the method of fundamental solution for ...inverse heat conduction problem, fundamental solution...

THE METHOD OF FUNDAMENTAL SOLUTION FOR SOLVINGMULTIDIMENSIONAL INVERSE HEAT CONDUCTION

PROBLEMS

Y.C. HON AND T.WEI

Abstract. We propose in this paper an effective meshless and integration-free

method for the numerical solution of multidimensional inverse heat conduc-

tion problems. Due to the use of fundamental solutions as basis functions,

the method leads to a global approximation scheme in both the spatial and

time domains. To tackle the ill-conditioning problem of the resultant linear

system of equations, we apply the Tikhonov regularization method by using

the L-curve criterion for the regularization parameter to obtain a stable ap-

proximation to the solution. The effectiveness of the algorithm is illustrated

by several numerical two- and three-dimensional examples.

1. Introduction

A standard inverse heat conduction problem (IHCP) is to compute the unknown

temperature and heat flux at an unreachable boundary from scattered temperature

measurements at reachable interior or boundary of the domain. In solving direct

heat conduction problems, the errors induced from boundary or interior measure-

ments are reduced due to the diffusive characteristic of heat conduction process.

These errors, however, are extrapolated and amplified due to the extremely ill-

posedness of the inverse heat conduction problems. In other words, a small error

in the measurement can induce enormous error in computing the unknown solution

at the unreachable boundary.

Several techniques have been proposed for solving a one-dimensional IHCP [4,

6, 32, 33, 34, 41]. Among the methods proposed for higher dimensional IHCP,

boundary element [7, 30], finite difference [13, 27] and finite element [24, 40] have

Key words and phrases. Inverse heat conduction problem, fundamental solution method,

Tikhonov regularization.

The work described in this paper was partially supported by a grant from CityU (Project No.

7001461).

1

2 Y.C. HON AND T.WEI

been widely adopted for problems in two-dimension. Besides, the sequential func-

tion specification method [4, 7] and mollification method [36] have also been used

in solving the IHCP. There is, however, still a need on numerical scheme for multi-

dimensional IHCP.

The traditional mesh-dependent finite difference and finite element methods re-

quire dense meshes, and hence tedious computational time, to give a reasonable

approximation to the solution of IHCP and suffers from numerical instability prob-

lem. The use of boundary element method (BEM) reduces the computational time

and storage requirement but the problem of numerical stability still persists. Since

there is no need on domain discretization in the BEM, the location of interior points,

where the temperature data are collected, can be chosen in a quite arbitrary way

[7].

In this paper a new meshless computational method is proposed to approximate

the solution of a multidimensional IHCP under arbitrary geometry. The proposed

method uses the fundamental solution of the corresponding heat equation to gen-

erate a basis for approximating the solution of the problem. Comparing with the

mesh-dependent methods like FEM and BEM, the proposed method does not re-

quire any domain or boundary discretization. This meshless advantage makes the

method feasible to solve high dimensional IHCPs under arbitrary geometry. Fur-

thermore, unlike BEM which requires an interpolation of the unknown function on

the boundary for complicated boundary integrations, the proposed method gives a

global numerical solution on the entire time-spatial domain. The required surface

temperature and heat flux on the unreachable boundary can easily be computed

without any extra quadratures. To tackle the numerical instability in the resul-

tant ill-conditioned linear system, we use Tikhonov regularization technique and

L-curve method [14, 15, 16, 17, 31] to obtain a stable numerical solution for the

multi-dimensional IHCP problem.

In the recent rapid development of meshless computational schemes, the radial

basis function (RBF) has been successfully developed as an efficient meshless scheme

for solving various kinds of partial differential equations (PDEs). Most of the

problems are still confined to direct problems [18, 19, 20, 23, 25]. In [21], Hon and

Wu first applied the meshless RBF to solve a Cauchy problem for Laplace equation,

METHOD OF FUNDAMENTAL SOLUTION FOR IHCP 3

which is a severely ill-posed problem. In [22], Hon and Wei combined the meshless

RBF with the method of fundamental solution (MFS) to successfully solve an one-

dimensional inverse heat conduction problem. This approach can be interpreted as

an extension of the MFS for treating elliptic problems, for examples, the Laplace

equation [5, 35], the biharmonic equations [26], elastostatics problems [38], and wave

scattering problems [28, 29]. The MFS was also applied to solve nonhomogeneous

linear and nonlinear Poisson equations [1, 2, 3, 11, 37]. More details of the MFS can

be found in the review papers of Fairweather and Karageorghis [10] and Golberg

and Chen [12]. It is noted here that most of these research works focus on well-

posed problems in which the Dirichlet or Neumann data are given on the whole

boundary. In the IHCP problem, the boundary conditions are usually complicated

and incomplete. We expect that the combination of the meshless RBF and MFS will

extend its application to solve higher dimensional inverse problems under irregular

geometry.

