advanced mathematical methods and software platform for ... · team: list of delivarables: 1-3 4-6...

23/7/2012

1

Advanced Mathematical Methods and Software Platform for Solving Multi-Physics,

Multi-Domain Problems on Modern Computer Architectures: Applications to Environmental

Engineering and Medical Problems

Work Package 2 - Task 2.2 Interface Relaxation Methods (IR)

Kick Off Meeting - 2012/07/21 – Chania, Crete, Greece

Yota Tsompanopoulou - Univ. of Thessaly

outline

Timetables - Deliverables

Scientific Description

21/7/2012 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou

2

23/7/2012

2

Timetables - Deliverables


3


4

Start Date: 01/01/02012 End Date: 31/09/2015

Month 2012 2013 2014 2015

jan-mar

apr-jun

jul - sep

oct - dec

jan-mar

apr-jun

jul - sep

oct - dec

jan-mar

apr-jun

jul - sep

oct - dec

jan-mar

apr-jun jul aug sep

WP Title: Research Team: List of delivarables: 1-3 4-6 7-9 9-12

13-15

16-18

19-21

22-24

25-27

28-30

31-33

34-36

37-39

40-42 43 44 45

1 WP 1 COORDINATION & PLANNING 1 (2, 3)

3 (ANNUAL)INTERMEDIATE TECHNICAL REPORTS

1 (2, 3) 1 FINAL TECHNICAL REPORT

2 WP 2

NUMERICAL & ANALYTICAL METHODS FOR DISCONTINUOUS

MULTIDOMAIN PROBLEMS (DISCONTINOUS COLLOCATION

METHODS, RELAXATION METHODS, STOCHASTIC-DETERMINISTIC HYBRID

METHODS, FOKAS METHODS)

1 2.1 DISCONTINUOUS COLLCATION: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE CODE

2 (1) 2.2 RELAXATION METHODS: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE

2 2.3 STOCHASTIC-DETERMINISTIC HYBRID METHODS: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE

1 (3) 2.4 FOKAS METHODS: 1 TECHNICAL REPORT / 2 PAPERS

3 WP 3

MAPPING & IMPLEMENTATION ON STATE-OF-THE-ART

COMPUTING ENVIROMENTS (CLUSTERS-GRIDS-CLOUDS-FPGAS-RECONFIGURABLE

ARRAYS)

3 (1) 3.1 CLUSTERS-GRIDS-CLOUDS: 1 TECHNICAL REPORT / 3 PAPERS / SOFTWARE

2

3.2 FPGAS-RECONFIGURABLE ARRAYS: 1 TECHNICAL REPORT / 2 PAPERS / SOFTWARE

4 WP 4

INTEGRATING, EVALUATING AND VALIDATING THE RESULTS ON

ENVIRONMENTAL AND MEDICAL PROBLEMS

2, 3 (1) 4.1 INTEGRATION: 1 TECHNICAL REPORT / SOFTWARE

1 (2, 3) 4.2 MEDICAL PROBLEMS: 1 TECHNICAL REPORT / 1 PAPERS

1 (2, 3) 4.3 ENVIROMENTAL PROBLEMS: 1 TECHNICAL REPORT / 1 PAPERS

5 WP 5 PROJECT EVALUATION 1 PROJECT EVALUATION BY EXTERNAL EVALUATORS

6 WP 6 DISSEMINATION OF THE RESULTS 1 (2, 3) WEB SITE, WORKSHOP: WORKSHOP PROCEEDINGS

Start End

23/7/2012

3

Timetable - Deliverables


5

Timetable

Start : 1/1/2012

End : 30/6/2015

Duration : 40 months

Deliverables

1 technical report (τεχνικές εκθέσεις)

3 research papers (επιστημονικά άρθρα)

Scientific Description

Interface Relaxation Methods


6

23/7/2012

4

Interface Relaxation (IR) Methods for General Elliptic PDEs

1. Introduction • Basic target. • A composed Problem. • Basic Methodology • General stuff.

2. IR Methods. • AVE. • GEO. • ROB.

3. Convergence Analysis. 4. Numerical Results.

21/7/2012 7 Kick Off Meeting - WP2 - Task 2.2 Yota Tsompanopoulou

Main Target The creation of a general methodology for the

solution of the composite PDE problems with the following properties:

