂τ= −

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ1 − d1)

∂Θ

∂ξ1

−(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ2 − d1)

∂Θ

∂ξ2

−(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ3 − d1)

∂Θ

∂ξ3

+∂Θ

∂τ. (48)

The terms right hand side of the equation (40), can be written as

∂2Θ

∂ξ21

=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ21

(49)

∂2Θ

∂ξ22

=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ22

(50)

∂2Θ

∂ξ23

=

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ23

. (51)


By using (48-51), we have the equation of the form

∂Θ

∂τ−

(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ1 − d1)

∂Θ

∂ξ1

−(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ2 − d1)

∂Θ

∂ξ2

−(η(τo) − d1)

(η(τ) − d1)2

dτ(η)

dτ(ξ3 − d1)

∂Θ

∂ξ3

=

(

η(τo) − d1

η(τ) − d1

)2

∂2Θ

∂ξ21

+∂2Θ

∂ξ22

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ22

+∂2Θ

∂ξ23

(

η(τo) − d1

η(τ) − d1

)2∂2Θ

∂ξ23

. (52)

The above equation (52) can be arranged as follows

1

β

∂Θ

∂τ+ v1

∂Θ

∂ξ 1

+ v2

∂Θ

∂ξ 2

+ v3

∂Θ

∂ξ 3

= β

(

∂2Θ

∂ξ21

+∂2Θ

∂ξ22

+∂2Θ

∂ξ23

)

. (53)

The important terms in equation (53), have the values

v1 = −1

β

dτ(η)

dτ

′

, v2 = −1

β

dτ(η)

dτ, v3 = −

1

β

dτ(η)

dτ

β =ξ1 − d1

τ(η) − d1

, β =ξ2 − d1

τ(η) − d1

, β =ξ3 − d1

τ(η) − d1

.

Conclusion:The most important feature of this formulation is that convective term arises during transforma-tion which is responsible for the motion of interface in the phase change problem with liquidas well as solid phase. The terms v1, v2, and v3, play an significant role in heat transfer andhence affect the progress of solidification as well melting front. This approach provides thefundamental governing equations (28), (39), and (53) in the form of transformation of dimen-sionless heat diffusion equation and can be used to analyze the complex geometries by usingdifferent numerical techniques. The correction factor β which is also arises naturally whichgives the correct transformed flux at the interface. Thus this formulation is a golden standardfor numerical analyst to analyze the moving boundary problems.


Nomonclature:T1, T2 = Temperature in the liquid and solid phaseTm,s = Melting or solidification temperature

ρ = Fluid densityΓ1, Γ2 = Liquid and solid boundaries

∇T = Temperature differenceCP = Specific heatvn = Interface velocityγ = Stefan numberL = Latent heat, amount of heat release or absorbed by interface

Θ1, Θ2 = Dimensionless liquid and solid domain temperaturek = Thermal Conductivity

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problem J. Comp. Phys., 141, pp. 1-21, 1998.[10] C. Vuik and G. Segal and F.J. Vermolen A conserving discretization for a Stefan problem with an interface

reaction at the free boundary Comput. Visual Sci., 3, pp. 109-114, 2000.[11] N.C.W. Kuijpers, F.J. Vermolen, K. Vuik, and S. van der Zwaag A model of the beta-AlFeSi to alpha-

Al(FeMn)Si transformation in Al-Mg-Si alloys Materials Transactions, 44, pp. 1448–1456, 2003[12] F.J. Vermolen and C. Vuik Solution of vector Stefan problems with cross-diffusion Journal of Computational

and Applied Mathematics, 176, pp. 179-201,2005[13] F.J. Vermolen, C. Vuik, E. Javierre and S. van der Zwaag Review on some Stefan Problems for Particle Disso-

lution Nonlinear Analysis: Modelling and Control, 10, pp. 257-292, 2005[14] E. Javierre and C. Vuik and F.J. Vermolen and S. van der Zwaag A comparison of models for one-dimensional

Stefan problems J. Comp. Appl. Math., 192, pp. 445-459, 2006.

J. KSIAM Vol.13, No.1, 41–53, 2009

A NEW PRIMAL-DUAL INTERIOR POINT METHOD FOR LINEAROPTIMIZATION

GYEONG-MI CHO

DEPARTMENT OF MULTIMEDIA ENGINEERING, DONGSEO UNIVERSITY, SOUTH KOREA


ABSTRACT. A primal-dual interior point method(IPM) not only is the most efficient methodfor a computational point of view but also has polynomial complexity. Most of polynomial-time interior point methods(IPMs) are based on the logarithmic barrier functions. Peng etal.([14, 15]) and Roos et al.([3]-[9]) proposed new variants of IPMs based on kernel functionswhich are called self-regular and eligible functions, respectively. In this paper we define a newkernel function and propose a new IPM based on this kernel function which has O(n

23 log n

ε)

and O(√

n log nε) iteration bounds for large-update and small-update methods, respectively.

1. INTRODUCTION

In this paper we propose a new primal-dual IPM for the following standard LO problem

min{cT x : Ax = b, x ≥ 0}, (1)

where A ∈ Rm×n with rank(A) = m, c, x ∈ Rn, b ∈ Rm, and its dual problem

max{bT y : AT y + s = c, s ≥ 0}, (2)

where y ∈ Rm and s ∈ Rn.Since Karmarkar’s paper [12] in 1984, IPMs have shown their efficiency in solving large-

scale linear programming problems with a wide variety of successful applications. In this paperwe propose a new primal-dual interior point algorithm based on a new kernel function. Most ofpolynomial-time interior point algorithms for LO are based on the logarithmic barrier function[2, 10, 11, 18]. Peng et al. [14, 15] proposed a new variant of IPMs based on self-regular bar-rier functions and achieved so far the best known complexity, i.e. O(

√n log n log n

ε ) for large-update methods with a specific self-regular function. Roos et al. [3, 4, 5, 6, 7, 8, 9] proposednew primal-dual IPMs for LO problems based on eligible barrier functions. They proposed the

Received by the editors February 10 2009; Accepted March 6 2009.2000 Mathematics Subject Classification. 90C05, 90C51.Key words and phrases. primal-dual interior point method, kernel function, complexity, polynomial algorithm,

linear optimization problem.This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-

2007-531-C00012).

41

42 GYEONG-MI CHO

scheme for analyzing the algorithm based on four conditions on the kernel function [4] and ob-tained O(

√n log n

ε ) iteration bound for small-update IPMs in all cases andO(√

n log n log nε )

iteration bound for large-update methods with a specific kernel function [3]. Recently, Aminiand Haseli [1] proposed a generalized version of the kernel function in [4] and achieved thebest known iteration bound for large-update methods by using the general scheme in [4] basedon four conditions on the kernel function.

Motivated by their works, we define a new kernel function and propose a new primal-dualinterior point algorithm for LO based on this kernel function. And we analyze the complexityfor large-update and small-update methods based on the kernel function. Furthermore, thecomplexity bounds obtained by the algorithm are O(n

23 log n

ε ) and O(√

n log nε ) for large-

update and small-update methods, respectively. For large-update methods the complexity resultimproves the best known bound since n

23 <

√n log n for 4 ≤ n ≤ 2.4128 × 107. For small-

update method this is currently the best known complexity result.The paper is organized as follows. In section 2 we recall the generic IPM and the motivation

of the new algorithm. In section 3 we define a new kernel function and give its properties whichare essential for complexity analysis. In section 4 we derive the complexity result for the newalgorithm. Finally, concluding remarks are given in section 5.

We use the following notations throughout the paper. Rn+ and Rn

++ denote the set of n-dimensional nonnegative vectors and positive vectors, respectively. For x, s ∈ Rn, xmin andxs denote the smallest component of the vector x and the componentwise product of the vectorsx and s, respectively. We denote X the diagonal matrix from a vector x, i.e. X = diag(x).e denotes the n-dimensional vector of ones and log n, the natural logarithm. For f(x), g(x) :R++ → R++, f(x) = O(g(x)) if f(x) ≤ c1g(x) for some positive constant c1 and f(x) =Θ(g(x)) if c2g(x) ≤ f(x) ≤ c3g(x) for some positive constants c2 and c3.

2. PRELIMINARIES

In this section we recall the generic IPM. Without loss of generality, we assume that both (1)and (2) satisfy the interior point condition (IPC) [16], i.e., there exists (x0, y0, s0) such that

Ax0 = b, x0 > 0, AT y0 + s0 = c, s0 > 0.

Using the duality theorem (Theorem II.2 in [16]), we can find an optimal solution by solvingthe following system:

Ax = b, x ≥ 0,

AT y + s = c, s ≥ 0,

xs = 0.

(3)

The basic idea of primal-dual IPMs is to replace the third equation in (3) by the parameterizedequation xs = µe with µ > 0. Now we consider the following system:

Ax = b, x > 0,

AT y + s = c, s > 0,

xs = µe.