In [9], Frankel et al. gave an unified treatment for the IHCP by using the

Chebyshev polynomials as basis functions. The use of fundamental solution as the

basis functions in this paper provides a global approximation to the solution in

both the spatial and time variables. Since IHCP problem is severely ill-posed and

its solution is extremely sensitive to any perturbation of given data, enormous error

will be accumulated from using any finite difference scheme for discretizing the time

variable [8]. The proposed method has a definite advantage over most of the existing

numerical methods in solving these kinds of time-dependent inverse problems. The

use of Laplace transform for the time variable will reduce this time discretization

error but result in solving a Cauchy problem for inhomogeneous modified Helmholtz

equation with an extra parameter in the resulting equation [8]. For a slightly

larger parameter, the Laplace transform method fails to give a stable and accurate

approximation. In fact, it is even more difficult to solve a Cauchy problem for

modified Helmholtz than solving the original IHCP. For large scale problems, an

efficient numerical method for treating the ill-conditioning discrete problem is still

needed for improvement. The recently developed iterative algorithms based on

Lanczos bidiagonalization will be a consideration for future work.


2. Methodology

Let Ω be a simply connected domain in Rd, d = 2, 3 and Γ1, Γ2, Γ3 be threeparts of the boundary ∂Ω. Suppose that Γ1 ∪Γ2 ∪Γ3 = ∂Ω, Γ1 or Γ2 can be emptyset. The IHCP to be investigated in this paper is to determine the temperature

and heat flux on boundary Γ3 from given Dirichlet data on Γ1, Neumann data on

Γ2 and scattered measurement data at some interior points.

Consider the following heat equation:

(2.1) ut(x, t) = a2∆u(x, t), x ∈ Ω, t ∈ (0, tmax),

under the initial condition

(2.2) u(x, 0) = ϕ(x), x ∈ Ω,

and the boundary conditions

(2.3) u(x, t) = f(x, t), x ∈ Γ1, t ∈ (0, tmax],

and

(2.4)∂u

∂ν(x, t) = g(x, t), x ∈ Γ2, t ∈ (0, tmax],

where ν is the outer unit normal with respect to Γ2.

Let {xi}Mi=1 ⊂ Ω be a set of locations with noisy measured data h̃(k)i of exact tem-perature u(xi, t

(k)i ) = h

(k)i , i = 1, 2, · · · ,M , k = 1, 2, · · · , Ii where t(k)i ∈ (0, tmax]

are discrete times. The absolute error between the noisy measurement and exact

data is assumed to be bounded, i.e. |h̃(k)i − h(k)i | ≤ δ for all measurement points atall measured times. Here, the constant δ is called the noisy level of input data.

The IHCP is now formulated as: Reconstruct u|Γ3 and ∂u∂ν |Γ3 from (2.1)–(2.4)and the scattered noisy measurements h̃(k)i , i = 1, 2, · · · ,M , k = 1, 2, · · · , Ii.

Remark 2.1. Most of the existing numerical methods for solving the IHCP problem

require a continuous time measurement u(xi, t) = hi(t), which is not realistic in

real life.

The fundamental solution of (2.1) is given by

(2.5) F (x, t) =1

(4πa2t)d2e−

|x|24a2t H(t),


where H(t) is the Heaviside function. Assuming that T > tmax is a constant, the

following function

(2.6) φ(x, t) = F (x, t + T ),

is a general solution of (2.1) in the solution domain Ω× [0, tmax].We denote the measurement points to be {(xj , tj)}mj=1,m =

∑Mi=1 Ii so that a

point at the same location but with different time is treated as two distinct points.

The corresponding measured noisy data and exact data are denoted by h̃j and

hj . The collocation points are then chosen as {(xj , tj)}m+nj=m+1 on the initial regionΩ × {0}, {(xj , tj)}m+n+pj=m+n+1 on surface Γ1 × (0, tmax] and {(xj , tj)}m+n+p+qj=m+n+p+1 onsurface Γ2 × (0, tmax]. Here, n, p, q denote the total number of collocation pointsfor the initial condition (2.2), Dirichlet boundary condition (2.3) and Neumann

boundary condition (2.4) respectively. The only requirement on the collocation

points are pairwisely distinct in the (d + 1)-dimensional space (x, t).

Following the idea of MFS, an approximation ũ to the solution of the IHCP

under the conditions (2.2)-(2.4) with the noisy measurements h̃j can be expressed

by the following linear combination:

(2.7) ũ(x, t) =n+m+p+q∑

j=1

λ̃jφ(x− xj , t− tj),

where φ(x, t) is given by (2.6) and λ̃j are unknown coefficients to be determined.

