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March 2000 1
ALADIN project
Algorithmes Adaptes au Calcul Intensif
Advanced Algorithms for Scientific Computing
Jocelyne Erhel
Project leader since 01/09/1997
J. Erhel, ALADIN
March 2000 2
Composition on 31/1/2000
Permanent researchers
Philippe Chartier (CR-INRIA)
Jocelyne Erhel (DR-INRIA)
Bernard Philippe (DR-INRIA)
Claude Simon (MdC, IUT de Lannion)
External collaborators , ingenieur expert , invited
Michel Crouzeix (Professeur, U. de Rennes 1)
Haiscam Abdallah (MdC, U. de Rennes 2)
Olivier Bertrand (completed)
Yann-Herve De Roeck (Ifremer)
M. Sadkane (ex-INRIA) professor at U. Brest since 1997
J-C. Paoletti (ex-SIMULOG) engineer at Transpac, Rennes, since 1998
Ph-D students
IRISA co-encadres co-tutelle
Frederic Guyomarc’h Ahmid Zaoui Dany Mezher
Eric Lapotre Hussein Hoteit Claude Tadonki
J. Erhel, ALADIN
March 2000 3
Completed Ph-D
Ph-D name current status location
March 1996 Robert Erra lecturer ESIEA, Paris
January 1997 Vincent Heuveline researcher U. Heidelberg
October 1997 Pierre-Francois Lavallee engineer CNRS, IDRIS, Paris
November 1997 Sophie Robert engineer SACET, Rennes
December 1997 Anne Aubry PRAG U. Marseille
February 1998 Stephane Chauveau engineer Philips, Eindhoven
March 1998 Nicolas Mallejac engineer CEA, Paris
September 1998 Stephanie Rault computer scientist department of finance
November 1998 Philippe Feat teacher secondary school
January 1999 Mathias Brieu maıtre de conferences U. Lille
January 1999 Olivier Beaumont maıtre de conferences ENS-Lyon
December 1999 Olivier Bertrand post-doc IFP
12 thesis - university:4 - national center:3 - industry:2 - other:3
J. Erhel, ALADIN
March 2000 4
Research context
applied mathematics electromagnetism
scientific computing geophysics
computer science mechanics
multi-disciplinary problems
collaboration between experts of different fields
various levels of involvement in the applications
J. Erhel, ALADIN
March 2000 5
Modelling of flow in a porous mediaH. Hoteit, B. Philippe, J. Erhel - 1999
with IMF in Strasbourg
Modelling Discretisation with Mixed Finite Elements
Partial Differential Equations Algebraic Differential Equation of index 1 (EDA-1)8>>>>><>>>>>:
s ∂p∂t
+ ∇q = f
q = −k∇p
Boundary Conditions (g)
Initial Condition
p : hydraulic load q : Darcy speed
8>><>>:
S dPdt
+ DP − W ∗ T = F
WP − M T = G
Initial Condition
S and D are diagonal
J. Erhel, ALADIN
March 2000 6
Numerical schemes involved
first step : numerical integration of DAE
implicit schemes ensure stability
objective : use a library such as DASSL with variable time step and variable order
second step :large scale linear problem (or nonlinear)
sparse solvers reduce the complexity and the storage
objective : combine a library such as SPARSKIT with DASSL
compare direct and iterative solvers
third step :more complex models
objective : simulate solute transport
J. Erhel, ALADIN
March 2000 7
Statistical error estimation with the toolbox Aquarelsthe result must have 9 significant digits
10−20
10−17
10−14
10−11
10−8
10−5
Amplitude
10−5
10−2
101
104
Est
imat
ion
d’er
reur
Code généré par le calcul formelTriangle [(2.01,2.01),(3+1e−6,.01),(3+1e−5,.01)]
1 2 3 4 5 67 8 9 10
1112
13
14
15
16
Regularite = 1.20
Correlation = 0.95
Conditionnement = 9.50 e+6
only 3 significant digits
The code generated by symbolic software
amplifies rounding errors
when computing the matrix
C =
0BB@
a b c
b d e
c e f
1CCA
−1
using the Cramer formulas
J. Erhel, ALADIN
March 2000 8
A numerically stable computation
10−20
10−17
10−14
10−11
10−8
10−5
Amplitude
10−15
10−12
10−9
10−6
10−3
100
103
Est
imat
ion
d’er
reur
Code avec factorisationTriangle [(2.01,2.01),(3+1e−6,.01),(3+1e−5,.01)]
12
34
56
78
910
1112
1314
15
16
Regularite = 1.05
Correlation = 0.99
Conditionnement = 6.15 e+6
9 significant digits
The Cholesky factorisation
is numerically stable
and is more CPU-efficient
J. Erhel, ALADIN
March 2000 9
Pre-stack depth migration of reflection seismicY-H. De Roeck - 1999
Source
Receivers
Wave front
RaysInterfacesGeological
0
50
100
150
200
250
Mili
-sec
onds
50 100 150 200 250 300 350 400Shots
smavh/Donnees/s8-inter-bmut.ita
Algorithmic goal: efficient sparse linear least squares solver
J. Erhel, ALADIN
March 2000 10
Waveform inversion
Set of parameters for the direct model:
– f : signal wavelet
– ν: slowness, linearized into
– r: reflectivity (fast variations)
– ν0: background propagator
– p: positioning parameters
f
0
p
ν
r
ν
J(f,ν,p) =12‖csynthetics(f,ν,p) − ddata‖2 (cost function)
=12
∑s∈shots
∑h∈receivers
∫ T
0
(cs,h(t) − ds,h(t)
)2dt
= ‖f ?t B(ν0,p) · r − d‖2
For given f , ν0, p : a linear least squares problem.