(4)

INTERIOR POINT METHOD 43

If the IPC holds, then the system (4) has a unique solution for each µ > 0 [13]. We denote thissolution as (x(µ), y(µ), s(µ)) and call it the µ-center. The set of µ-centers (µ > 0) is calledthe central path [17]. The limit of the central path (as µ goes to zero) exists and since the limitpoint satisfies (3), it yields the optimal solution for (1) and (2) [16]. Primal-dual IPMs followthe central path approximately and approach the solution of (1) and (2) as µ goes to zero.For given (x, y, s) := (x0, y0, s0) by applying Newton method to the system (4) we have thefollowing Newton system

A∆x = 0,

AT ∆y + ∆s = 0,

s∆x + x∆s = µe− xs.

(5)

Since A has full row rank, the system (5) has a unique search direction (∆x,∆y, ∆s) [16].By taking a step along the search direction (∆x,∆y, ∆s), one constructs a new positive iterate(x+, y+, s+),

x+ = x + α∆x, y+ = y + α∆y, s+ = s + α∆s,

for some α > 0.For the motivation of the new algorithm we define the following scaled vectors: for (x, s) >

0 and µ > 0,

v :=√

xs

µ, dx :=

v∆x

x, ds :=

v∆s

s. (6)

Using (6), we can rewrite the system (5) as follows:

Adx = 0,

AT ∆y + ds = 0,

dx + ds = v−1 − v,

(7)

where A := 1µAV −1X, V := diag(v), and X := diag(x). Note that the right side of the third

equation in (7) equals the negative gradient of the logarithmic barrier function Ψl(v), i.e.

dx + ds = −∇Ψl(v), (8)

where

Ψl(v) :=n∑

i=1

ψl(vi), ψl(t) =n∑

i=1

(t2 − 1

2− log t

).

We call ψl the kernel function of the logarithmic barrier function Ψl(v). In this paper wereplace ψl(t) with a new kernel function ψ(t) which will be defined in section 3.

The generic interior point algorithm works as follows: Assume that we are given a strictlyfeasible point (x, y, s) which is in a τ−neighborhood of the given µ−center. Then we decreaseµ to µ+ = (1 − θ)µ, for some fixed θ ∈ (0, 1) and then we solve the Newton system (5) toobtain the unique search direction. The positivity condition of a new iterate is ensured withthe right choice of the step size α which is defined by some line search rule. This procedure isrepeated until we find a new iterate (x+, y+, s+) that is in a τ−neighborhood of the µ+−centerand then we let µ := µ+ and (x, y, s) := (x+, y+, s+). Then µ is again reduced by the factor

44 GYEONG-MI CHO

1− θ and we solve the Newton system targeting at the new µ+-center, and so on. This processis repeated until µ is small enough, say until nµ < ε.

Generic Primal-Dual Algorithm for LO

Input:a threshold parameter τ > 0;an accuracy parameter ε > 0;a fixed barrier update parameter θ, 0 < θ < 1;

(x0, s0) and µ0 := 1 such that Ψl

(√x0s0

µ0

)≤ τ ;

beginx := x0; s := s0; µ := µ0;while nµ ≥ ε dobegin

µ := (1− θ)µ;while Ψl(v) > τ dobegin

Solve the system (5) for ∆x,∆y, ∆s;Determine a step size α;x := x + α∆x;s := s + α∆s;y := y + α∆y;v :=

√xsµ ;

endend

end

Remark 2.1. If θ is a constant independent of the dimension of the problem n, e.g. θ = 12 ,

then we call the algorithm a large-update method. If θ depends on n, e.g. θ = 1√n

, then thealgorithm is called a small-update method.

3. THE NEW KERNEL FUNCTION

In this section we define a new kernel function and give its properties which are essential toour complexity analysis. We call ψ : R++ → R+ a kernel function if ψ is twice differentiableand satisfies the following conditions:

ψ′(1) = ψ(1) = 0,

ψ′′(t) > 0, t > 0,

limt→0+

ψ(t) = limt→∞ψ(t) = ∞.

(9)


Now we define a new function ψ(t) as follows:

ψ(t) := 8t2 − 12t + 2 + 2t−2, t > 0. (10)

Then we have the following:

ψ′(t) = 16t− 12− 4t−3, ψ′′(t) = 16 + 12t−4, ψ′′′(t) = −48t−5. (11)

From (11), ψ(t) is a kernel function and

ψ′′(t) > 16, t > 0. (12)

In this paper, we replace the function Ψl(v) in (8) with the function Ψ(v) as follows:

dx + ds = −∇Ψ(v), (13)

where Ψ(v) =∑n

i=1 ψ(vi), where ψ(t) is defined in (10). Hence the new search direction(∆x,∆y, ∆s) is obtained by solving the following modified Newton system:

A∆x = 0,

AT ∆y + ∆s = 0,

s∆x + x∆s = −µv∇Ψ(v).(14)

Note that dx and ds are orthogonal because the vector dx belongs to null space and ds to therow space of the matrix A. Since dx and ds are orthogonal, we have

dx = ds = 0 ⇔ ∇Ψ(v) = 0 ⇔ v = e ⇔ Ψ(v) = 0 ⇔ x = x(µ), s = s(µ).

We use Ψ(v) as the proximity function to measure the distance between the current iterateand the µ-center for given µ > 0. We also define the norm-based proximity measure δ(v) asfollows:

δ(v) :=12||∇Ψ(v)|| = 1

2||dx + ds||. (15)

Lemma 3.1. For ψ(t) we have(i) ψ(t) is exponentially convex for all t > 0,(ii) ψ′′(t) is monotonically decreasing for all t > 0,(iii) tψ′′(t)− ψ′(t) > 0 for all t > 0,(iv) ψ

′′(t)ψ

′(βt)− βψ

′(t)ψ

′′(βt) > 0, t > 1, β > 1.

Proof: For (i), by Lemma 2.1.2 in [15], it suffices to show that the function ψ(t) satisfiestψ′′(t) + ψ′(t) ≥ 0 for all t > 0. Using (11), we have

tψ′′(t) + ψ′(t) = t(16 + 12t−4) + (16t− 12− 4t−3) = 32t− 12 + 8t−3.

Let g(t) = 32t − 12 + 8t−3. Then g′(t) = 32 − 24t−4 and g′′(t) = 96t−5 > 0, t > 0.

Letting g′(t) = 0, we have t = (34)

14 . Since g(t) is strictly convex and has a global minimum

g((34)

14 ) > 27. Hence we have the result.

For (ii), from (11), we have ψ′′′(t) < 0.For (iii), from (11), we have

tψ′′(t)− ψ′(t) = t(16 + 12t−4)− (16t− 12− 4t−3) = 12 + 16t−3 > 0.

46 GYEONG-MI CHO

For (iv), using Lemma 2.4 in [4], (ii) and (iii), we have the result. This completes the proof.2

Lemma 3.2. For ψ(t) we have(i) 8(t− 1)2 ≤ ψ(t) ≤ 1

32(ψ′(t))2, t > 0,(ii) ψ(t) ≤ 14(t− 1)2, t ≥ 1.

Proof: For (i), using the first condition of (9) and (12), we have

ψ(t) =∫ t

1

∫ ξ

1ψ′′(ζ)dζdξ ≥ 16

∫ t

1

∫ ξ

1dζdξ = 8(t− 1)2.

which proves the first inequality. The second inequality is obtained as follows:

ψ(t) =∫ t

1

∫ ξ

1ψ′′(ζ)dζdξ ≤ 1

16

∫ t

1

∫ ξ

1ψ′′(ξ)ψ′′(ζ)dζdξ

=116

∫ t

1ψ′′(ξ)ψ′(ξ)dξ =

116

∫ t

1ψ′(ξ)dψ′(ξ) =

132

(ψ′(t))2.

For (ii), using Taylor’s Theorem, ψ(1) = ψ′(1) = 0, ψ

′′′< 0, and ψ

′′(1) = 40, we have

ψ(t) = ψ(1) + ψ′(1)(t− 1) +

12ψ′′(1)(t− 1)2 +

13!

ψ′′′

(ξ)(ξ − 1)3

=12ψ′′(1)(t− 1)2 +

13!

ψ′′′

(ξ)(ξ − 1)3

<12ψ′′(1)(t− 1)2 = 14(t− 1)2,

for some ξ, 1 ≤ ξ ≤ t. This completes the proof. 2

Let % : [0,∞) → [1,∞) be the inverse function of ψ(t) for t ≥ 1 and ρ : [0,∞) → (0, 1] theinverse function of −1

2ψ′(t) for t ∈ (0, 1]. Then we have the following lemma.

Lemma 3.3. For ψ(t) we have(i)

√s8 + 1 ≤ %(s) ≤ 1 +

√s8 , s ≥ 0,

(ii) ρ(s) ≥ ( 2s+2)

13 , s ≥ 0.

Proof: For (i), let s = ψ(t), for t ≥ 1, i.e. %(s) = t, t ≥ 1. By the definition of ψ(t),s = 8t2 − 12t + 2 + 2t−2. This implies that

8t2 = s + 12t− 2− 2t−2 ≥ s + 8

because 12t− 2− 2t−2 is monotone increasing with respect to t and t ≥ 1. Hence we have

t = %(s) ≥√

s

8+ 1.

Using Lemma 3.2 (i), we have s = ψ(t) ≥ 8(t− 1)2. Then we have

t = %(s) ≤ 1 +√

s

8, s ≥ 0.