For this choice of basis function φ, the approximated solution ũ automatically

satisfies the original heat equation (2.1). Using the initial condition (2.2) and

collocating at the boundary conditions (2.3) and (2.4), we then obtain the following

system of linear equations for the unknown coefficients λ̃j :

(2.8) Aλ̃ = b̃,

where

(2.9) A =

φ(xi − xj , ti − tj)

∂φ∂ν (xk − xj , tk − tj)


and

(2.10) b̃ =

h̃i

ϕ(xi, ti)

f(xi, ti)

g(xk, tk)

where i = 1, 2, · · · , (m + n + p), k = (m + n + p + 1), · · · , (m + n + p + q), j =1, 2, · · · , (n + m + p + q) respectively.

The solvability of the system (2.8) depends on the non-singularity of the matrix

A, which is still an open research problem. It is not surprise that the resultant

matrix A is extremely ill-conditioned due to the ill-posed nature of the IHCP. The

following sections shows that the use of regularization technique can produce a

stable and accurate solution for the unknown parameters λ̃ of the matrix equa-

tion (2.8), and hence the solution for the IHCP is obtained.

3. Regularization Techniques

The ill-conditionedness of the coefficient matrix A indicates that the numerical

result is sensitive to the noise of right hand side b̃ and the number of collocation

points. In fact, the condition number of the matrix A increases dramatically with

respect to the total number of collocation points. The singular value decomposition

(SVD) usually works well for direct problem [39] but still fails to provide a stable

and accurate solution to the system (2.8). A number of regularization methods

have been developed for solving this kind of ill-conditioning problem [14, 15, 17].

In our computation we apply the Tikhonov regularization technique [42] based on

SVD with L-curve criterion for choosing a good regularization parameter to solve

the matrix equation (2.8).

Denote N = n+m+ p+ q. The SVD of the N ×N matrix A is a decompositionof the form:

(3.1) A = WΣV ′ =N∑

i=1

wiσiv′i

where W = (w1, w2, · · · , wN ) and V = (v1, v2, · · · , vN ) satisfying W ′W = V ′V =IN . Here, the superscript ′ represents the transpose of a matrix. It is known that


Σ = diag(σ1, σ2, · · · , σN ) has non-negative diagonal elements satisfying:

(3.2) σ1 ≥ · · ·σN ≥ 0.

The values σi are called the singular values of A and the vectors wi and vi are

called the left and right singular vectors of A respectively.

For the matrix A arising from the discretization of MFS, the singular values

decay rapidly to zero and the ratio between the largest and the smallest nonzero

singular values is often huge. Thus the linear system (2.8) is a discrete ill-posed

problem in the sense defined in Hansen’s paper [14].

Based on the singular value decomposition, it is easy to know that the solution

for the linear equations (2.8) is given by

(3.3) λ̃ =N∑

i=1

w′ib̃σi

vi.

The corresponding solution with exact data is

(3.4) λ =N∑

i=1

w′ibσi

vi

where b is calculated by (2.10) from the exact data hj . The difference between

these two solutions is then given by

(3.5) λ̃− λ =N∑

i=1

w′ieσi

vi,

where e = b̃ − b, ‖e‖ ≤ δ. The terms in the difference with small values of σi willbe large since the noisy level is generally greater than the smallest nonzero singular

value. This is the reason why the following Tikhonov regularization method is

proposed.

The Tikhonov regularized solution λ̃α for equation (2.8) is defined to be the

solution to the following least square problem:

(3.6) mineλ {‖Aλ̃− b̃‖2 + α2‖λ̃‖2},where ‖ · ‖ denotes the usual Euclidean norm and α is called the regularizationparameter. The Tikhonov regularized solution based on SVD can the be expressed

as:

(3.7) λ̃α =N∑

i=1

fiw′ib̃σi

vi,


where fi = σ2i /(σ2i +α

2) are called the filter factors, i = 1, 2, · · · , N . The differencebetween λ̃α and λ is then given by

(3.8) λ̃α − λ =N∑

i=1

(fi − 1)w′ib

σivi +

N∑

i=1

fiw′ieσi

vi.

The norm of this error vector is

(3.9) ‖λ̃α − λ‖ =(

N∑

i=1

((fi − 1)w

′ib

σi

)2+

(fi

w′ieσi

)2)1/2.

Hence, the approximated solution ũα with noisy measurement for the IHCP is given

by

(3.10) ũα(x, t) =N∑

j=1

(λ̃α)jφ(x− xj , t− tj),

where (λ̃α)j is the i-th entry of vector λ̃α.

Let uN =∑N

i=1 λjφ(x− xj , t− tj) be an approximated solution with exact datagiven in the IHCP. An error estimation is then given as follow:

(3.11) ‖ũα − uN‖L2(D) ≤ ‖λ̃α − λ‖N∑

j=1

‖φ(x− xj , t− tj)‖L2(D),

where D = Ω× (0, tmax) and the value ‖λ̃α − λ‖ is given by formula (3.9).The determination of a suitable value for the regularization parameter α is crucial

and is still under intensive research [42]. In this paper, we use the L-curve criterion

to choose a good regularization parameter. This method was firstly developed by

Lawson and Hanson [31] and more recently by Hansen and O’Leary [15]. The

L-curve method is sketched in the following.