J. Erhel, ALADIN
March 2000 11
Use of Truncated Pivoted QR (Stewart)
B(: ,E) = Q · (i,i)
@@
@
@@
@
@@
@
@@
@
@@
@
@@@
@@
@@@
0 3000 6000 900010−25
10−20
10−15
10−10
10−5
100
Diagonal elements of the R−factor of a pivoted QR on B’B
pivot number
abso
lute
am
plitu
de |R
ii|
diag(|R(B’B)|)diag(|R(B’C’CB)|)
1.e−11 threshold: r(B’B) = 7792
r(B’C’CB) = 7645
1.e−20 threshold: r(B’B) = 7895
r(B’C’CB) = 7892
approximates TSVD for σi > σ1 ∗ |Rii||R11| .
– Quasi Gram Schmidt algorithm;
– Q-less form;
– convolution on the fly
⇒ handles B only;
– R factor upper triangular n × n
⇒ not very sparse;
– E, handy permutation vector.
Still not a true rank-revealing QR.
J. Erhel, ALADIN
March 2000 12
Initial reflectivity projected on the orthonal of Ker(B)
Horizontal extent of the grid (m)
Ver
tical
ext
ent o
f the
grid
(m
)
10 20 30 40 50 60 70 80 90
10
20
30
40
50
60
70
80
90
100
Raw noisy data
TPQR on Bf: # zeros & quasi−zeros = 387
20 40 60 80
20
40
60
80
100
Regularisation
– Synthetic reflectivity to be retrieved
– TPQR at different thresholds
|R11||Rii|
≥
8<:
1011
102
Raw noisy data
TPQR on Bf: # zeros & quasi−zeros = 146 3
20 40 60 80
20
40
60
80
100
J. Erhel, ALADIN
March 2000 13
Conjugate Gradient for Least Squares
Regularisation by limited convergence
0 20 40 60 80
0
20
40
60
80
100
CGLS with Bf after 1000 iterations CGLS with B
f after 250 iterations
0 20 40 60 80
0
20
40
60
80
100
left : Resulting reflectivity after too many iterations.
right: Stopped according to an error estimate.