For (ii), let z = −12ψ′(t), for all t ∈ (0, 1]. Then by the definition of ρ, ρ(z) = t, t ∈ (0, 1]

and 2z = −ψ′(t). So we have 2z = −16t + 12 + 4t−3.

4t−3 = 2z + 16t− 12 ≤ 2z + 4,

since 16t− 12 is monotone increasing in t and 0 < t ≤ 1. Hence we have

ρ(z) = t ≥ (2

z + 2)

13 , z ≥ 0.

2

Using Lemma 3.1 (iv), we have the following. The reader can refer to Theorem 3.2 in [4] forthe proof.

Lemma 3.4. Let % : [0,∞) → [1,∞) be the inverse function of ψ(t), t ≥ 1. Then we have

Ψ(βv) ≤ nψ

(β%

(Ψ(v)

n

)), v ∈ R++, β ≥ 1.

2

In the following theorem we obtain an estimate for the effect of a µ-update on the value ofΨ(v).

Theorem 3.5. Let 0 ≤ θ < 1 and v+ = v√1−θ

. If Ψ(v) ≤ τ, then we have

Ψ(v+) ≤ 141− θ

(√nθ +

√τ

8

)2

.

Proof: Since 1√1−θ

≥ 1 and %(

Ψ(v)n

)≥ 1, we have

%(

Ψ(v)n

)√

1−θ≥ 1. Using Lemma 3.4 with

β = 1√1−θ

, Lemma 3.2 (ii), Lemma 3.3 (i), and Ψ(v) ≤ τ , we have

Ψ(v+) ≤ nψ

(1√

1− θ%

(Ψ(v)

n

))≤ 14n

%

(Ψ(v)

n

)−√1− θ

√1− θ

2

≤ 14n

(1 +

√τ8n −

√1− θ√

1− θ

)2

≤ 14n

(θ +

√τ8n√

1− θ

)2

=14

1− θ

(√nθ +

√τ

8

)2

,

where the last inequality holds from 1 −√1− θ = θ1+√

1−θ≤ θ, 0 ≤ θ < 1. This completes

the proof. 2

Denote

Ψ0 :=14

1− θ

(√nθ +

√τ

8

)2

. (16)

Then Ψ0 is an upper bound for Ψ(v) during the process of the algorithm.

48 GYEONG-MI CHO

Remark 3.6. For large-update method by taking τ = O(n) and θ = Θ(1), Ψ0 = O(n). Forsmall-update method with τ = O(1) and θ = Θ( 1√

n), Ψ0 = O(1).

4. COMPLEXITY ANALYSIS

In this section we compute a proper step size and the decrease of the proximity function duringan inner iteration and give the complexity results of the algorithm. For fixed µ, taking a stepsize α, we have new iterates x+ = x + α∆x, s+ = s + α∆s. Using (6), we have

x+ = x

(e + α

∆x

x

)= x

(e + α

dx

v

)=

x

v(v + αdx)

and

s+ = s

(e + α

∆s

s

)= s

(e + α

ds

v

)=

s

v(v + αds).

Thus we have

v+ :=√

x+s+

µ=

√(v + αdx)(v + αds).

Define for α > 0f(α) = Ψ(v+)−Ψ(v).

Then f(α) is the difference of proximities between a new iterate and a current iterate for fixedµ. By Lemma 3.1 (i), we have

Ψ(v+) = Ψ(√

(v + αdx)(v + αds)) ≤ 12(Ψ(v + αdx) + Ψ(v + αds)).

Hence we have f(α) ≤ f1(α), where

f1(α) :=12(Ψ(v + αdx) + Ψ(v + αds))−Ψ(v). (17)

We havef(0) = f1(0) = 0.

Taking the derivative of f1(α) with respect to α, we have

f ′1(α) =12

n∑

i=1

(ψ′(vi + α[dx]i)[dx]i + ψ′(vi + α[ds]i)[ds]i),

where [dx]i and [ds]i denote the i-th components of the vectors dx and ds, respectively. Using(13) and (15), we have

f ′1(0) =12∇Ψ(v)T (dx + ds) = −1

2∇Ψ(v)T∇Ψ(v) = −2δ(v)2. (18)

Differentiating f ′1(α) with respect to α, we have

f ′′1 (α) =12

n∑

i=1

(ψ′′(vi + α[dx]i)[dx]2i + ψ′′(vi + α[ds]i)[ds]2i ). (19)

Since f ′′1 (α) > 0, f1(α) is strictly convex in α unless dx = ds = 0.


Lemma 4.1. Let δ(v) be as defined in (15). Then we have

δ(v) ≥ 2√

2Ψ(v).

Proof: Using Lemma 3.2 (i), we have

Ψ(v) =n∑

i=1

ψ(vi) ≤ 132

n∑

i=1

ψ′(vi)2 =132||∇Ψ(v)||2 =

18δ(v)2.

Hence we have δ(v) ≥ 2√

2Ψ(v). 2

Remark 4.2. Throughout the paper we assume that τ ≥ 1. Using Lemma 4.1 and the assump-tion Ψ(v) ≥ τ, we have

δ(v) ≥ 2√

2Ψ(v) ≥ 2√

2. (20)2

For notational convenience we denote δ := δ(v) and Ψ := Ψ(v).

Lemma 4.3. (Modification of Lemma 4.1 in [4]) Let f1(α) be as defined in (17) and δ be asdefined in (15). Then we have

f ′′1 (α) ≤ 2δ2ψ′′(vmin − 2αδ). (21)

Lemma 4.4. (Modification of Lemma 4.2 in [4]) If the step size α satisfies the inequality

−ψ′(vmin − 2αδ) + ψ′(vmin) ≤ 2δ, (22)

then we havef ′1(α) ≤ 0.

Lemma 4.5. (Modification of Lemma 4.3 in [4]) Let ρ : [0,∞) → (0, 1] denote the inversefunction of −1

2ψ′(t) for all t ∈ (0, 1]. Then, in the worst case, the largest step size α satisfying(22) is given by

α :=12δ

(ρ(δ)− ρ(2δ)).

Lemma 4.6. (Modificatiion of Lemma 4.4 in [4]) Let ρ and α be as defined in Lemma 4.5.Then we have

α ≥ 1ψ′′(ρ(2δ))

.

Defineα :=

1ψ′′(ρ(2δ))

. (23)

Then we have α ≤ α.

Lemma 4.7. Let α be as defined in (23). If Ψ(v) ≥ τ ≥ 1, then we have

α ≥ 1

45δ43

.

50 GYEONG-MI CHO

Proof: Using the definition of ψ′′(t), Lemma 3.3 (ii), and (20), we have

α =1

ψ′′(ρ(2δ))=

116 + 12 1

(ρ(2δ))4

≥ 1

16 + 12(δ + 1)43

≥ 1

4√

2δ + 12(δ + 1)43

≥ 1

(12 + 4√

2)(δ + 1)43

≥ 1

243 (12 + 4

√2)δ

43

≥ 1

45δ43

.

2

For notational convenience we denote

α =1

45δ43

. (24)

Note that α ≤ α. We will use α as the default step size.

Lemma 4.8. (Lemma 1.3.3 in [15]) Suppose that h(t) is a twice differentiable convex functionwith

h(0) = 0, h′(0) < 0

and h(t) attains its (global) minimum at t∗ > 0 and h′′(t) is increasing with respect to t. Thenfor any t ∈ [0, t∗], we have

h(t) ≤ th′(0)2

.

Lemma 4.9. (Modification of Lemma 4.5 in [4]) If the step size α is such that α ≤ α, then

f(α) ≤ −αδ2.

Proof: Let the univariate function h be such that

h(0) = f1(0) = 0, h′(0) = f ′1(0) = −2δ2, h′′(α) = 2δ2ψ′′(vmin − 2αδ).

Then h(t) is twice differentiable, h(0) = 0, and h′(0) < 0. Since h′′(α) > 0, h(t) is strictlyconvex and hence has a global minimum at some α∗ > 0. From (21), we have f ′′1 (α) ≤ h′′(α).As a result, we get f ′1(α) ≤ h′(α) and f1(α) ≤ h(α). Taking α ≤ α, we have

h′(α) = h′(0) +∫ α

0h′′(ζ)dζ

= −2δ2 + 2δ2

∫ α

0ψ′′(vmin − 2ξδ)dξ

= −2δ2 − 2δ2

2δ

∫ α

0ψ′′(vmin − 2ξδ)d(vmin − ξδ)

= −2δ2 − δ(ψ′(vmin − 2αδ)− ψ′(vmin))≤ −2δ2 + 2δ2 = 0,


where the inequality follows from (22). Since h′′′(α) = −4δ2ψ′′′(vmin − 2αδ) > 0, h′′(α) ismonotonically increasing in α. Thus, using Lemma 4.8, we have

f1(α) ≤ h(α) ≤ 12αh′(0) = −αδ2.

Since f(α) ≤ f1(α), the lemma is proved. 2

Theorem 4.10. Let α be as defined in (24) and Ψ(v) ≥ 1. Then

f(α) ≤ −2Ψ(v)13

45.