Define a curve L by:

(3.12) L = { (log(‖λ̃α‖2), log(‖Aλ̃α − b̃‖2)), α > 0}.

The curve is known as L-curve and a suitable regularization parameter α is the one

near the ”corner” of the L-curve [14, 17]. In the numerical computation, a point

with maximum curvature is seek to be the ”corner” as given in [17] by

(3.13) K(α) = 2ηρ

η

α2ηρ + 2αηρ + α4ηη(α2η2 + ρ2)3/2


where

η = ‖λ̃α‖2 =N∑

i=1

(fiw′ib̃σi

)2, ρ = ‖Aλ̃α − b̃‖2 =N∑

i=1

((1− fi)w′ib̃)2,

and

η = − 4α

N∑

i=1

(1− fi)f2i (w′ib̃σi

)2.

In our computation, the Matlab code developed by Hansen [16] was used to

obtain the optimal choice of regularization parameter α∗ for solving the discrete ill-

conditioned system (2.8). The corresponding approximated solution for the problem

(2.1)–(2.4) with noisy measurement data is then given by

(3.14) ũα∗(x, t) =N∑

j=1

(λ̃α∗)jφ(x− xj , t− tj).

The temperature and heat flux at surface Γ3 can also be calculated respectively.

4. Numerical Examples

For numerical verification, we assume that the heat conduction coefficient a = 1

and tmax = 1 for all the following examples. In the cases when the input data

contain noises, we use the function rand given in Matlab to generate the noisy data

h̃i = hi + δrand(i) where hi is the exact data and rand(i) is a random number in

[−1, 1]. The magnitude δ indicates the noisy level of the measurement data.To test the accuracy of the approximated solution, we compute the Root Mean

Square Error (RMSE) by

(4.1) E(u) =

√√√√ 1Nt

Nt∑

i=1

((ũα∗)i − ui)2,

at sufficiently total Nt number of testing points in the domain Γ3×[0, 1] where(ũα∗)iand ui are respectively the approximated and exact temperature at a test point.

The RMSE for the heat flux E(∂u∂ν ) is also similarly defined.

Consider the following three examples:

Two-dimensional IHCPs:

Example 1: The exact solution of (2.1) is given by

(4.2) u(x1, x2, t) = 2t +12(x21 + x

22).


Example 2: The exact solution of (2.1) is given by

(4.3) u(x1, x2, t) = e−4t(cos(2x1) + cos(2x2)).

Both examples are computed under the following three different domains and

boundary conditions:

Case 1: Let

Ω = { (x1, x2) | 0 < x1 < 1, 0 < x2 < 1},Γ1 = { (x1, x2) | x1 = 1, 0 < x2 < 1},Γ2 = { (x1, x2) | 0 < x1 < 1, x2 = 1},Γ3 = ∂Ω \ {Γ1 ∪ Γ2}.

Case 2: Let

Ω = { (x1, x2) | x21 + x22 < 1},Γ1 = ∅,Γ2 = { (x1, x2) | x21 + x22 = 1, x1 > 0, }.Γ3 = ∂Ω \ {Γ1 ∪ Γ2}.

Case 3: Let Ω be the same as Case 1, Γ1 = Γ2 = ∅, Γ3 = ∂Ω \ {Γ1 ∪ Γ2}.Locations of the internal measurements and collocation points over Ω for the three

cases are shown in Figure 1.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x 2 Ω

Γ2

Γ1

Γ3

Γ3

Case 1.

Ω Γ3

Γ2

Case 2.

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Case 3.

Ω

Γ3

Γ3 Γ3

Γ3

Figure 1. Distribution of measurement points and collocation

points. Here, star represents collocation point matching Dirich-

let data, square represents collocation point matching Neumann

data, dot represents collocation point matching initial data, and

circle represents point with sensor for internal measurement.

The boundary data f(x, t), g(x, t) and initial temperature ϕ(x, t) are obtained

from the given exact solutions. The values of temperature at measurement points


will be used respectively by the exact and noisy data. Numerical results are ob-

tained by taking the constant T = 1.8 for Example 1 and T = 1.6 for Example

2 in all different cases. The total numbers of various collocation points and mea-

surement points are n = 36, m = 100, p = 55 , q = 50 in Case 1 for Examples

1-2. In this situation the total number of points and testing points are N = 241

and Nt = 882. For Case 2, the total numbers are n = 96, m = 85, p = 0, q = 85,

N = 266, Nt = 693 and for Case 3, n = 36, m = 30, p = 0, q = 0, N = 66,

Nt = 1764 respectively.

