Various ways to use a second level preconditioner

Transcription

1 Various ways to use a second level preconditioner C. Vuik 1, J.M. Tang 1, R. Nabben 2, and Y. Erlangga 3 1 Delft University of Technology Delft Institute of Applied Mathematics 2 Technische Universität Berlin Institut für Mathematik 3 University of British Columbia Department of Earth and Ocean Sciences SIAM CSE 2009 March 2-6, / 28

2 Outline 1 Introduction 2 Two level PCG 3 Theoretical comparison 4 Numerical comparison 5 Conclusions 2 / 28

3 Introduction Tomography scans of bitumen species Mechanical problem Collaboration with: Tom Jönsthövel, Martin van Gijzen and Tom Scarpas 3 / 28

4 Introduction 2D simplification 4 / 28

5 Introduction Convergence of PCG with various size of the jumps 5 / 28

6 Introduction Convergence of Deflated PCG 6 / 28

7 Introduction Preconditioned Conjugate Gradient Method Preconditioned Conjugate Gradients Method (PCG) 1 Solve iteratively: M 1 Ax = M 1 b where M is a traditional preconditioner that resembles A Requirements for Preconditioner M Mz = y is relatively easy to solve M 1 A has a smaller condition number than A Theorem 2 Exact error of PCG after iteration j:! j p κ(m x x j A 2 x x 0 1 A) 1 A p κ(m 1 A) / 28

8 Introduction Preconditioned Conjugate Gradient Method Problem of PCG The spectrum of M 1 A contains a number of small eigenvalues Consequence κ `M 1 A is large Slow convergence of the iterative process Question Can the convergence of PCG be improved by eliminating those small eigenvalues in some way? 8 / 28

9 Introduction Deflated PCG History CG Ar 1950 Preconditioned CG M 1 Ar 1980 Block Preconditioned CG Block Preconditioned Deflated CG rp (M 1 )Ar i i= rp (M 1 )PAr i i= / 28

10 Two level PCG Definitions Given are SPD coefficient matrix, A R n n projection subspace matrix, Z R n k, with full rank Projection operator Define E R k k, Q R n n, and the projection matrix, P R n n, as follows: P := I AQ, Q := ZE 1 Z T, E := Z T AZ. In addition, M R n n is an SPD matrix that is called the preconditioner. 10 / 28

11 Two level PCG Properties of the projection operators Lemma Let A, Z, Q and P be as in the previous definition. Then, the following equalities hold: (a) P = P 2 ; (b) PA = AP T ; (c) P T Z = 0, P T Q = 0; (d) PAZ = 0, PAQ = 0; (e) QA = I P T, QAZ = Z, QAQ = Q; (f) Q T = Q. 11 / 28

12 Two level PCG Two-Level PCG methods List of considered methods Name Method Operator References PREC Traditional Preconditioned CG M 1 Go96L, Mei77V AD Additive Coarse-Grid Correction M 1 + Q Bra86PS,Tos05W DEF1 Deflation Variant 1 M 1 P Vui99SM DEF2 Deflation Variant 2 P T M 1 Kol98,Nic87 Sa00YEG A-DEF1 Adapted Deflation Variant 1 M 1 P + Q Smi96BG, Tro01OS A-DEF2 Adapted Deflation Variant 2 P T M 1 + Q Smi96BG, Tro01OS BNN Abstract Balancing P T M 1 P + Q Man93 R-BNN1 Reduced Balancing Variant 1 P T M 1 P R-BNN2 Reduced Balancing Variant 2 P T M 1 Man93,Tos05W 12 / 28

13 Two level PCG General Two-Level PCG Method for solving Ax = b Two-Level PCG 1: Select arbitrary x and V start, M 1, M 2, M 3, V end 2: x 0 := V start, r 0 := b Ax 0 3: y 0 := M 1 r 0, p 0 := M 2 y 0 4: for j := 0, 1,..., until convergence do 5: w j := M 3 Ap j 6: α j := (r j, y j )/(p j, w j ) 7: x j+1 := x j + α j p j 8: r j+1 := r j α j w j 9: y j+1 := M 1 r j+1 10: β j := (r j+1, y j+1 )/(r j, y j ) 11: p j+1 := M 2 y j+1 + β j p j 12: end for 13: x it := V end 13 / 28

14 Two level PCG Choices of V start, M 1, M 2, M 3, V end for each method Choices Method V start M 1 M 2 M 3 V end PREC x M 1 I I x j+1 AD x M 1 + Q I I x j+1 DEF1 x M 1 I P Qb + P T x j+1 DEF2 Qb + P T x M 1 P T I x j+1 A-DEF1 x M 1 P + Q I I x j+1 A-DEF2 Qb + P T x P T M 1 + Q I I x j+1 BNN x P T M 1 P + Q I I x j+1 R-BNN1 Qb + P T x P T M 1 P I I x j+1 R-BNN2 Qb + P T x P T M 1 I I x j+1 14 / 28