Proof: Using Lemma 4.9, (24), and Lemma 4.1 we have

f(α) ≤ −αδ2 = −δ23

45≤ −((8Ψ(v))

12 )

23

45= −2Ψ(v)

13

45.

This completes the proof. 2

Lemma 4.11. (Lemma 1.3.2 in [15]) Let t0, t1, · · ·, tK be a sequence of positive numbers suchthat

tk+1 ≤ tk − γt1−βk , k = 0, 1, · · ·,K − 1,

where γ > 0 and 0 < β ≤ 1. Then K ≤⌊

tβ0γβ

⌋.

Denote the value of Ψ(v) after the µ−update as Ψ0 and the subsequent values in the same outeriteration as Ψk, k = 1, 2, · · · . Then we have

Ψ0 ≤ Ψ0, (25)

where Ψ0 is defined in (16). Let K denote the total number of inner iterations per outer itera-tion. Then we have

ΨK−1 > τ, 0 ≤ ΨK ≤ τ.

Lemma 4.12. Let Ψ0 be as defined in (16) and K the total number of inner iterations in theouter iteration. Then we have

K ≤ 34Ψ230 .

Proof: Using Theorem 4.10 and Lemma 4.11 with γ := 245 and β := 2

3 , we have

K ≤(

452

)(32

)Ψ

230 ≤ 34Ψ

230 .


Theorem 4.13. Let a LO problem be given, Ψ0 as defined in (16) and τ ≥ 1. Then the totalnumber of iterations to have an approximate solution with nµ < ε is bounded by⌈

34θ

Ψ230 log

n

ε

⌉.

52 GYEONG-MI CHO

Proof: If the central path parameter µ has the initial value µ0 := 1 and is updated by multiply-ing 1− θ with 0 ≤ θ < 1, then after at most

⌈1θ

logn

ε

⌉

iterations we have nµ < ε [16]. For the total number of iterations, we multiply the number ofinner iterations by that of outer iterations. Hence the total number of iterations is bounded by

⌈34θ

Ψ230 log

n

ε

⌉.


Remark 4.14. Taking τ = O(n) and θ = Θ(1), we haveO(n23 log n

ε ) iteration complexity forlarge-update IPMs which improves the best known complexity result for 4 ≤ n ≤ 2.4128×107.For small-update methods, we haveO(

√n log n

ε ) iteration complexity which is the best knowncomplexity result so far.

5. CONCLUDING REMARKS

Motivated by recent work of Amini and Maseli [1], we propose a new primal-dual interiorpoint algorithm for LO problems based on a new kernel function and analyze the iterationcomplexity of the algorithm. For large-update methods we have O(n

23 log n

ε ) iteration bound.Since n

23 <

√n log n for 4 ≤ n ≤ 2.4128×107, this complexity result is better than the one in

[1] which is currently the best known complexity, that is O(√

n log n log nε ). For small-update

methods the iteration bound is O(√

n log nε ) which is the best known iteration bound.

Future research might focus on the extension to semidefinite optimization and second ordercone optimization. Numerical tests will be another topic for future research.

REFERENCES

[1] K. Amini and A. Haseli, A new proximity function generating the best known iteration bounds for both large-update and small-update interior point methods, ANZIAM J. 49 (2007), 259-270.

[2] E.D. Andersen, J. Gondzio, Cs. Meszaros, and X. Xu, Implementation of interior point methods for largescale linear programming, in: T. Terlaky(Ed.), Interior point methods of mathematical programming, KluwerAcademic Publisher, The Netherlands, 189-252, 1996.

[3] Y.Q. Bai, M. El Ghami, and C. Roos, A new efficient large-update primal-dual interior-point method basedon a finite barrier, Siam J. on Optimization 13 (2003), 766-782.

[4] Y.Q. Bai, M. El Ghami, and C. Roos, A comparative study of kernel functions for primal-dual interior-pointalgorithms in linear optimization, Siam J. on Optimization 15 (2004), 101-128.

[5] Y.Q. Bai and C. Roos, A primal-dual interior point method based on a new kernel function with linear growthrate, in: Proceedings of the 9th Australian Optimization Day, Perth, Australia, 2002.

[6] Y.Q. Bai and C. Roos, A polynomial-time algorithm for linear optimization based on a new simple kernelfunction, Optimization Methods and Software 18 (2003), 631-646.


[7] Y.Q. Bai, G. Lesaja, C. Roos, G.Q. Wang, and M. El Ghami, A class of large-update and small-update primal-dual interior-point algorithms for linear optimization, J. Optim. Theory and Appl., DOI 10.1007/s10957-008-9389-z., 2008.

[8] M. El Ghami, I. Ivanov, J.B.M. Melissen, C. Roos, and T. Steihaug, A polynomial-time algorithm for linearoptimization based on a new class of kernel functions, Journal of Computational and Applied Mathematics,DOI 10.1016/j.cam.2008.05.027., 2008.

[9] M. El Ghami and C. Roos, Generic primal-dual interior pont methods based on a new kernel function,RAIRO-Oper. Res. 42 (2008), 199-213.

[10] C.C. Gonzaga, Path following methods for linear programming, Siam Review 34 (1992), 167-227.[11] D. den Hertog, Interior point approach to linear, quadratic and convex programming, Mathematics and its

Applications 277, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1994.[12] N.K. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984), 373-

395.[13] M. Kojima, S. Mizuno, and A. Yoshise, A primal-dual interior-point algorithm for linear programming, in: N.

Megiddo(Ed.), Progress in mathematical programming: Interior point and related methods, Springer-Verlag,New York, 29-47, 1989.

[14] J. Peng, C. Roos, and T. Terlaky, Self-regular functions and new search directions for linear and semidefiniteoptimization, Mathematical Programming 93 (2002), 129-171.

[15] J. Peng, C. Roos, and T. Terlaky, Self-Regularity, A new paradigm for primal-dual interior-point algorithms,Princeton University Press, 2002.

[16] C. Roos, T. Terlaky, and J. Ph. Vial, Theory and algorithms for linear optimization, An interior approach,John Wiley & Sons, Chichester, U.K., 1997.

[17] G. Sonnevend, An analytic center for polyhedrons and new classes of global algorithms for liner (smooth,convex) programming, in: A. Prekopa, J. Szleezsan, and B. Strazicky(Ed.), System modeling and optimization: Proceeding of the 12th IFIP-Conference, Budapest, Hungary, September 1985, Volume 84, Lecture Notes inControl and Information Sciences, Springer Verlag, Berlin, West-Germany, 866-876, 1986.

[18] N.J. Todd, Recent developments and new directions in linear programming, in: M. Iri and K. Tanabe(Ed.),Mathematical Programming : Recent developments and applications, Kluwer Academic Publishers, Dor-drecht, 109-157, 1989.

[19] S.J. Wright, Primal-dual interior-point methods, SIAM, Philadelphia, USA, 1997.[20] Y. Ye, Interior-point algorithms, John Wiley & Sons, Chichester, UK, 1997.

J. KSIAM Vol.13, No.1, 55-72, 2009

In this paper we are concerned with the situation where sometimes the value (c +1) is reported erroneously as c in relation to size-biased MPSD. As we know, weighted distributions arise when the observations generated from a stochastic process are not given equal chance of being recorded; instead, they are recorded according to some weight function.

Received by the editors August 2 2008; Accepted in revised form March 15 2009. 2000 Mathematics Subject Classification. 62E15 Key words and phrases. Misclassification, size biased, modified power series distribution, raw moments,

central moments, factorial moments, variance ratio, inverted parabola, generalized Poisson, generalized negative binomial, Borel distribution. † Corresponding author.

55

MISCLASSIFICATION IN SIZE-BIASED MODIFIED POWER SERIES DISTRIBUTION AND ITS APPLICATIONS

ANWAR HASSAN1† AND PEER BILAL AHMAD2

1 DEPARTMENT OF STATISTICS, UNIVERSITY OF KASHMIR, SRINAGAR, INDIA E-MAIL address: [email protected], [email protected] 2DEPARTMENT OF STATISTICS, UNIVERSITY OF KASHMIR, SRINAGAR, INDIA E-MAIL address: [email protected]

ABSTRACT. A misclassified size-biased modified power series distribution (MSBMPSD) where some of the observations corresponding to x = c + 1 are misclassified as x = c with probability α, is defined. We obtain its recurrence relations among the raw moments, the central moments and the factorial moments. Discussion of the effect of the misclassification on the variance is considered. To illustrate the situation under consideration some of its particular cases like the size-biased generalized negative binomial (SBGNB), the size-biased generalized Poisson (SBGP) and size-biased Borel distributions are included. Finally, an example is presented for the size-biased generalized Poisson distribution to illustrate the results.

1. INTRODUCTION

In certain experimental investigations involving discrete distributions external factors may induce a measurement error in the form of misclassification. For instance, a situation may arise where certain values are erroneously reported; such a situation termed as modified or misclassified has been studied by Cohen ([1],[2],[3] for the Poisson and the binomial random variables, Jani and Shah [4] for modified power series distribution (MPSD) where some of the value of one are sometimes reported as zero, and recently by Patel and Patel ([5], [6]) incase of generalized power series distribution (GPSD) and MPSD for a more general situation where sometimes the value (c +1) is reported erroneously as c.