Numerical results by using only SVD for the solution (3.4) and uN are presented

in Table 1. The computed RMSEs from the experiments show that even for exact

input data the direct method cannot produce an acceptable solution to this kind

of ill-conditioned linear systems. In fact, the condition numbers in Case 1 and

Case 2 lie between 1033-1035 which are too large to compute an accurate solution

without the use of any regularization. For Case 3, the condition number 4.5 · 1019

in Example 1 and 7.9 ·1018 in Example 2 are much smaller and hence the RMSEs inCase 3 looks better. The numerical results given in Table 1 indicate that the IHCP

is an severely ill-posed problem. The discretization by using MFS also leads to a

highly ill-conditioned discrete problem. The use of the regularization technique as

shown in the following numerical results will give a stable and much more accurate

approximation to the solution of the IHCP.

Table 2 gives the RMSEs on the temperature and heat flux in domain Γ3× [0, 1]with exact data by using Tikhonov regularization method with L-curve choice α∗

for the regularization parameter. It can be observed from Table 2 that the RMSEs

have much been reduced compared to Table 1. It is remarked here that the L-curve

method works well in searching the crucial regularization parameter but can further

be improved if a scaling factor is used.

Numerical results for the three examples with noisy data (noisy level δ = 0.01 in

all cases) are shown in Table 3. It can be observed that the regularization method

with L-curve technique provides an acceptable approximation to the solution of

the IHCP whilst the direct method completely fails. The numerical result for the

Example 1 with Case 2 looks bad but can be much improved if a scaling factor

0.0001 is multiplied to the L-curve regularized parameter α∗. In this case the


RMSEs for E(u) and E(∂u∂ν ) are 0.0248 and 0.0600 respectively. The problem to

choose an optimal regularization parameter is still an open question to researchers.

It is noted here that the setting for Case 3 comes from a real-life problem pro-

duced by a steel company. In fact, the solution for the IHCP may not unique but

our computational results demonstrate that the proposed method is flexible enough

to give a reasonable approximation to the solution of the IHCP under insufficient

information.

Table 1. RMSEs in domain Γ3 × [0, 1] with exact data. No reg-ularization technique

Example 1 Example 2

Case 1E(u)=5.9122 E(u)= 0.4909

E(∂u∂ν )=2.2155 E(∂u∂ν )=0.2325

Case 2E(u)= 2.2742 E(u)=7.9616

E(∂u∂ν )=0.5040 E(∂u∂ν )= 1.7736

Case 3E(u)=0.0102 E(u)= 0.0011

E(∂u∂ν )= 0.0543 E(∂u∂ν )=0.0063

Table 2. RMSEs in domain Γ3 × [0, 1] with exact data by usingTikhonov regularization technique with L-curve parameter α∗

Example 1 Example 2

Case 1α∗= 2.7404e-12 E(u)= 2.7087e-4 α∗=3.4429e-10 E(u)=9.1315e-5

E(∂u∂ν )=6.6912e-4 E(∂u∂ν )=3.5104e-4

Case 2α∗=4.1551e-13 E(u)=0.0010 α∗= 8.6630e-6 E(u)= 0.0110

E(∂u∂ν )= 0.0029 E(∂u∂ν )=0.0313

Case 3α∗= 1.0370e-8 E(u)= 8.9921e-4 α∗=2.0310e-12 E(u)=4.8012e-4

E(∂u∂ν )=0.0030 E(∂u∂ν )=0.0025

The relationship between the RMSEs and the value of constant T for Examples

1-2 under Case 2 with noisy data (δ = 0.01) is displayed in Figure 2. Here, the

regularization parameter α∗ is obtained again by using the L-curve method. The


Table 3. RMSEs in domain Γ3× [0, 1] with noisy data (δ = 0.01)by using Tikhonov regularization with L-curve parameter α∗

Example 1 Example 2

Case 1α∗=2.770e-3 E(u)= 0.0729 α∗=3.6608e-4 E(u)= 0.0146

E(∂u∂ν )=0.1348 E(∂u∂ν )=0.0405

Case 2α∗= 0.0028 E(u)=0.2778 α∗=1.2103e-5 E(u)=0.0104

E(∂u∂ν )=0.4615 E(∂u∂ν )= 0.0318

Case 3α∗=2.7527e-5 E(u)= 0.0221 α∗= 4.9821e-6 E(u)=0.0120

E(∂u∂ν )=0.0494 E(∂u∂ν )= 0.0578

numerical results indicate that the choice of the parameter T plays a minor but

strange role for obtaining an accurate approximation. For Example 1, the RMSEs

decrease with respect to the increasing value of T but decrease in Example 2. This

interesting behavior is one of the focuses of our future research works.

1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

parameter T

RM

S(u

) an

d R

MS

(du)

Example 1 for case 2. σ=0.01

1 2 3 4 5 6 7 8 9 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

parameter T

RM

S(u

) an

d R

MS

(du)

Example 2 for case 2. σ=0.01

Figure 2. RMSEs of temperature and heat flux on Γ3×[0, 1] withrespect to parameter T

To further extend the application of the proposed method, we investigate the fol-

lowing sample problem given in [7]:

Example 3: Let

Ω be defined as in Case 1.