J. Erhel, ALADIN
March 2000 14
How to stop with CGLS
0 200 400 60010
−5
10−4
10−3
10−2
10−1
100
τ = 10%
τ = 5%
τ = 1%
log 10
(|r/
r 0|)
relative residual vs iteration0 200 400 600
0.2
0.4
0.6
0.8
1
τ = 10%
τ = 5%
τ = 1%
|e/e
0|
relative true error vs iteration
– residual
– true error
– Hanke’s error estimate
0 200 400 6000
0.2
0.4
0.6
0.8
1
error estimate vs iteration
τ = 10%
τ = 5%
τ = 1%
ξ / ξ
max
J. Erhel, ALADIN
March 2000 15
Objectives of the project
design, analysis, implementation of
numerical algorithms and methods
speed accuracy
convergence order of approximation
parallelism stability
complexity invariants
applications of scientific computing
J. Erhel, ALADIN
March 2000 16
Domains of research
Differential equations - P. Chartier
Linear algebra
– eigenvalue problems - B. Philippe
– linear systems and least-square problems - J. Erhel
J. Erhel, ALADIN
March 2000 17
Ordinary differential equations y′ = f(y)Differential algebraic equations y′ = f(y,z) and g(y) = 0
Aims and results
general linear methods and Runge-Kutta methods
conservation of invariants
with U. Auckland, U. Geneve, U. Arizona, U. Trieste
optimisation problems (P. Chartier with MOCOA)
molecular dynamics (with NUMATH)
free software RADAU5M and RKPS63
Pseudo-symplectic method
[BIT-3 1998,BIT-4 1998]
-4
-3
-2
-1
0
1
2
3
4
-6 -4 -2 0 2 4 6
q
p
PS63EXACT
BUTCHER6
Research in other groups Aladin is the only group in France on this topic
multistep methods - stochastic DE - delay DE - Lie methods - software
New Talent Discovery prize - A. Aubry - 1997
J. Erhel, ALADIN
March 2000 18
Algebraic differential equations of index 2 (EDA2)
A. Aubry, P. Chartier - [App.Num.Math. 1996,SINUM 1998]
P. Chartier with U. Auckland [BIT 1996] , with U. San Sebastian
y′ = f(y,z) ∈ R
m ,
0 = g(y) ∈ R
n(1)
The exact solution (y(x),z(x)) lies on the manifold
V = (y,z) ∈ R
m × R
n,0 = g(y),0 = gy(y)f(y,z) (2)
Examples of application
mechanical systems, molecular biology, electrical networks, astronomy, etc
J. Erhel, ALADIN
March 2000 19
Vitesse : (u,v)
mg
λ
(u,v)
Longueur de la corde : l
Constante gravitationnelle : g
(p,q)
p
q
l
Masse : m
Coordonnees du point : (p,q)
Tension de la corde : λ
mv′ = −2qλ − mg
mu′ = −2pλ
q′ = v
p′ = u
H = m2 (u2 + v2) + mgq est constant le long de toute solution
0 = p2 + q2 − l2
0 = pu + qv
− − − − − − −
Example : pendulus behaviour
J. Erhel, ALADIN
March 2000 20
y0y1 y2
y3
y4
yn−2yn−1
yn
Φh Φh Φh
Φh
Φh
Φh
ΦhΦh
Φh
P
PP
PP
PP
P
P P
Solution propagee apres projection
Solution propagee avant projection
y0
yn−1 yn
y1 y2y3
y4
yn−2yn−1
yn
PΦh
Φh
y1 y2
y3
y4
y1 y2
y3
y4
yn−2
ynΦh
yn−2
yn−1
P
g(y) = 0
g(y) = 0
Runge-Kutta methods with projection (Radau I or Gauss for example)
propagation after projection : existing methods
propagation before projection : new method
J. Erhel, ALADIN
March 2000 21
0 100 200 300 400 500 600−0.93
−0.92
−0.91
−0.9
−0.89
−0.88
−0.87
Temps
Ham
ilton
ien
Hamiltonien du pendule
Solution propagée avant projection
Solution propagée après projection
Gauss s=2
The new method preserves the Hamiltonian of the system.
J. Erhel, ALADIN
March 2000 22
Method global error on y (EDO) global error on y (EDA2)
Gauss h2s hs or hs+1
propagation after projection ∗ h2s
propagation before projection ∗ h2s−2 or h2s
Convergence of Gauss methods
J. Erhel, ALADIN
March 2000 23
10−3
10−2
10−1
100
10−10
10−8
10−6
10−4
10−2
100
102
Solution propagée avant projection
Solution propagée après projection
Test Asher−Petzold − Méthode de Gauss s=2 − De 4 à 512 pas
Err
eur
glob
ale
en é
chel
le lo
garit
hmiq
ue
Pas intégration en échelle logarithmique
Both methods have the same order (same slope),
with a larger constant for the new one.
J. Erhel, ALADIN
March 2000 24
Linear systems : Ax = b or min ‖Ax − b‖, with A sparse
Aims and results
projection iterative methods :
CG and GMRES
to speed-up convergence
with Minneapolis and Queensland
Examples of application
electromagnetism (with M3N)
behaviour of composite structures (with
LM2S)
image analysis (with VISTA)
DEFLATED GMRES
[JCAM 1996, NLAA 1998]
0 50 100 150 200 250 300 350 40010
−14
10−12
10−10
10−8
10−6
10−4
10−2
100
matvec
resi
dual
n=100; beta=0.9; circulant(−1.5,2,30), D(31:100)
FULL−GMRES
GMRES(50)
MORGAN(25,15)
DEFLATION(25,15,2)DEFLATION(25,15,4)
Research in other groups
Aladin is one of the few groups in France studying linear solvers
dense matrices (REMAP) - direct methods (CERFACS, U. Bordeaux)
iterative methods (U. Lille, U. Littoral, U. Paris 6) - software
J. Erhel, ALADIN
March 2000 25
Solving a sequence of linear systemsF. Guyomarc’h - J. Erhel
A symmetric positive definite (SPD)
Solve
Ax(1) = b(1)
Ax(2) = b(2)
...