Cohen [2] altered data from Bortkiewicz’s [7] classical example on deaths from the kick of a horse in the Prussian Army, to illustrate the practical application of his results. He assumed that twenty of 200 given records which should have shown one death were in error by reporting no deaths. The same example was considered by Williford and Bingham [8].

56 ANWAR HASSAN, PEER BILAL AHMAD

When the weight function depends on the lengths of the units of interest, the resulting distribution is called length biased. More generally, when the sampling mechanism selects units with probability proportional to some measure of the unit size, the resulting distribution is called size-biased. Such distributions arise in life length studies and were studied by various authors (see Blumenthal [9], Scheaffer [10], Gupta ([11], [12], [13], [14]), Gupta and Tripathi ([15], [16])).

Gupta [17] defined the MPSD with probability function given by ( ) T x ,

)(f)(g)x(a]xX[P

x

1 ∈θθ

== (1)

where a(x)>0 and T is a subset of the set of non-negative integers, g(θ), f(θ) are positive, finite and differentiable, is the parameter. In case g(θ) is invertible it reduces to Patils[18] generalized power series distribution and if in additions T is the entire set of non-negative integers it reduces to power services distribution (PSD) given by Noack [19]. The class of distributions (1) includes among others the generalized negative binomial, generalized Poisson and Borel distributions.

θ

Gupta [17] obtained the mean ( )( )θμ and variance (μ2) of MPSD (1) given by

)(g)(f)(f)(g)(θ′θθ′θ

=θμ (2)

and θθμ

θ′θ

=μd

)(d )(g)(g

2 (3)

A size-biased MPSD is obtained by taking the weight of MPSD (1) as X, given by

( ) ( ) ( )

)(f)(g)x(b

)(f)()(g)x(b

)(f)()(g)x(xa]xX[P *

xxx

2 θθ

=θθμθ

=θθμθ

== (4)

where b(x) = xa(x) and f*(θ) = μ(θ) f(θ). As stated above we have studied the situation such that the probabilities in the

distribution (4) are modified by a constant quantity α(0 ≤ α ≤ 1) by increasing the probability of c value of the variable.

In this paper we are concerned with the situation where sometimes the value c+1 is reported erroneously as one in relation to SBMPSD. We obtain the recurrence relations between the raw moments, the central moments and the factorial moments of misclassified SBMPSD. Discussion of the effect of the misclassification on the variance is considered. To illustrate the situation under consideration some of its particular cases like the size-biased generalized negative binomial (SBGNB), the size-biased generalized Poisson (SBGP) and size-biased Borel distributions are included. Finally, an example is presented for the size-biased generalized Poisson distribution to illustrate the results.

2. MISCLASSIFIED SIZE-BIASED MODIFIED POWER SERIES DISTRIBUTION

Suppose X is a random variable having the SBMPSD (4) from which a random sample is

MISCLASSIFICATION IN SIZE-BIASED MPSD AND ITS APPLICATIONS 57

drawn. Assume that some of the observations corresponding to x = c +1 are erroneously reported as x = c, and let this probability of misclassifying be α. Then the resulting distribution of X, the so called misclassified size-biased modified power series distribution (MSBMPSD), can be written in the form

(5) ⎪⎩

⎪⎨

⎧

∈μ+=μα=μα+

=θα== ++

+

S x , f/gb1c x, f/g)b-(1

c x ; f/)gbb(g),;x(p]xX[P

x(x)

1c1c

1ccc

3

where S=T− (c, c+1) is a subset of the set I of non-negative integers not containing c and c +1, b(x), f = f(θ), g = g(θ), μ = μ(θ) are stated above and 0 ≤ α ≤1, the α being the proportion of misclassified observations. It is interesting to note that for α = 0 the distribution (5) reduces to simple size-biased MPSD. Further if g(θ) is invertible, it reduces to the misclassified size-biased generalized power series distribution and in addition if T is regarded as an entire set I of non-negative integers, it will be called the misclassified size biased power series distribution.

In this section we obtain the mean and the variance and establish certain recurrence relations for the raw, the central and the factorial moments of the distribution (5). Here notations with * corresponds to size-biased MPSD.

2.1. Mean of the distribution. For mean, by definition, we have

Mean= ∑∑ μ+

μα−+

+μα+

=θα=μ′+

++

∈ fgxb

fgb)1)(1c(

f)gbb(cg

),,x(xpx

x1c

1c1ccc

Tx31

where ∑ stands for the sum over x ∈S here and onwards. Also, from (5) we have ∑+α−+α+=μ +

++x

x1c

1c1ccc gbgb)1()gbb(gf

Differentiating w.r.t. θ and multiplying both sides by g/g′, we get

∑+α−++α+α+=μ′+μ′′

++

++

++

xx

1c1c

1c1c

1c1c

cc gxbgb)1)(1c(gbgcbgcb)ff(

gg

∑+α−++α+=α−μ′+μ′′

+++

++

xx

1c1c1cc

c1c1c gxbgb)1)(1c()gbb(cggb)ff(

gg

where ( )(θμθ∂∂

=μ′ ) and )(ff θθ∂∂

=′ and )(gg θθ∂∂

=′ .

After simplification we get mean of (5) as

Mean = μ

α−⎟

⎠⎞

⎜⎝⎛

θ∂μ∂

′+μ=μ⎥

⎦

⎤⎢⎣

⎡α−μ′+μ′

′=μ′

+++

+ fgb

fffgb)ff(

gg 1c

1c1c1c1

( ) μα−μ= ++ fgb 1c

1c*1 (6)


where θ∂μ∂

′+μ=μ

ff*

1 is mean of size-biased MPSD(4).

2.2. Recurrence relation among raw moments. The rth raw moment of (5) is given as

∑∑ =++=++==θα=μ′∈

]xX[Px]1cX[P)1c(]cX[Pc),,x(px rrr3

Tx

rr

[ ]∑+α−++α+μ

= +++

xx

r1c1c1cc

cr gbxgb)1)(1c()gbb(gcf1

(7)

Differentiating (7) w.r.t. θ, we get

[ ggb)1c)(1(bgg)1c(cbggc)f(

1 c1c

1r1c

crc

1c1r2

r ′+α−+′+α+′μ

=θ∂μ′∂

++

+−+

] [ 1c1c

r1cc

cr1xx

1r gb)1c)(1()gbb(gcggbx +++

−+ +α−+α+−′+∑ ] ⎥

⎦

⎤⎢⎣

⎡μμ′+μ′

+∑ 2x

xr

)f(ffgbx

Multiplying both sides by g and after simplification we get

gffbgc

fggg r1c

1cr1r

r μ′⎟⎟⎠

⎞⎜⎜⎝

⎛μμ′

+′

−αμ′

+μ′′=θ∂μ′∂

++

+

r1c

1crr

1r gg

fgfg

fbgc

gg

μ′⎟⎟⎠

⎞⎜⎜⎝

⎛μμ′′

+′′

+μ

α−

θ∂μ′∂

′=μ′ +

+

+

r*1

1c1cr

r

fbgc

gg

μ′μ+μ

α−

θ∂μ′∂

′= +

+

This, upon using (6), reduces to

r11c

1cr

rr

1r fbg

)c(gg

μ′μ′+μ

−μ′α+θ∂μ′∂

′=μ′ +

+

+ (8)

Higher moments can be obtained with r = 2, 3, ⋅⋅⋅ etc. From (7), it is easy to establish a relation between rth moment (μ′r) of misclassified size-biased MPSD (5) and the rth moment (μ*r) of the size-biased MPSD (4) as

μ

+−α+μ=μ′+

+

fgb))1c(c(

1c1crr*

rr (9)

from which the higher order moments (r>1) μ′2, μ′3 …… are obtained.

2.3. Recurrence relation among central moments. The rth central moment of (5) is given as


[ ] ∑∑ μ′−+μ′−++μ′−=μ′−=μ +∈

xr

11cr

1cr

1Tx

xr

1r P)x(P)1c(P)c(P)x(

[ ]∑ μ′−+μ′−+α−+α+μ′−μ

= +++

xx

r1

1c1c

r11cc

cr1 gb)x(gb)1c)(1()gbb(g)c(

f1

Differentiating w.r.t. θ and simplifying, we get

( )μ

′μ−μ′−α+⎟

⎠⎞

⎜⎝⎛

θ∂μ′∂

μ−μ⎟⎟⎠

⎞⎜⎜⎝

⎛ ′=

θ∂μ∂ +

−+ fggb

)c(rgg c

1cr

r1

11r1r

r

This will yield

[ ] μμ′−−μα+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛

θ∂μ′∂

μ+θ∂μ∂

⎟⎟⎠

⎞⎜⎜⎝

⎛′

=μ ++−+ f)c(gbr

gg r

1r1c

1c1

1rr

1r (10)

Putting r =1 in (10) and noting μ0 = 1 and μ1 = 0, we get the variance (μ2) of (5) as

μ

−μ′α+⎟⎠⎞

⎜⎝⎛

θ∂μ′∂

′=μ

+

+ fgb)c(

gg 1c

1c11

2 (11)

Using (11) in (10) we obtain a recurrence relation for central moments as

[ ]μ

μ′−−μα+⎥

⎦

⎤⎢⎣

⎡μ

−μ′α−μμ+

θ∂μ∂

′=μ

++

++

−+ f)c(gb

fgb)c(

rgg r

1r1c

1c1c

1c121r

r1r

[ ]

μ−μ′μ−μ′−−μ

α+μμ+θ∂μ∂

′= −+

+− f)c(r)c(gbr

gg 11r

r1r1c

1c21rr (12)

where r = 2, 3, ⋅⋅⋅ etc for higher order moments.