Γ1 = ∅,Γ2 = { (x1, x2) | x1 = 1, 0 < x2 < 1} ∪ { (x1, x2) | x2 = 1, 0 < x1 < 1},Γ3 = ∂Ω \ {Γ1 ∪ Γ2}.


The locations of measurement points are shown in Figure 3. The temperature

distribution is given by

u(x1, x2, t) =

U(x1, x2, t), for t ≤ 0.5,

U(x1, x2, t)− 2U(x1, x2, t− 0.5), for 0.5 < t ≤ 1.0,

U(x1, x2, t)− 2U(x1, x2, t− 0.5) + U(x1, x2, t− 1.0), for t > 1.0,

where

U(x1, x2, t) = 1.5t2 + t(0.5x21 − x1 + x22 − 2x2 + 1)−4 ∑∞j=1 1(jπ)4 (0.5cos(jπx1) + cos(jπx2))(1− e−j

2π2t).

The exact temperature data at the sensor locations are given by u(x1, x2, t) and

the noisy data are randomly generated as before. In this computation, we take

n = 121,m = 189, p = 0, q = 189, N = 499, Nt = 882. The plots of temperature

and heat flux with respect to time at points (0, 0), (1, 0), (0, 1) are displayed in Fig-

ure 4 in which T = 1.2 and δ = 0.01. As shown in Figure 4, although the basis

functions generated by MFS are sufficient smooth over the solution domain and

the solution to Example 3 does not have a continuous second order time derivative,

the computed temperature match the exact data excellently. The computed heat

flux looks less accurate but is also comparable to the results given in paper [7]. It

is also noted that the measurement points in this example are far away from the

unspecified boundary whereas these points are close to the boundary in [7].

Finally, we consider the following three-dimensional IHCP:

Example 4: Let

Ω = { (x1, x2, x3) | 0 < xi < 1, i = 1, 2, 3},Γ1 = { (x1, x2, x3) | 0 < x1 < 1, 0 < x2 < 1, x3 = 1},Γ2 = { (x1, x2, x3) | 0 < x1 < 1, 0 < x2 < 1, x3 = 0},Γ3 = ∂Ω \ {Γ1 ∪ Γ2}.The locations of measurement points and collocation points in the domain Ω are

shown in Figure 5. In this computation, we take n = 245,m = 250, p = 180, q =

180, N = 855, Nt = 5324.

The exact solution for Example 4 is given by

(4.4) u(x1, x2, x3, t) = e(−4t)(cos(2x1) + cos(2x2) + cos(2x3)).


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Example 3.

x1

x 2

Ω

Γ2

Γ2 Γ

3

Γ3

Figure 3. Distribution of measurement and collocation points.

Square represents point with Neumann data, dot represents col-

location point for initial temperature, and circle represents point

with sensor.

In the computation, the value of the parameter T is 2.3 and the noisy level is

set to be δ = 0.01. The errors between the exact solution and the approximated

solution for the temperature and heat flux on boundary Γ3 at time t = 1 are

shown in Figure 6 and Figure 7 respectively. Note that the L2 norm of u and∂u∂ν over Γ3 × [0, 1] are about 0.62 and 0.52 respectively. Their relative errors areapproximately double the values given in Figures 6 and Figure 7. These small

relative errors show that the proposed scheme is effective for solving the three-

dimensional inverse heat conduction problem of which very little numerical result

has been given so far.

5. Conclusions

The universal approach for solving time-dependent problems involves a time-

marching procedure, i.e., advancing each time step and solving the remained prob-

lem in the spatial domain. The approach proposed in this paper gives a global

approximated solution in both time and spatial domain. Numerical results indicate

that the method of fundamental solution MFS combined with the regularization

technique provides an efficient and accurate approximation to this highly ill-posed

inverse heat conduction. The use of L-curve criterion for a suitable regularization


0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

u(0,0,

t)

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

t

dudy

(0,0,t

)

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

t

u(1,0,

t)

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

t

dudy

(1,0,t

)

0 0.2 0.4 0.6 0.8 1−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

t

u(0,1,

t)

0 0.2 0.4 0.6 0.8 1−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

t

dudx

(0,1,t

)

Figure 4. Plots of temperature and heat flux versus time at

points (0, 0), (1, 0), (0, 1). Solid line represents the exact value and

dotted-line represents the computed value

parameter stabilizes the resultant ill-conditioned system but still needed to be fur-

ther investigated for optimal convergence. The proposed approach is potentially

valuable to solve the inverse problems in multidimensional space but for large-scale

problems, other iterative algorithms based on Lanczos bidiagonalization may be


0

0.5

1

0

0.5

10

0.2

0.4

0.6

0.8

1

x1

Example 4.

x2

x 3

Γ1

Γ2

Ω

Figure 5. Locations of measurement and collocation points. Star

represents measurement point with Dirichlet data, square repre-

sents measurement point with Neumann data, and circle represents

point with sensor.

more suitable. This will be our future work.