15 Two level PCG Choices for Z Choices Z is sufficiently sparse, so that Z and AZ can be stored in two vectors; Z is full, so that Z and AZ are full matrices. Motivation DDM, the columns of Z correspond to subdomains (approximated) eigenvector deflation methods Costs to compute Z T y or (AZy) one inner product one matrix vector multiplication 15 / 28

16 Two level PCG Extra computational cost per iteration of the two-level PCG methods IP = inner products, MVM = matrix-vector multiplications, VU = vector updates and CSS = coarse system solves Theory Implementation Method Py, P T y Qy IP / MVM VU CSS AD DEF DEF A-DEF A-DEF BNN R-BNN R-BNN / 28

17 Theoretical comparison Comparison Theorem The following two statements hold: σ `M 1 PA = σ `P T M 1 A = σ `P T M 1 PA ; σ `(P T M 1 P + Q)A = σ `(M 1 P + Q)A = σ `(P T M 1 + Q)A. Interpretation DEF1, DEF2, R-BNN1, and R-BNN2 have identical spectra, and the same is true for BNN, A-DEF1, and A-DEF2 17 / 28

18 Theoretical comparison Comparison Theorem Let the spectra of DEF1 and BNN be given by σ(m 1 PA) = {λ 1,..., λ n}, σ(p T M 1 PA + QA) = {µ 1,..., µ n}, respectively. Then, the eigenvalues within these spectra can be ordered such that the following statements hold: λ i = 0 and µ i = 1, for i = 1,..., k; λ i = µ i, for i = k + 1,..., n. 18 / 28

19 Theoretical comparison Comparison Theorem Suppose that the spectrum of DEF1, DEF2, R-BNN1, or R-BNN2 is {0,..., 0, λ k+1,..., λ n}, with λ k+1 λ k+2... λ n and the spectrum of BNN, A-DEF1, or A-DEF2 is {1,..., 1, µ k+1,..., µ n}, with µ k+1 µ k+2... µ n. Then, λ i = µ i for all i = k + 1,..., n. 19 / 28

20 Theoretical comparison Equivalent methods Theorem Let x R n be an arbitrary vector. The following methods produce exactly the same iterates in exact arithmetic: BNN with V start = Qb + P T x; DEF2, A-DEF2, R-BNN1 and R-BNN2 (with V start = Qb + P T x); DEF1 (with V start = x) whose iterates are based on Qb + P T x j / 28

21 Numerical comparison Porous media problem Composition Permeability Shale Sandstone Shale Sandstone Shale / 28

22 Numerical comparison Bubbly flow problem Composition Density air 10 3 air 10 3 water air air 10 3 air / 28

23 Numerical comparison Geometry of subdomains Ω 1 Ω 2 Ω 1 Ω 2 Ω 3 Ω 4 Ω 3 Ω 4 Ω 5 (a) Porous media problem. (b) Bubbly flow problem. Figure: Geometry of subdomains Ω j. Number of subdomains is fixed in the porous media problem, whereas it can be varied in the bubbly flow problem. 23 / 28

24 Numerical comparison Standard parameters x j x 2 / x AD DEF1 DEF2 A DEF1 A DEF2 BNN R BNN1 R BNN Iteration 24 / 28 Figure: Relative errors in 2 norm during the iterative process, for the porous media problem with

25 Numerical comparison Approximate coarse solves x j x 2 / x AD DEF1 DEF2 A DEF1 A DEF2 BNN R BNN1 R BNN Iteration 25 / 28 Figure: Relative errors in 2 norm during the iterative process for the porous media problem with

26 Numerical comparison Severe termination criteria x j x 2 / x AD DEF1 DEF2 A DEF1 A DEF2 BNN R BNN1 R BNN Iteration 26 / 28 Figure: Relative 2 norm errors during the iterative process for the porous media problem with

27 Numerical comparison Perturbed starting vectors DEF2 A DEF2 R BNN1 R BNN x j x A / x A /

28 Conclusions Conclusions Conclusions Various methods from deflation, additive coarse grid correction and balancing can be written in one framework DEF1, DEF2, R-BNN1, and R-BNN2 have identical spectra BNN, A-DEF1, and A-DEF2 have identical spectra Theoretically the methods are very close with respect to convergence With respect to cost and robustness of implementation there are serious differences A-DEF2 seems to be the most robust and fastest method Further reading it def.html accepted for publication in Journal of Scientific Computing 28 / 28