Master Thesis Literature Study Presentation

Master Thesis Literature Study Presentation Delft University of Technology The Faculty of Electrical Engineering, Mathematics and Computer Science January 29, 2010

Plaxis Introduction Plaxis Finite Element Method Plaxis B.V. is a company specialized in finite element software intended for 2D and 3D analysis of deformation, stability and groundwater flow in geotechnical engineering.

Plaxis Finite Element Method

Finite Element Method Plaxis Finite Element Method The Finite Element Method (FEM) is a numerical technique for finding approximate solutions of partial differential equations (PDE).

Example Introduction Plaxis Finite Element Method

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Main problem Solve: Ax = b, (1) where A R n n is SPD.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Basic Iterative Methods Let us take x i+1 = x i + M 1 (b Ax i ) (2) Which generates a sequence with the following property x i x 0 + span{m 1 r 0, M 1 A(M 1 r 0 ),..., (M 1 A) i 1 (M 1 r 0 )}. (3)

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Set M = I.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Set M = I. x x i A = min y K i (A;r 0 ) x y A, for all i.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Set M = I. x x i A = min y K i (A;r 0 ) x y A, for all i. Solution of the problem leads to the.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Conjugate Gradient Algorithm Choose x 0, set i = 0, r 0 = b Ax 0. WHILE r k =0 DO i := i + 1 IF i = 0 DO p 1 = r 0 ELSE β i = r T i 1 r i 1 r T i 2 r i 2 p i = r i 1 + β i p i 1 ENDIF α i = r i 1 T r i 1 p i T Ap i x i = x i 1 + α i p i r i = r i 1 α i Ap i END WHILE (4)

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Theorem Let A and x be the coefficient matrix and the solution of (1), and let (x i, i = 0, 1, 2...) be the sequence generated by the CG method. Then, elements of the sequence satisfy the following inequality: x x i A 2 ( κ(a) 1 κ(a) + 1 ) i x x 0 A, (5) where κ(a) is the condition number of A in the 2 - norm.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Speed up? Yes, with preconditioning.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Transform the system (1) into A x = b, (6) where A = P 1 AP T, x = P T x and b = P 1 b, where P is a non-singular matrix. The SPD matrix M defined by M = PP T is called the preconditioner.

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Preconditioned Conjugate Gradient Algorithm Choose x 0, set i = 0, r 0 = b Ax 0. WHILE r i =0 DO z i = M 1 r i i := i + 1 IF i = 0 DO p 1 = z 0 ELSE β i = r T i 1 z i 1 r T i 2 z i 2 p i = z i 1 + β i p i 1 ENDIF α i = r i 1 T z i 1 p i T Ap i x i = x i 1 + α i p i r i = r i 1 α i Ap i END WHILE (7)

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration There are several possibilities for the preconditioner. Examples Jacobi, M - diag(a).

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration There are several possibilities for the preconditioner. Examples Jacobi, M - diag(a). SSOR, M = 1 2 ω ( 1 ω D + L)( 1 ω D) 1 ( 1 ω D + L)T

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration Second Simple Problem Figure: Second Simple Problem Representation

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration 4 2 0 log 10 (i th residue) 2 4 6 8 CG CG with Jacobi Official Approach with PreSSOR 10 0 100 200 300 400 500 600 700 Iteration step Figure: Plot of log 10 of i-th residue

Main Problem and Basic Iterative Methods Convergence of CG Precondtioned Numerical Ilustration 10 1 10 0 10 1 10 2 10 3 10 4 0 50 100 150 200 Figure: Plot of log 10 of eigenvalues

Main idea Deflated Preconditoned vector DEFLATION

Main idea Deflated Preconditoned vector How to get rid of unfavorable eigenvalues that deteriorate the convergence of PCG?

Main idea Deflated Preconditoned vector Split x = (I P T )x + P T x, (8) where, P = I AQ, Q = ZE 1 Z T E = Z T AZ and Z R n m. (9)

Main idea Deflated Preconditoned vector After some transformation we get an equivalent, deflated system: PA x = Pb, (10) which can be solved with CG or PCG.

Main idea Deflated Preconditoned vector Deflated Preconditioned Conjugate Gradient Algorithm Choose x 0, set i = 0, r 0 = P(b A x 0 ). WHILE r k =0 DO i := i + 1 IF i = 1 DO y 0 = M 1 r 0 p 1 = y 0 ELSE y i 1 = M 1 r i 1 β i = r T i 1 y i 1 r T i 2 y i 2 p i = y i 1 + β i p i 1 ENDIF α i = r i 1 r T i 1 p i T PAp i x i = x i 1 + α i p i r i = r i 1 α i PAp i END WHILE x orginal = Qb + P T x last (11)

Main idea Deflated Preconditoned vector How to choose Z?

Main idea Deflated Preconditoned vector Several choices for Z: Approximated Eigenvector

Main idea Deflated Preconditoned vector Several choices for Z: Approximated Eigenvector Subdomain

Main idea Deflated Preconditoned vector Several choices for Z: Approximated Eigenvector Subdomain Rigid Body Mode

Main Idea Schwarz Alternating Procedure Numerical Experiments

Main Idea Schwarz Alternating Procedure Numerical Experiments Definition We will call a method a method, if its main idea will be based on the principle of divide and conquer applied on the domain of the problem.