Acknowledgment

We wish to thank the referees for their constructive comments which helped us

much in improving the presentation of this paper.


00.5

1

0

0.5

1−5

0

5

x 10−3

x1

x3

Err

or o

f u(x

1,0,

x 3,1

)

00.5

1

0

0.5

1−2

0

2

4

x 10−3

x2

x3

Err

or o

f u(0

,x2,

x 3,1

)

00.5

1

0

0.5

1−5

0

5

x 10−3

x1

x3

Err

or o

f u(x

1,1,

x 3,1

)

00.5

1

0

0.5

1−4

−2

0

x 10−3

x2

x3

Err

or o

f u(1

,x2,

x 3,1

)

Figure 6. Surface plots of errors to temperature on boundary Γ3

References

[1] Balakrishnan, K. and Ramachandran, P.A., A particular solution Trefftz method for non-

linear Poisson problems in heat and mass transfer, Journal of computational physics,

150(1999), 239–267.

[2] Balakrishnan, K. and Ramachandran, P.A., The method of fundamental solutions for linear

diffusion-reaction equations, Mathematical and Computer Modelling, 31(2000), 221–237.

[3] Balakrishnan, K. and Ramachandran, P.A., Osculatory Interpolation in the Method of

Fundamental Solution for Nonliear Possion Problems, Journal of Computational Physics,

172(2001), 1–18.

[4] Beck, J.V., Blackwell, B. and Clair, Ch.R.St., Inverse heat conduction, Ill-posed Problems,

Wiley-Interscience Publication, New York, 1985.

[5] Bogomonlny, A., Fundamental solutions method for elliptic boundary value problems, SIAM

Journal on Numerical Analysis, 22 (1985), 644–669.

[6] Chantasiriwan, S., Comparison of three sequential function specification algorithms for the

inverse heat conduction problem, International Communications in Heat and Mass Transfer,

26(1999), no.1 , 115-124.


00.5

1

0

0.5

1−0.01

0

0.01

x1

x3

Err

or o

f dud

y(x 1

,0,x

3,1)

00.5

1

0

0.5

1−5

0

5

10

x 10−3

x2

x3

Err

or o

f dud

x(0,

x 2,x

3,1)

00.5

1

0

0.5

1−0.01

0

0.01

x1

x3

Err

or o

f dud

y(x 1

,1,x

3,1)

00.5

1

0

0.5

1−0.02

−0.01

0

x2

x3

Err

or o

f dud

x(1,

x 2,x

3,1)

Figure 7. Surface plots of errors to heat flux on boundary Γ3

[7] Chantasiriwan, S., An algorithm for solving multidimensional inverse heat conduction prob-

lem, International Journal of Heat and Mass Transfer, 44 (2001), 3823–3832.

[8] Cho, H.A., Golberg, M.A., Muleshkov, A.S. and Li. X, Trefftz methods for time dependent

partial differential equations (preprint).

[9] Frankel, J.I. and Keyhani, M., A global time treatment for inverse heat conduction problems,

Journal of Heat Transfer, 119 (1997), 673–683.

[10] Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic

boundary value problems, Advances in Computational Mathematics, 9(1998), 69–95.

[11] Golberg, M.A., The method of fundamental solutions for Poisson’s equation, Engineering

Analysis with Boundary Elements, 16(1995), 205–213.

[12] Golberg, M.A. and Chen, C.S., The method of fundamental solutions for potential, Helmholtz

and diffusion problems. In: M. A. Golberg editor. Boundary integral methods-numerical and

mathematical aspects, Sounthampton: Computational Mechanic Publications, 1998, 103–176.

[13] Guo, L. and Murio, D.A, A mollified space-marching finite-difference algorithm for the two-

dimensional inverse heat conduction problem with slab symmetry, Inverse problem, 7 (1991),

247–259.


[14] Hansen, P.C., Analysis of discrete ill-posed problems by means of the L-curve, SIAM Review,

34 (1992), 561–580.

[15] Hansen, P.C. and O’Leary, D.P., The use of the L-curve in the regularization of discrete

ill-posed problems. SIAM Journal of Scientific Computing, 14 (1993), 1487–1503.

[16] Hansen, P.C., Regularization tools: a Matlab package for analysis and solution of discrete

ill-posed problems, Numerical Algorithms, 6 (1994), 1–35.

[17] Hansen, P.C., The L-curve and its use in the numerical treatment of inverse problems, Com-

putational Inverse Problems in Electrocardiology, 4, edited by Johnston, P., Advances in

Computational Bioengineering Series, WIT Press, Southampton, 2000.

[18] Hon, Y.C. and Mao, X.Z., A radial basis function method for solving options pricing models,

The Journal of Financial Engineering, 8(1999), 31–49.