Preconditioned Conjugate Gradient for the first system (PCG)
Acceleration of convergence in the second system
J. Erhel, ALADIN
March 2000 26
A and M SPD - PCG=CG applied to B = M1/2AM1/2
Initialisation
r0 = b − Ax0
z0 = Mr0
p0 = z0
For k = 0,1 . . .
αk = (rk,zk)(Apk,pk)
xk+1 = xk + αkpk
rk+1 = rk − αkApk
zk+1 = Mrk+1
βk+1 =(rk+1,zk+1)
(rk,rk)
pk+1 = zk+1 + βk+1pk
Endfor
minimisation
‖rk‖B−1 = minx∈x0+Span(p0,... ,pk−1) ‖b − Ax‖B−1
eigenvalues of B and MA
0 < λ1 ≤ . . . ≤ λn
condition number κ = λn/λ1 ≥ 1
asymptotic convergence
‖rk‖B−1 ≤ 2‖r0‖B−1
√κ−1√κ+1
k
choose M such that κ ' 1
J. Erhel, ALADIN
March 2000 27
Acceleration of convergence
Projections
W = (w1, . . . ,wm) such that D = W T AW non singular
H = I − WD−1(AW )T
HT = I − AWD−1W T
Preconditioning
PCG with M = HHT and r0 = HT r−1 ⊥ W
M positive semi-definite and Ker(M) = Ker(HT ) = Span(W ) but
rk ⊥ W
Theorem : results for PCG are valid here
J. Erhel, ALADIN
March 2000 28
AUGCG [SIMAX 2000]
W = (p0,p1, . . . ,pm−1) thus D = W T AW is diagonal
Initialisation
x0 = x−1 + WD−1W T r−1
r0 = b − Ax0 = HT r−1
z0 =(I − WD−1(AW )T
)r0 = Hr0
p0 = z0
Projection
zk = HHT rk = Hrk = rk − (rk,Awm)(wm,Awm)wm
(zk,rk) = (rk,rk)
I/O on W can be used in the initialisation
J. Erhel, ALADIN
March 2000 29
DEFCGwith U. Minneapolis [SISC 2000]
Exact Deflation
W = (v1,v2, . . . ,vm) with Avj = λjvj thus H = I − WW T
κ =λn
λm+1
Invariant subspace approximation
Harmonic Ritz vectors
(AZ)T AZyi − θi(AZ)T Zyi = 0, yi ∈ R
m+k
with Z = [W (j),P (j)]
W (j+1) = Z(y1, . . . ,ym)
Accuracy of Ritz vectors increases at each new system
J. Erhel, ALADIN
March 2000 30
AUGCG and DEFCG : example - Matrix S2RMQ4M1
0 10 20 30 40 50 60 70 80 90−12
−10
−8
−6
−4
−2
0
2
Dotted: system one(CG); Dashed: systems two to four(DefCG); Solid: system two(AugCG)
log(
rela
tive
erro
r)
Preconditioning IC(1)
0 5 10 15 20 25 30 35 40−12
−10
−8
−6
−4
−2
0
2
Dotted: system one(CG); Dashed: systems two to four(DefCG); Solid: system two(AugCG)
log(
rela
tive
erro
r)
Preconditioning IC(4)
J. Erhel, ALADIN
March 2000 31
An application of AUGCGwith LM2S - M. Brieu - J. Erhel - [Comp. Str. 1999, Int.J.Eng.Sc. 2000]
Nonlinear behaviour
of composite structures
homogenisation approach
Newton-type method
parallel domain decomposition
4 linear interface problems
Equi-biaxial tension
Uniaxial tension
1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Equi-triaxial Tension
Pure shear
0
Number of iterations in CG
Problem
20
10
5
15
25
J. Erhel, ALADIN
March 2000 32
Eigenvalue problems and singular value decompositionAx = λx or Ax = λBx or A = UΣV T , with A sparse
Aims and results
projection methods :Arnoldi and Davidson
to speed-up convergence
to control accuracy
coordination of european project STABLE
thesis with CERFACS
fluid dynamics (with LIMSI)
Research in other groups
Very few groups in France
dense matrices - projection methods (CER-
FACS,U. Brest) - solid mechanics (LM2S) -
data mining - software
pseudo-spectrum
[LAA 1996,Num. Alg. 1997,Comp. 1998]
2nd prize Leslie Fox - V. Heuveline - 1999
J. Erhel, ALADIN
March 2000 33
Parallelism
Aims and results
design and implementation
of parallel algorithms
for ODE and linear algebra
to solve large scale problems
Examples of application
electromagnetism (with U. Rennes 1, IPSIS)
high-speed networks (NSF)
parallel tensor product :
x −→ x ⊗ni=1 A(i)
[PDCP-I 2000, PDCP-II 2000]
0 5 10 15 20 25 30 350
5
10
15
20
25
30
35
# processors
spee
d−up
Parallel run on T3E
n = 12 = 2,985,984 6
Research in other groups