2.4. Recurrence relation among factorial moments. The rth factorial moment of (5) is given by

∑∑ +++==μ′ +∈

x]r[

1c]r[

c]r[

x]r[

Tx]r[ PxP)1c(PcPx

μ

+μα−+

+μα+

= ∑+1c]r[c]r[++

fgbx

fgb)1()1c(

f)gbb(gc x

x]r[

1c1cc (13)

Differentiating (13) w.r.t. θ, we get

[ ggb)1)(1c()1c()gbb(ggccf1 c

1c]r[

1cc1c]r[]r[ ′α−+++α+′

μ=⎟⎟

⎠

⎞⎜⎜⎝

⎛θ∂

μ′∂++

−

]μ

′α+⎥

⎦

⎤⎢⎣

⎡μμ′

+′

μ′−′+ +−∑ fggcb

ffggbxx

c]r[1c

]r[1x

x]r[

Using the identity x.x[r] = x[r+1] + rx[r], we get


{ }[ { }]r[]1r[1cc

1c]r[]1r[]r[ )1c(r)1c()gbb(ggrccf1

++++α+′+μ

=θ∂

μ′∂ ++

−+

{ } ]μ

′α+⎥

⎦

⎤⎢⎣

⎡μμ′

+′

μ′−′++′α− +−++ ∑ f

ggcbffggbrxxggb)1(

c]r[1c

]r[1x

x]r[]1r[c

1c

⎥⎦

⎤⎢⎣

⎡θ∂

μ′∂′

+′′

μ′+μ

α−μ′−

θ∂

μ′∂′

=μ′+

++

]1[]r[

1c1c

]r[

]r[]r[

]1r[ ff

fggf

fgbc

rgg

[ ]μ

α−μ′−μ′−

θ∂

μ′∂′

=+

+

fgbc

rgg 1c

1c]r[

]r[*]r[

]r[ (14)

where is first factorial moments of size biased MPSD(4). *]1[μ′

Again in view of (6), this can be put in the form

[ ] ( )μ

−μ′α−μ′−μ′+⎟⎟

⎠

⎞⎜⎜⎝

⎛θ∂

μ′∂⎟⎟⎠

⎞⎜⎜⎝

⎛′

=μ′+

++ f

gbcr

gg 1c

1c]r[

]r[]r[]1[

]r[]1r[ (15)

where r = 2,3, ⋅⋅⋅ and μ′[1] , μ′[2] are given by ∑ ∑

∈ ∈

μ′===μ′Tx Tx

1xx]r[

]1[ xPPx

and ∑ ∑∈ ∈

μ′−μ′=−==μ′Tx Tx

12xx]r[

]2[ P)1x(xPx

where μ′1 and μ′2 are obtained from (8) with r = 0 and r = 1

3. VARIANCE COMPARISON

In this section we discuss how the variance of (5) is effected when the reporting observations are erroneously misclassified. From (9) we have

( ) μ+−α+μ′=μ′ ++ fgb)1c(c 1c

1c*

11 (16)

and ( ) μ+−α+μ′=μ′ ++ fgb)1c(c 1c

1c22*

22 (17) Hence the variance of the distribution (5) is given by

( ) ( )21c1c

*1

1c1c

22*22 fgb))1c(c(fgb)1c(c μ+−α+μ′−μ+−α+μ′=μ +

++

+

( ) ( )μ

μα−μ′α+μ−−α+μ′−μ′=+

+++

++ f

gbfgb2fgb1c2)(

1c1c1c

1c*

11c

1c2*

1*

2

( )μ

μα−−−μ′α+μ=μ+

+++ f

gbfgb1c22

1c1c1c

1c*

1*22


where ⎟⎟⎠

⎞⎜⎜⎝

⎛θ∂μ′∂

′=μ

*1*

2 gg

is the variance of the size-biased MPSD (4).

This gives a relation between the variance μ2 of (5) and the variance μ*2 of (4) as

(18) ),(*22 θαφ+μ=μ

where φ(α, θ) is a function of α and θ only given by

[ ]μ

μα−−−μ′α=θαφ+

+++ f

gbfgb1c22),(

1c1c1c

1c*

1 (19)

with

⎟⎟⎠

⎞⎜⎜⎝

⎛θ∂θμ∂

′+′′

=μ′)(

ff

gfgf*

1 and θ∂μ′∂

′=μ

*1*

2 gg

The above relation between the two variances shows that the variance of the size-biased MPSD has been affected by a term φ(α,θ) and is due to reporting the observations erroneously. We note that this misclassification has reasonably moderate effect on the variance . Now if we take the ratio of both the variances we have

*2

*2

2 ),(1Zμθαφ

+=μμ

= (20)

This ratio Z shows that μ2 may be equal to, greater than or less than μ*2 depending upon

the value of α and θ; that is, on the term φ(α, θ). This term would be useful in studying the effect of misclassification on the variance. This effect can be studied in two ways

i) When the value of θ is fixed. ii) When the value of α is fixed. Here we discuss the case (i) only. Similarly case (ii) can be dealt with. Thus when the

value of θ is fixed, the ratio Z will be a function of α only. That is to say Z = φ(α). Hence we have

*2

1c1c

*1

1c1c )fgb21c22(gbZ

μμα−−−μ′

=α∂∂ +

++

+ (21)

and 0 fgb

2Z *2

21c1c

2

2

<μ⎟⎟⎠

⎞⎜⎜⎝

⎛μ

−=α∂∂ +

+

From this, we note that the curve Z = φ(α) seems to be a concave towards the origin.

Equating α∂∂Z

to zero gives the value of α as

)f/gb2()1c22( 1c1c

*1 μ−−μ′=α +

+ (22) and hence the maximum value of Z becomes as


)4(

)1c22(1Z *2

2*1

max μ−−μ′

+= (23)

It is clear from (23) that the ratio Z is always greater than unity. This indicates that misclassification has reasonably a moderate effect on the variance. Further, it would be interesting to see that this ratio, when graphed on the (α, Z) axis, will provide an invertible parabola very concave to origin having its vertex at the point

( ) ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

μ

−−μ′+

μ−−μ′

++

2*2

2*1

1c1c

*1

2

)1c2(21 ;

)f/gb2()1c22(

(24)

when the value of θ is fixed . This parabola would be more useful in studying the behavioral effect of misclassification on the variance. Of course, one has to treat separately various possible specific cases with respect to their situations being observed in practice.

4. SOME APPLICATIONS

We illustrate here the situation under consideration defined by (5) in which some of the observations corresponding to the value (c+1) are sometimes reported erroneously as c, for some of its special cases like the size-biased generalized negative binomial distribution (SBGNBD), the size-biased generalized Poisson distribution (SBGPD) and size-biased Borel distribution, and apply them to the results obtained in section 2 and 3.

4.1. Misclassified size-biased GNBD. Jain and Consul [20] defined the generalized negative binomial distribution (GNBD) as

T x; )1( )1xxm(!x

)xm(m]xX[P xxmx ∈θ−θ+−β+Γ

β+Γ== −β+ (25)

where 0 < θ < 1; m > 0; |βθ| < 1

It reduces to (1) with)1xxm(!x

)xm(m)x(a+−β+Γ

β+Γ= , g (θ) = θ(1− θ)β−1, f(θ) = (1 −θ)−m

Suppose X has a size- biased GNBD given by

T x; )1xxm()!1x(

)1()1)(xm(]xX[Pxxm1x

∈+−β+Γ−θ−θθβ−β+Γ

==−β+−

(26)

which is size-biased MPSD (4) with

)(1

m)( and )1xxm()!1x(

)xm(m)x(ax)x(bθβ−θ

=θμ+−β+Γ−

β+Γ=⋅=

m1 )1()(f ,)1()(g −−β θ−=θθ−θ=θLet us assuming that the value (c +1) is sometimes reported as c in (26) and let the

probability of misclassifying these observations be α. Then the resulting distribution of X, the so called a misclassified size -biased GNBD, can be defined as


[ ]

[ ]

[ ]⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧ ⎡ +Γαβ+Γ−β mm)cm(mc1

∈θβ−θθ−

θ−θ+−β+Γ−

β+Γ

+=θβ−θθ−

θ−θ−+β+Γ+β+Γα−

=θβ−θθ−

⎥⎦

⎤⎢⎣

θ−θ−+β+Γ

+β+

+−β+Γ−θ−θ

=θα

−

−β

−

+−β

−

−β

s x 1/m)1(

)1( )1xxm()!1x(

)xm(m

1 c x ; 1/m)1(

)1( c)1c(m!c

)1c(mm)1(

c x; )1/(m)1(

)1(c)1c(m!c

)1c()1ccm()!1c(

)1(

],:x[P

m

x1

m

1c1

m

1

(27)