1. Consists of a set of collaboration solvers for local (simple) problems.

2. Allow the choice of the most appropriate discretization method of the local problem.

3. Simplifies the geometry and the physics of the global problem.

4. Software reuse. 5. Wide applicability and high efficiency. 6. Increased adaptivity and inherent parallelism.


23/7/2012

5

A Composed PDE Problem

Uxx + Uyy = 0

U = 1.5Un

Uxx+Uyy= -1.0 Uxx+Uyy + yUx = -1.0

Uxx+Uyy + xUx = -1.0

Heat Radiaton Region

Mounting Region

Heat Producing Region


Basic Methodology

1. Split the problem in the PDE level.

2. Use solution’s properties to set conditions on the common boundaries (interfaces)

3. Give initial values on the interfaces.

4. Solve each PDE locally.

5. Use relaxation method to smooth the solution/derivatives or/and operators on the interfaces.

6. Check convergence criteria and go back to 4.


23/7/2012

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Basic Methodology


11

, in

, on b

Lu f Ω\ Ω

u u

Initial global composed PDE problem

Basic Methodology


12

, in

0 , on \

, on

i i i i

ij i i

b

i i i

L u f Ω

G u i j

u u

Set of simple PDE subproblems

23/7/2012

7

Basic Methodology


13

Overview in IR Methods Primitive Relaxers: About 10 proposed/implemented/analysed.

Smooth values and normal derivatives in various ways

Advanced Relaxers: Smooth additional operators (continuity of mass,

temperature, conservation of energy/momentum, equilibrium conditions, Lagrange multipliers, Steklov-Poincare operators, etc.)

Differences in convergence and applicability.


23/7/2012

8

AVE

For k=0,1,2,…

Define:

Solve the Neumann problems:

1,...,1,12

1

2

pidx

du)β(

dx

duβg

ii xx

k)(

ii

xx

k)(

iii

,1

)12(

11 fuL k

00

)12(

1

xx

ku

1

)12(

1

1

gdx

du

xx

k

στο Ω1 ,)12(

p

k

pp fuL

1

)12(

1

p

xx

k

pg

dx

du

p

0)12(

pxx

k

pu

στο Ωp ,)12(

i

k

ii fuL

1

)12(

1

i

xx

k

i gdx

du

i

i

xx

k

i gdx

du

i

)12(

στο Ωi

i=2,…,p-1


AVE (cont’d)

Define:

Solve the Dirichlet problems:

1...,11 12

1

12

p,,i)uα(uαhii xx

)k(

iixx

)k(

iii

,1

)22(

11 fuL k

00

)22(

1

xx

ku

1

)22(

11

huxx

k

στο Ω1 ,)22(

p

k

pp fuL

1

)22(

1

pxx

k

p hup

0)22(

pxx

k

pu

στο Ωp ,)22(

i

k

ii fuL

1

)22(

1

ixx

k

i hui

ixx

k

i hui

)22(

στο Ωi

i=2,…,p-1


23/7/2012

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GEO

For k=0,1,2,…

Define:

Solve the Dirichlet problems:

1,...,1,2

)(

1

)()(

1

)(

pidx

du

dx

duρ

uug

ii xx

k

i

k

ii

xx

k

i

k

ii

,1

)1(

11 fuL k

00

)1(

1

xx

ku

1

)1(

11

guxx

k

στο Ω1 ,)1(

p

k

pp fuL

1

)1(

1

pxx

k

p gup

0)1(

pxx

k

pu

στο Ωp ,)1(

i

k

ii fuL

1

)1(

1

ixx

k

i gui

ixx

k

i gui

)1(

στο Ωi

i=2,…,p-1


ROB

For k=0,1,2,…

Define:

Solve the problems:

1,...,11

11

pi

uλdx

dug

uλdx

dug

i

i

i

i

i

i

xx

(k)

ii

xx

(k)

ii

xx

(k)

ii

xx

(k)

ii

,1

)1(

11 fuL k

00

)1(

1

xx

ku

στο Ω1

1

1

1

11

1

1

11

guλdx

du

xx

)(k

xx

)(k

,)1(

i

k

ii fuL

i

i

xx

)(k

ii

xx

)(k

i guλdx

du

ii

1

1

στο Ωi

i=2,…,p-1

i

i

xx

)(k

ii

xx

)(k

i guλdx

du

ii

1

1

1

1

11

,)1(

p

k

pp fuL

0)1(

pxx

k

pu

στο Ωp

p

p

xx

)(k

pp

xx

)(k

pguλ

dx

du

pp

1

1

1

1

11


23/7/2012

10

Convergence Analysis

The model problem.

Convergence conditions and “optimum” values for the relaxation parameters of AVE.

Optimum values for the relaxation parameters of ROB.

Convergence conditions for GEO.