domain decomposition (LM2S, 4B) - parallel libraries
IEEE best paper award at RENPAR 11 - C. Tadonki - 1999
J. Erhel, ALADIN
March 2000 34
Numerical reliability : rounding errors and condition numbers
Aims and results
interval algorithms
interaction symbolic/numerical
to control numerical quality
coordination of FIABLE
Examples of application
chemistry (BERTIN/ANDRA)
computational geometry
(GENIE2, FIABLE)
numerical GCD of two polynomials - 1999
Linear algebra hidden
k = maxi∃∆p ‖∆p‖ ≤ ε,∃∆q ‖∆q‖ ≤ ε,
deg(GCD(p + ∆p,q + ∆q)) = ip =
Q10i=1(X − i) and q =
Q15i=9(X − i)
before refining after refining
root 1 10.000038 9.99999999997
root 2 8.999986 9.00000000003
Research in other groups
stable algorithms - condition number estimation -
control of numerical quality (CERFACS, U. Paris 6) - interval library
J. Erhel, ALADIN
March 2000 35
Plans
free software environment
numerical algorithms
industrial contacts health and medicine
Manpower
3 INRIA researchers : P. Chartier, J. Erhel, B. Philippe
C. Simon external collaborator?
Y-H. De Roeck associate researcher?
Needs
one software engineer
one more researcher
J. Erhel, ALADIN
March 2000 36
Technology transfer
Objectives
increase further the national visibility of the project
increase further the collaboration with industries
Software development
free software (like other teams in the domain)
integration into the Scilab package
parallel versions
Industrial collaborations prospected
parallel and reliable versions of scientific softwares
hydro-geology and cardiology
J. Erhel, ALADIN
March 2000 37
Domains of application
Aladin will be strongly involved in three applications,
including modelling and discretisation.
Hydro-geology - H. Hoteit, B. Philippe, J. Erhel
modelling of flow and transport of solute - DAE and linear systems
cooperation with IMF, ESTIME, Cameroon (Campus project), Andra?
Acoustic and seismic image processing - Y-H. De Roeck, B. Philippe, J. Erhel
modelling of geotechnic problems - large scale 3D ill-conditioned problems
cooperation with Ifremer - Contrat de Plan Etat Region
Cardiology? - P. Chartier, J. Erhel
3D modelling of electric heart activity - DAE
grant of CIFRE type? - interaction with ICEMA action?
J. Erhel, ALADIN
March 2000 38
Differential equationsE. Lapotre, P. Chartier
Objectives
shooting methods
numerical resolution of Hamilton-Jacobi-Bellman equations
cooperation with MOCOA
solving EDA of index 2
Gauss-based methods without explicit use of projection step
analysis using geometric invariants
cooperation with U. San Sebastian, U. Auckland
J. Erhel, ALADIN
March 2000 39
Eigenvalue and singular value problemsD. Mezher, B. Philippe
Objectives
utility of balancing to improve the speed and accuracy of methods to solve large scale
nonsymmetric eigenvalue problems
cooperation with Berkeley? (proposal submitted)
0.9
1
1.1
1.2
1.3
−0.1
−0.05
0
0.05
0.10
2
4
6
8
10
12
14
16
18
pseudo-spectrum before balancing
0.9
1
1.1
1.2
1.3
−0.1
−0.05
0
0.05
0.10
2
4
6
8
10
12
14
16
18
pseudo-spectrum after balancing
J. Erhel, ALADIN
March 2000 40
General linear models : y = Ax + ε,ε ∼ (0,Ω)Y-H. De Roeck, J. Erhel, F. Guyomarc’h, B. Philippe
Objectives
regularisation with polynomial filter functions
iterative preconditioned methods
cooperation with Ifremer and U. Minneapolis - NSF proposal submitted
50 100 150 200 250
50
100
150
200
250
polynomial filter
50 100 150 200 250
50
100
150
200
250
blurred image
50 100 150 200 250
50
100
150
200
250
Tychonov filter
J. Erhel, ALADIN
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