θ) in the foregoing results we obtain results for (2

where 0 < θ<1, 0 ≤ α≤ 1, m > 0 and |θβ|<1. Using the values of b(x), μ(θ), g(θ) and f (7) as under

4.1.1. Mean )1()1(gbm)1(

)1()1(

m 1mc1c21 θβ−θ−

α−

θβ−θ−

+θβ−θ

=μ′= −+β+ (28)

4.1.2. Variance [ ] c1c42 gb

m12mm

)1()1(

+α

+−β+θβ−θβ−θβ−θ−θ

=μ=

⎥⎦⎤

⎢⎣⎡ θβ−θ−

α−θβ−+−θ−+θβ−θ −+β

+31mc

1c2 )1()1(gb

m)1)(1c2()1(2)1(m2

(29) 4. ce relation among raw moments

11m )1()1( −−+β θβ−θ−1.3. Recurren

r11mc

1c1rr1 m)c()1)(1( −μ′α+

μ′∂θβ−θ−θ=μ′ −−

r1r )1)(1(gb μ′μ′+θ−θβ−θ∂

−+β++ (30)

. Recurrence relation among central moments 4.1.41m1c

1c21rr1 r)1)(1( − α+μμ+

μ∂θβ−θ−θ=μ 1r )1)(1(mgb −+β−

+−+ θ−θβ−θ∂

(31) 4. ce relation among factorial moment

[ ])c(r)c( 11rr

1r −μ′μ−μ′−−μ −

1.5. Recurren s.

[ ]]r[1− + 1c1c]r[]1[]1r[ mgbr)1)(1( −++ α−μ′−μ′

θ∂

μ′∂θβ−θ−θ=μ′

[ ]]r[]r[

1m c)1)(1( −μ′θ−θβ− −+β (32)


4.1.6. The variance ratio *2

2Zμμ

= becomes

[ ]12mm

)1()1(gbm

)1)(1c2()1(2)1(m2)1()1(gb

m1Z

31mc1c

2

32mc1c

−β+θβ−θβ−θ

⎢⎢⎢

⎣

⎡

⎥⎦⎤θβ−θ−

α−

θβ−+−θ−+θβ−θθβ−θ−

α

+=

−+β+

−+β+

(

For

33)

a fixed value of θ, the value α will be

[ ]

1mc1c

3

2

)1(gb)1(2)1)(1c2()1(2)1(m2m

−+β+ θ−θβ−

θβ−+−θ−+θβ−θ=α (34)

for which the maximum value Z comes out as

[ ]

)m12m)(1(4)1)(1c2()1(2)1(m2 −θ−+θβ−θ1Z

22

max θβ−θβ−−β+θ−θθβ−+

+= (35)

The invertible parabola when graphed on (α, Z) axis will have the vertex at the point

[ ] ;

)1(gb)1(2)1)(1c2()1(2)1(m2m

1mc1c

3

2

−+β+ θ−θβ−

θβ−+−θ−+θβ−θ

[ ])m2m)(1(4

)1)(1c2()1(2)1(m2122

θβ−θβ+β+θ−θθβ−+−θ−+θβ−θ

+ (36)

4.2. Misclassified size-biased Generalized Poisson distribution. Consul and Jain [21]

defined the generalized Poisson distribution (GPD) as

L 1,2 0, x !x

e)x(]xX[P)x(1x 21λ+λλ λ+λ−−

211 === (37)

and Consul [22] modified the form of (37) to

Shoukri

L 1,2 0, x 1,|| 0, ; !x

e)x1( )x1(x1x θβ+ β+θ−−

]xX[P =<θβ>θ== (38)

where 1

21 and

λλ

=βλ=θ

It is a particular case of Gupta [17] MPSD with

θθβ−−

θθ=θβ+

=)x1( 1x

= e)f( , e)g( , !x

)x(a

biased GPD given by Now suppose X has a size-


L 1,2 x ; )!1x(

e)1()x1(]xX[P)1x(1x1x

=−

θθβ−β+==

+βθ−−−

(39)


( ) θβ−θ−

θ=θ=θθβ−θ

=θμ−β+

== e)g( ,e)f( , )1(

, )!1x()x1()x(xa)x(b

1x

Assume that some of the values c + 1 are erroneously reported as c and let α be the probability of misclassifying them. Then the resulting distribution of X, the so called misclassified size-biased GPD, can be defined as

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

∈θβ−θ

θ−β+

+=θβ−θθ+β+α−

=θβ−θ

⎥⎦

⎤⎢⎣

⎡ θ+β+α+

−β+

θ

==

θ

θβ−−

θ

+θβ−

θθβ−−θβ−

s x )1/(e

)e()!1x()x1(

1 c x )1/(e!c

)e()]1c(1)[1(

c x ;)1(

e!c

)e()]1c(1[)!1c()c1()e(

]xX[P

x1x

1cc

c1cc

(40)

where S is defined earlier; 0 ≤ α ≤ 1, θ > −1 and |βθ| < 1. Using the values of b(x), μ(θ), g(θ) and f(θ) in the forgoing results we obtain the results

for (40):

4.2.1. Mean )1(c1c21 egb)1(

)1()1(1 +βθ−

+θβ−α−θβ−θ

+θβ−

=μ′= (41)

4.2.2. [ c1c

24

2

2 gb)1)(1c2()1(22)1(

)2(+α−θβ−+−θβ−θ+α+

θβ−θ+βθ−βθ

=μ

(42) ][ ])1(1c1c

)1(3 e)1(gb e)1( +βθ−−+

+βθ− θβ−θβ−4.2.3. Recurrence relation among raw moments

[ ] r1r

r)1(c

1cr

1r ce)1(gb)1(

μ′μ′+−μ′βθ−α+θ∂μ′∂

θβ−θ

=μ′ +βθ−++ (43)

4.2.4. Recurrence relation among central moments

)1(gbr)1(

c1c1r2

r1r θβ−α+μμ+⎟

⎠⎞

⎜⎝⎛

θ∂μ∂

θβ−θ

=μ +−+

(44) [ ])1(11r

r1r e)c(r)c( +βθ−

− −μ′μ−μ′−−μ4.2.5. Recurrence relation among factorial moments

[ ] [ ] )1(]r[]r[

c1c]r[]1[

]r[]1r[ ec)1(gbr

)1(+βθ−

++ −μ′θβ−α−μ′−μ′+θ∂

μ′∂

θβ−θ

=μ′ (45)


4.2.6. The variance ratio *2

2Zμμ

= becomes

[ ])1(3c1c

2 e)1(gb)1)(1c2()1(221Z +βθ−+ θβ−α−θβ−+−θβ−θ+α+=

(46) 12)1(c1c )2)(1(egb −β+θ−+ θ+βθ−βθθβ−×

For the fixed value of θ, the value of α is

[ ] 3)1(c1c

2 )1(egb2)1)(1c2()1(22 θβ−θβ−+−θβ−θ+=α β+θ−+ (47)

for which the maximum value Z will be

[ ]

)2(4)1)(1c2()1(221Z 2

22

max βθ−θ+βθθβ−+−θβ−θ+

+= (48)

The invertible parabola when graphed on (α, Z) axis will have the vertex at the point

[ ] [ ]]2[4

)1)(1c2()1(22 1 ; )1(egb2

)1)(1c2()1(222

22

3)1(c1c

2

βθ−θ+βθθβ−+−θβ−θ+

+θβ−

θβ−+−θβ−θ+β+θ−

+

(49)

4.3. Misclassified Size-biased Borel distribution. Borel [23] has defined a Borel distribution with one parameter for the random variable X as

L 1,2, x ,10 ; e )!x()x1(),x(p]xX[P )x1(x

1x

=<θ<θ+

=θ== θ+−−

(50)

(This distribution describes a distribution of the number of customers served before a queue vanishes under condition of a single queue with random arrival times (at constant rate) of customers and a constant time occupied in serving each customer).

It is a particular case of modified power series distribution (1) with

θθ−−

=θθ=θ+

= e)f( , )e()g( , !x)x1()x(a

1x

Now suppose X has a size-biased Borel distribution with p.d.f given by

L 1,2, x ; e)1( )1x(

)x1(]xX[P )x1(1x!