Model problem (1-dim)

0xa 1x1ix ix 1px bxp

1 ip

(a,b)x

0)()( buau

,2 fuγuxx


23/7/2012

11

It is known that:

The solution of -uxx + γ2u = 0 ,

with

is given by

(a,b)x

,υu(a)c(a)uc 121

243 υu(b)c(b)uc

1α)γ(b

4321

α)γ(b

4321

x)γ(b

43

x)γ(b

43 υ)ecγc)(cγ(c)ecγ)(ccγc(

)ecγc()ecγ(c

.υ)ecγc)(cγ(c)ecγ)(ccγc(

)ecγ(c)ecγc(2α)γ(b

4321

α)γ(b

4321

α)γ(x

21

α)γ(x

21


22

D.E. for the error functions (AVE) i=1,…,p , k=0,1,…

Neumann steps (similar for the Dirichlet steps):

1...,112

1

2

p,,idx

dε)β(

dx

dεβgε

ii xx

k)(

ii

xx

k)(

iii

,012

11 )k(εL

00

12

1

xx

)k(ε

1

12

1

1

gεdx

dε

xx

)k(

in Ω1 ,εL )k(

pp 012

1

)12(

1

p

xx

k

pg

dx

d

p

012

pxx

)k(

pε

in Ωp ,εL )k(

ii 012

1

12

1

i

xx

)k(

i gεdx

dε

i

i

xx

)k(

i gεdx

dε

i

12

in Ωi

i=2,…,p-1

,uuε i

(k)

i

(k)

i


23/7/2012

12

23

Error functions (AVE)

,1

1

2

1

2

sinh

1cosh

2

11

2

111sinh

cosh12

k)(

,iii

k)(

i,ii)ii

(γiγ

))i

x(xi

(γ

k)(

i,ii

k)(

,iii)ii

(γiγ

x))i

(xi

(γ)k(

i

)dεβ(dεβ

)dεβ(dεβ(x)ε

12 ,...,pi

Neumann:

Dirichlet:

,1

1

12

1

12

)sinh(

1sinh

12

11

12

111sinh

sinh22

)k(

,iii

)k(

i,iiii

γ

))i

x(xi

(γ

)k(

i,ii

)k(

,iii)ii

(γ

x))i

(xi

(γ)k(

i

)εα(εα

)εα(εα(x)ε

12 ,...,pi

,

2

2

dx

)(xdεdε

j

k)(

ik)(

i,j where .1212 )(xεε j

)k(

i

)k(

i,j


24

Relation between two consecutive errors (AVE)

,

112211

T,...,ε,εεε (k)

,pp

(k)

,

(k)

,

(k)

,112211

T,...,dε,dεdεdε (k)

,pp

(k)

,

(k)

,

(k)

,12)22( )k(Dk

dεMε

k)(Nk

εMdε2)12(

,2

1

ii

iD

,iiγm

αM

,1

11

1

ii

ii

ii

iiD

i,iγm

)nα(

γm

nαM

,12

11

1

ii

iD

i,iγm

)α(M ,12 ,...,pi,2,1 ...,pi

22 ,...,pi

,2

1

i

iiN

,iim

γβM

,1

1

11

i

iii

i

iiiN

i,im

γ)nβ(

m

γnβM

,12

1

11

i

iiN

i,im

)γβ(M ,12 ,...,pi,2,1 ...,pi

22 ,...,pi

•

•

•


23/7/2012

13

Convergence, optimum values (AVE)

Theorem: Let of length

and γi=γ, i=1,...,p. If then

is minimized by

,...,pii 1 ,

DN MM

iiii

iii

nmnm

nmα

11

1

.22

11

1 ,...,p-i,nmnm

nmβ

iiii

iii

and

,1 ,21ln

,...,piγ

)(

i

iii

p

i

i ,xx, ΩΩΩ 1

1


26

D.E. for the error functions (GEO)

1,...,1,2

)(

1

)()(

1

)(

pidx

dε

dx

dερ

εεgε

ii xx

k

i

k

ii

xx

k

i

k

ii

,0)1(

11 kεL

00

)1(

1

xx

kε

1

)1(

11

gεεxx

k

στο Ω1 ,0)1( k

ppεL

1

)1(

1

pxx

k

p gεεp

0)1(

pxx

k

pε

στο Ωp ,0)1( k

iiεL

1

)1(

1

ixx

k

i gεεi

ixx

k

i gεεi

)1(

στο Ωi

i=2,…,p-1


23/7/2012

14

27

Error functions (GEO)

1,...,2 ,2

2

1

1

1111

1111

11

pi(k)