Main Idea Schwarz Alternating Procedure Numerical Experiments Figure: An example of domain decomposition

Main Idea Schwarz Alternating Procedure Numerical Experiments There are several ways to do it. The most known are: Schwarz Alternating Procedure Schur Complement

Main Idea Schwarz Alternating Procedure Numerical Experiments Lets consider a domain as shown in the last figure with two overlapping subdomains Ω 1 and Ω 2 on which we want to solve a PDE of the following form: { Lu = b, in Ω u = g, on Ω. (12)

Main Idea Schwarz Alternating Procedure Numerical Experiments Schwarz Alternating Procedure Choose u 0 WHILE no convergence DO FOR i = 1,...s DO Solve Lu = b in Ω i with u = u ij in Γ ij Update u values on Γ ij, j END FOR END WHILE (13) In our case, s = 2.

Main Idea Schwarz Alternating Procedure Numerical Experiments Within the SAP there are two distinguished variants Multiplicative Schwarz Method (MSM) Additive Schwarz Method (ASM)

Multiplicative Schwarz Method Main Idea Schwarz Alternating Procedure Numerical Experiments [ u n+1/2 = u n A 1 + Ω 1 0 0 0 u n+1 = u n+1/2 + [ 0 0 0 A 1 Ω 2 ] (b Au n ) ] (b Au n+1/2 ) (14) where A Ωi stays for the discrete form of the operator L restricted to Ω i

Additive Schwarz Method Main Idea Schwarz Alternating Procedure Numerical Experiments ([ u n+1 = u n A 1 + Ω 1 0 0 0 ] [ 0 0 + 0 A 1 Ω 2 ]) (b Au n ) (15)

Main Idea Schwarz Alternating Procedure Numerical Experiments Additive Schwarz Method Or in a more general: Additive Schwarz Method Choose u 0, i = 0, WHILE no convergence DO r i = b Au n FOR i = 1,...s DO δ i = B i r i END FOR s u n+1 = u n + δ i i=1 i = i + 1 END WHILE (16) where B i = R t i A 1 Ω i R i

Numerical Experiments Main Idea Schwarz Alternating Procedure Numerical Experiments Figure: Domain Ω split into two subdomains, Ω 1 and Ω 2.

MSM Introduction Main Idea Schwarz Alternating Procedure Numerical Experiments 10 0 9 8 2 Size of the overlap 7 6 5 4 4 6 8 3 10 2 1 5 10 15 20 25 30 35 40 Iteration step 12 Figure: Contour plot of the log 10 (i-th residue) for MSM.

ASM Introduction Main Idea Schwarz Alternating Procedure Numerical Experiments 10 2 9 0 Size of the overlap 8 7 6 5 4 3 2 2 4 6 8 10 1 5 10 15 20 25 30 35 40 Iteration step 12 Figure: Contour plot of the log 10 (i-th residue) for ASM

Choice of method Test Problems Conclusions Goals

Choice of the method Choice of method Test Problems Conclusions Goals DPCG with Additive Schwarz as the preconditoner and physics based decomposition.

First Simple Problem Introduction Choice of method Test Problems Conclusions Goals Figure: First Simple Problem Representation

Choice of method Test Problems Conclusions Goals 4 2 0 Log 10 (j th residuum) 2 4 6 8 10 12 0 10 20 30 40 50 60 70 80 Iteration count Figure: First Simple Problem convergence behavior for PCG

Choice of method Test Problems Conclusions Goals Analysis of the decomposition position 80 2 70 0 60 2 Size of first subdomain 50 40 30 4 6 20 8 10 10 0 0 10 20 30 40 50 60 70 80 Iteration count Figure: First Simple Problem distribution of error in PCG

Second Simple Problem Choice of method Test Problems Conclusions Goals Figure: Second Simple Problem Representation

Choice of method Test Problems Conclusions Goals 4 2 0 Log 10 (j th residuum) 2 4 6 8 10 0 100 200 300 400 500 600 700 800 Iteration count Figure: Second Simple Problem convergence behavior for PCG with 2 subdomains

Choice of method Test Problems Conclusions Goals 705 600 2 500 0 400 2 300 4 200 100 6 0 20 40 60 80 100 120 140 160 180 200 Iteration count 8 Figure: Second Simple Problem distribution of error in PCG

Choice of method Test Problems Conclusions Goals The choice of the method is right.

Choice of method Test Problems Conclusions Goals The choice of the method is right. Number and size of the subdomains matters.

Choice of method Test Problems Conclusions Goals The choice of the method is right. Number and size of the subdomains matters. Introduction of deflation.

Goals Introduction Choice of method Test Problems Conclusions Goals

Questions Introduction Choice of method Test Problems Conclusions Goals Subdomain overlap.

Choice of method Test Problems Conclusions Goals Questions Subdomain overlap. Performance of.

Choice of method Test Problems Conclusions Goals The End