[19] Hon, Y.C., Cheung, K.F., Mao, X.Z. and Kansa, E.D., Multiquadric solution for shallow

water equations, ASCE Journal of Hydraulic Engineering, 125(1999) , 524–533.

[20] Hon, Y.C., Lu, M.W., Xue, W.M. and Zhou, X , A new formulation and computation of the

triphasic model for mechano-electrochemical mixtures, Computational Mechanics, 24 (1999),

155–165.

[21] Hon, Y. C. and Wu, Z.M., A numerical computation for inverse boundary determination

problem, Engineering Analysis with Boundary Elements, 24(2000), 599–606.

[22] Hon, Y.C. and Wei, T., A Meshless Computational Method for Solving Inverse Heat Conduc-

tion Problem, International Series on Advances in Boundary Elements, 13(2002), 135–144.

[23] Hon, Y.C. and Chen, W., Boundary knot method for 2D and 3D Helmholtz and convection-

diffusion problems with complicated geometry, International Journal for Numerical Methods

in Engineering, 56(2003), 1931–1948.

[24] Hsu, T.R., Sun, N.S., Chen, G.G. and Gong, Z.L., Finite element formulation for two-

dimensional inverse heat conduction analysis, Transactions of the ASME, Journal of Heat

Transfer, 114 (1992), 553–557.

[25] Kansa, E.J. and Hon, Y.C., Circumventing the ill-conditioning problem with multiquadric

radial basis functions: applications to elliptic partial differential equations, Computers and

Mathematics with Applications, 39(2000), 123–137.

[26] Karageorghis A. and Fairweather G., The method of fundamental solutions for the numerical

solution of the biharmonic equation, J. Comput. Phys., 69(1987) 433–459.

[27] Khalidy, N.A., A general space marching algorithm for the solution of two-dimensional bound-

ary inverse heat conduction problems, Numerical Heat Transfer, Part B, 34 (1998), 339–360.

[28] Kondapalli, P.S., Shippy, D.J. and Fairweather, G., Analysis of acoustic scattering in fluids

and solids by the method of fundamental solutions, J. Acoust. Soc. Amer., 91(1992), 1844–

1854.

[29] Kondapalli, P.S., Shippy, D.J. and Fairweather, G., The method of fundamental solutions for

transmission and scattering of elastic waves, Comput. Methods Appl. Mech. Engrg., 96(1992),

255–269.


[30] Kurpisz, K. and Nowak, A.J., BEM approach to inverse heat conduction problems, Engineer-

ing Analysis with Boundary Elements, 10 (1992), 291–297.

[31] Lawson, C.L. and Hanson, R.J., Solving Least Squares Problems, SIAM, Philadelphia, PA,

1995. First published by Prentice-Hall, 1974.

[32] Lesnic, D., Elliott, L. and Ingham, D.B., Application of the boundary element method to

inverse heat conduction problems, International Communications in Heat and Mass Transfer

39 (1996), No. 7, 1503–1517.

[33] Lesnic, D. and Elliott, L, The decomposition approach to inverse heat conduction, Journal

of Mathematical Analysis and Application, 232(1999), 82–98.

[34] Jonas, P and Louis, A.k., Approximate inverse for a one-dimensional inverse heat conduction

problem, Inverse Problems, 16 (2000), 175–185.

[35] Mathon, R. and Johnston, R.L., The approximatical solution of elliptic boundary-value prob-

lems by fundamental solutions, SIAM J. Numer. Anal., 14(1977), 638–650.

[36] Murio, D.A., The mollification method and the numerical solution of ill-posed problems, A

Wiley-Interscience Publication, 1993.

[37] Partridge, P.W. and Sensale, B., The method of fundamental solutions with dual reciprocity

for diffusion and diffusion-convection using subdomains, Engineering Analysis with Boundary

Elements, 24(2002), 633–641.

[38] Poullikkas, A., Karageorghis, A. and Georgiou, G., The method of fundamental solutions for

three-dimensional elastostatics problems, Computers and Structures, 80(2002) 365–370.

[39] Ramachandran, P.A., Method of fundamental solutions: singular value decomposition anal-

ysis, Communications in numerical methods in engineering, 18(2002), 789–801.

[40] Reinhardt, H.J., A numerical method for the solution of two-dimensional inverse heat con-

duction problems, International Journal for Numerical Methods in Enginerring, 32 (1991),

363–383.

[41] Shen, S.Y., A numerical study of inverse heat conduction problems, Computers and Mathe-

matics with Applications, 38(1999), 173–188.

[42] Tikhonov, A. N. and Arsenin, V.Y., On the solution of ill-posed problems , John Wiley and

Sons, New York, 1977.

Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon,

HongKong

E-mail address: [email protected]

Department of Mathematics, City University of Hong Kong & Department of Math-

ematics, Lanzhou University, Lanzhou, Gansu Province 730000, China

E-mail address: [email protected]

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