,ii

(k)

i,ii

(k)

,ii

(k)

i,i

γγ

)x(xγ)x(xγ

(k)

i,i

(k)

,iii

(k)

i,i

(k)

,ii

γγ

x)(xγx)(xγ)(k

i

dεdερεε

ee

ee

dεdερεε

ee

ee(x)ε

iiii

iiii

iiii

iiii

p=3, T,dεdεdεdεεε,εεε (k)

,

(k)

,

(k)

,

(k)

,

(k)

,

(k)

,

(k)

,

(k)

,

(k)

2322121123221211 ,,,,,

,,1,0 ,)1(

kεMε(k)k ,88RM

,

tanh

1

ii

iγ

A

2,1 ,sinh

1 i

γB

ii

i


28

Structure of iteration matrix (GEO)

3323323333

22222222122122222222

22222222122122222222

1111111111

22

22

11

11

0022

00

2222

2222

000022

002

1

2

100

002

1

2

100

00002

1

2

1

00002

1

2

1

γAργAργAγA

γAργAργBργBργAγAγBγB

γBργBργAργAργBγBγAγA

γAργAργAγA

ρρ

ρρ

ρρ

ρρ

M


23/7/2012

15

Region of convergence (GEO)

Lemma: The non-identically zero eigenvalues of Matrix M,

are equal to the non-identically zero eigenvalues of ,

where

M~

.1

1~

33222221

22222111

)AγA(γρBγρ

Bγρ)AγA(γρM

Theorem: If of length

, GEO converges to the solution of the original

problem iff

3,21 , ,ii

)BγC(CρC

Cρρ,

Cρ

2

2

2

22112

112

1

12

220

20

where .2,1 ,11 iAγAγC iiiii

iii ,xx ΩΩΩΩΩ 1321 ,


30

D.E. for the error functions (ROB)

111

11

,...,pi

ελdx

dεgε

ελdx

dεgε

i

i

i

i

i

i

xx

(k)

ii

xx

(k)

ii

xx

(k)

ii

xx

(k)

ii

,01

11 )(kεL

00

1

1

xx

)(kε

in Ω1

1

1

1

11

1

1

11

gεελdx

dε

xx

)(k

xx

)(k

,01 )(k

iiεL

i

i

xx

)(k

ii

xx

)(k

i gεελdx

dε

ii

1

1

in Ωi

i=2,…,p-1

i

i

xx

)(k

ii

xx

)(k

i gεελdx

dε

ii

1

1

1

1

11

,01 )(k

ppεL

01

pxx

)(k

pε

in Ωp

p

p

xx

)(k

pp

xx

)(k

pgεελ

dx

dε

pp

1

1

1

1

11


23/7/2012

16

31

Error functions (ROB)

•

•

1,...,2 ,))(())((

)()(

))(())((

)()(

11

11

11

11111

11

1

11

piελdεeλγλγeλγλγ

eλγeλγ

ελdεeλγλγeλγλγ

eλγeλγ(x)ε

(k)

,iii

(k)

,iiγ

iiii

γ

iiii

)x(xγ

ii

)x(xγ

ii

(k)

,iii

(k)

,iiγ

iiii

γ

iiii

x)(xγ

ii

x)(xγ

ii)(k

i

iiii

iiii

iiii

iiii

T,dε,ε,εdεdε,εεdεε (k)

p,p

(k)

p,p

(k)

,pp

(k)

,pp

(k)

,

(k)

,

(k)

,

(k)

,

(k)

11111112121111 ,,,,

,,1,0 ,)1(

kεMε(k)k

,)()(

)()(1

1

1

1)1(2),1(2

iiiiiiiiiii

iiiiiiiii

i

iinλmγλγnλmγγ

mλnγλmλnγ

dM


32

Structure of iteration matrix (ROB)

000000

00000

00000

00000

00000

00000

00000

000000

11212

12112312112

12212312212

6535

6434

4313

4212

21

)(p),(p

)(p,)(p)(p,)(p

)(p,)(p)(p,)(p

,,

,,

,,

,,

,

M

MM

MM

MM

MM

MM

MM

M

M


23/7/2012

17

Optimal parameters (ROB)

where has the same structure with Μ and

,0~

0 )\Mσ(σ(M)\

,)()(~ 11

1)1(2),1(2

i

iiiiiiiiiiii

d

mλnγλmλnγλM

M~

,1,,2 pi

Theorem: Consider the decomposition of Ω into p subdomains Ωi

of length

The spectral radious of the iterative matrix is zero if the parameters λi , are selected as