1x

=θ−θ−

+== +θ−−

−

(51)


θθ−−

=θθ=θθ−

θ=θμ

−+

=⋅= e)f( ,e)g( , 1

)( , )!1x(

)x1()x(ax)x(b1x

Assume that some of the values (c + 1) are erroneously reported as c, and let the proportions of these observations be α. Then the resulting distribution of X, the so called misclassified size-biased Borel distribution can be defined as


⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

∈θ−θ

θ−

+

+=θ−θθ+α−

=θ−

θ⎥⎦

⎤⎢⎣

⎡ θ+α+

−+

θ

==

θ

θ−−

θ

+θ−

θθ−−θ−

s x)1/(e

)e()!1x(

)x1(

1 c x )1/(e!c

)e(]2c)[1(

c x )1(

e!c

)e()c2()!1c(

)c1()e(

]xX[P

x1x

1cc

c1cc

(52)

where S is defined earlier, o ≤ α ≤ 1, 0 < θ < 1 and x = 1, 2, 3, ⋅⋅⋅ Using values of b(x), μ(θ), g(θ) and f(θ) in the forgoing results we obtain the results for

(52) as

4.3.1. Mean θ−+θ−α−

θ−θ

+θ−

=μ′= 2c1c21 egb)1(

)1()1(1

(53)

4.3.2. Variance [ c1c

24

2

2 gb)1)(1c2()1(22)1(

)3(+α−θ−+−θ−θ+α+

θ−θ−θ

=μ=

][ ]θ−−+

θ− θ−θ− 21c1c

23 e)1(gb e)1( (54) 4.3.3. Recurrence relation among raw moments

[ ] r1r

r2c

1cr

1r ce)1(gb)1(

μ′μ′+−μ′θ−α+θ∂μ′∂

θ−θ

=μ′ θ−++ (55)

4.3.4. Recurrence relation among central moments

)1(gbr)1(

c1c1r2

r1r θ−α+μμ+

θ∂μ′∂

θ−θ

=μ +−+ [ ]θ−− −μ′μ−μ′−−μ 2

11rr

1r e)c(r)c( (56)

4.3.5. Recurrence relation among factorial moments

[ ] [ ] θ−++ −μ′θ−α−μ′−μ′+

θ∂

μ′∂

θ−θ

=μ′ 2]r[]r[

c1c]r[]1[

]r[]1r[ ec)1(gbr

)1( (57)

4.3.6 The variance ratio *2

2Zμμ

= will be

[ ] θ−+

θ−+ θ−α−θ−+−θ−θ+α+= 2c

1c23c

1c2 egbe)1(gb)1)(1c2()1(221Z

(58) 123 )3()1( −βθ−θθ−For fixed value of θ, the value of α is

[ ] 32c1c

2 )1(egb2)1)(1c2()1(22 θ−θ−+−θ−θ+=α θ−+ (59)

for which the maximum value Z will be


[ ]

)3(4)1)(1c2()1(221Z 2

22

max θ−θθ−+−θ−θ+

+= (60)

The invertible parabola when graphed on (α,Z) axis will have the vertex at the point

[ ] [ ]

]3[4)1)(1c2()1(22 1 ;

)1(egb2)1)(1c2()1(22

2

2

32c1c

2

θ−θθ−+−θ−θ+

+θ−

θ−+−θ−θ+θ−

+

(61)

5. SOME NUMERICAL RESULTS

To see the effect of misclassification on variance a case of misclassified size-biased Borel distribution and misclassified size-biased GNB distribution with m=4 and is considered. It is assumed that some of the observations, which correspond to the value two, are reported erroneously as value one. The mean and the variances of the misclassified size-biased Borel distribution and misclassified size-biased GNBD have been calculated for different values of

1=β

α and θ and tabulated in the Tables 1 and 2 respectively, which shows the comparative study of the change in the mean and the variance of the distribution due to a change in , i.e. the proportion of misclassifying the observation corresponding to

by reporting it as one corresponding to α

2x = 1x = . The values in the first row of these two tables corresponding to are the values of the mean and the variance of the non-misclassified or usual distributions i.e. size-biased Borel distribution and Size-biased GNBD respectively. The tables shows that as

0=α

α increases, the mean of the distribution decreases while the variance decreases for small values of θ , while it increases for large values of . This can also be seen from the Tables 3 and 4, which gives the values of the

ratio (

θ

*2

2Zμμ

= ) of the variance of the misclassified distribution to that of the usual

distribution.

TABLE 1. The Values of Mean and Variance for Misclassified Size-Biased Borel Distribution

1.0=θ 5.0=θ 9.0=θ 1μ′ 2μ 1μ′ 2μ 1μ′ 2μ

0.0=α 1.3457 0.4420 5.0000 20.0000 109.0000 18900.00001.0=α 1.3257 0.4354 4.9832 20.1170 108.9982 18900.39015.0=α 1.2457 0.4011 4.9163 10.5787 108.9909 18901.95059.0=α 1.1657 0.3540 4.8494 21.0316 108.9837 18903.51080.1=α 1.1457 0.3403 4.8326 21.1434 108.9818 18903.9001


TABLE 2. The Values of Mean and Variance for Misclassified Size-Biased Generalized Negative Binomial Distribution with m=4, β=1

1.0=θ 5.0=θ 9.0=θ

1μ′ 2μ 1μ′ 2μ 1μ′ 2μ 0.0=α 0.4444 0.6173 4.0000 10.0000 36.0000 450.0000 1.0=α 0.4149 0.6152 3.9922 10.0702 35.9999 450.0004 5.0=α 0.2968 0.6119 3.9609 10.3500 35.9999 450.0020 9.0=α 0.1787 0.5762 3.9297 10.6279 35.9999 450.0036 0.1=α 0.1492 0.5629 3.9219 10.6970 35.9999 450.0040

TABLE 3. Variance Ratio ( *2

2Zμμ

= ) Values for Size-Biased Borel Distribution

θ=0.05 θ=0.1 θ=0.3 θ=0.5 θ=0.7 θ=0.9 α=0.1 0.9532 0.9851 1.0145 1.0058 1.0009 1.00001 α=0.3 0.8546 0.9499 1.0424 1.0174 1.0028 1.00004 α=0.5 0.7494 0.9075 1.0687 1.0289 1.0046 1.00007 α=0.7 0.6375 0.8578 1.0935 1.0403 1.0065 1.00010 α=0.9 0.5190 0.8009 1.1167 1.0516 1.0083 1.00013 α=1.0 0.4573 0.7700 1.1277 1.0572 1.0092 1.00015

TABLE 4. Variance Ratio ( *2

2Zμμ

= ) Values for GNBD with m=4 and β =1

θ=0.05 θ=0.1 θ=0.3 θ=0.5 θ=0.7 θ=0.9 α=0.1 0.9656 1.0039 1.0269 1.0070 1.0005 1.0000 α=0.3 0.8886 1.0032 1.0793 1.0210 1.0015 1.0000 α=0.5 0.8008 0.9913 1.1301 1.0350 1.0024 1.0000 α=0.7 0.7023 0.9680 1.1792 1.0489 1.0034 1.0000 α=0.9 0.5929 0.9334 1.2267 1.0638 1.0044 1.0000 α=1.0 0.5341 0.9119 1.2498 1.0697 1.0049 1.0000

6. AN ILLUSTRATIVE EXAMPLE (GOODNESS OF FIT)

To illustrate the practical application of results obtained in this paper, an example considered by Mishra and Singh [24] has been suitably altered. The example chosen is a


distribution of number of papers in Review of Applied Entomology, Vol.I (1913) observed by Williams [25]. Mishra and Singh [24] has shown that size-biased GPD gives a satisfactory good fit to the unaltered data. For our purpose of illustration it has been assumed that ten of the observations which correspond to the value two are reported erroneously as value one. We fitted the misclassified size-biased GPD (40) to the altered data. We considered θ known and obtained moment estimators of α (probability of misclassification) and β .Both the original and the altered data together with expected values of the altered data are given in Table 5.

TABLE 5.Number of authors (fx) according to number of papers (X) in the Review of Applied Entomology, Vol.I (1913) (Williams [25])

X Original sample frequencies

Altered data Observed frequencies

Expected frequencies

1 2 3 4 5 6 7 8 9

10

285 70 32 10 4 3 3 1 2 1

295 60 32 10 4 3 3 1 2 1

298.56 55.20 27.97 13.72 7.15 3.84 2.10 1.17 0.70 0.59

Total 411.0 411.0 411.0 d.f 3 χ2 3.741

P(χ2) 0.30 For the altered (misclassified data), we have 572.1X = and S2=1.581,

and , 124.0ˆ =α

9855.0ˆ −=β 3.0−=θ (known)

The estimate , obtained by using the altered data is to be compared with

obtained by Mishra and Singh [24] , when calculations are based on the original unaltered sample. The estimate

9855.0ˆ −=β

981407.0ˆ −=β124.0ˆ =α is to be compared with 10/70=0.142,

which is the proportion of twos that were misclassified in the process of altering the original data for this illustration.

As indicated by the low value obtained by χ2, the agreement between observed and expected frequencies in the altered sample is satisfactory. With 3 degrees of

freedom, . The value χ2(3)=3.741 for the altered sample is to be [ ] 30.0741.3P 2 =>χ


compared with χ2(3)=3.888 obtained by Mishra and Singh[24], in comparing frequencies

of the original sample with expected frequencies based on the unaltered sample estimate of and β . θ

ACKNOWLEDGMENTS

The authors thank very much the anonymous referee and editor for his/her many helpful suggestions, which substantially simplified the paper.

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j. ksiam vol.13, no.1, 1 11, 2009 a structured modeling ...€¦ · cases of peptic ulcer disease...

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