,1

p

pp

pm

nγλ

.2,...,1 ,)(

1

pi

mλnγ

nλmγγλ

iiii

iiiiii

,...,pii 1 ,

Lemma: Let Μ the iteration matrix of the method, then

.11

2

iiiiiiiii nλλγmλλγd


Numerical experiments

4 composed problems

History of convergence

Verification of “optimum” parameters

Effect of descritization of the domain

Effect of descritization of the operator


23/7/2012

18

4 composed problems

PDE1: Helmholtz equations with Cartesian decomposition of domain

7,..,0, ,2 iinfuγΔu I

iii

,20 ,16 ,10 ,exp2

1 2

3

2

2

2

1

2

0 γγγγ yx

)8

exp(5 ,25

,15 ,4)(sin2

2

7

2

6

2

5

2

4

yxγγ

γπyxγ

PDE2: Helmholtz equations with general decomposition of domain

3,..,0, ,2 iinfuγΔu II

iii

,25 ,16 ,exp2

1 2

2

2

1

2

0 γγγ yx

4)8

exp(2

5)(sin2

4

yx

πyxγ


4 composed problems PDE3: Linear operator with general decomposition of domain

IVin

yxuΔu

0

22 ,0)2(602.0

PDE4: Non-Linear operator with general decomposition of domain

IVIVinuΔu 31 , ,04.0 IV

y

u

x

uinuΔu 2 ,03.0)(10

IV

u

in

yxuΔu

0

22

1000

,0)2(60)1(2.0

IVIVinuΔu 31 , ,04.0

IV

u

inu

y

u

x

u

y

uΔu

2

500

1

2

2

1000

,03.0

31


23/7/2012

19

History of convergence for PDE4 (ROB)

3η επαν. 2η επαν. 1η επαν.


Verification of “optimum” parameters (PDE1)

ROB AVE


23/7/2012

20

Verification of “optimum” parameters (PDE3)

ROB


Effect of descritization of the operator History of convergence for ROB (PDE2)

Finite Difference (5 point star)

Collocation

Bi-linear Finite Elements 21/7/2012 40 Kick Off Meeting - WP2 - Task 2.2

Yota Tsompanopoulou

23/7/2012

21

Effect of discretization of the domain

History of convergence for ROB with bi-linear finite elements

for PDE2.

h=0.10

h=0.20 21/7/2012 41 Kick Off Meeting - WP2 - Task 2.2

Yota Tsompanopoulou

Conclusions


42

The IR methodology is suitable for composed PDE problems.

There are theoretical results for convergence, with “optimum” relaxation parameters.

Theoretical results are verified with numerical experiments (model and general problems).

23/7/2012

22

References


43

Analysis of an Interface Relaxation Method for Composite Elliptic

Differential Equations, P. Tsompanopoulou and E. Vavalis, Journal of

Computational and Applied Mathematics, 226(2), (Apr. 2009), pp 370-387.

An Experimental Study of Interface Relaxation Methods for Composite

Elliptic Differential Equations, P. Tsompanopoulou and E. Vavalis, Applied

Mathematical Modelling, 32, (Aug. 2008), pp 1620-1641.

Fine Tunning Interface Relaxation Methods for Elliptic Differential

Equations, J.R. Rice, P. Tsompanopoulou and E.A. Vavalis, Applied

Numerical Mathematics, 43(4), (Nov. 2002), pp 459-481.

Interface Relaxation Methods for Elliptic Differential Equations, J.R.

Rice, P. Tsompanopoulou and E.A. Vavalis, Applied Numerical Mathematics,

32 (2000), pp 219-245.


44

Thanks!

23/7/2012

23

Possible project acronyms


45

ACAdEMy Advanced mathematiCAl Engineering Medical

ACCEPt AdvanCed mathematiCal Engineering Problems

ADVISE ADVanced mathematIcal Software Engineering

AMALTHEA Advanced MAthematicaL meTHods EnviromentAl

AMETHyST Advanced Mathematical mETHods SofTware

ANiMATED AdvaNced MAThematical Engineering meDical

ANiMATE AdvaNced MAthematical meThods Engineering

ANAEMIA AdvaNced mAthematical Engineering MedIcAl

ANEMONE AdvaNcEd Mathematical envirOmeNtal Engineering

ACHIEVE AdvanCed matHematIcal EnViromental Engineering

CoMMENd advanCed Mathematical Methods ENgineering

CoMPLETE advanCed Mathematical PLatform EnviromenTal Engineering

advanced mathematical methods and software platform for ... · team: list of delivarables: 1-3 4-6...

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