Matrix-equation-based strategies for convection-diusion equations
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1 Matrix-equation-based strategies for convection-diusion equations Davide Palitta and Valeria Simoncini Dipartimento di Matematica, Università di Bologna CIME-EMS Summer School 2015 Exploiting Hidden Structure in Matrix Computations. Algorithms and Applications Cetraro, June 2015 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
2 The convection-diusion equation The convection-diusion equation Convection-diusion equation: ɛ u + w u = f, in Ω R d with d = 2, 3, ɛ > 0 (viscosity parameter), w R d (convection vector) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
3 The convection-diusion equation The convection-diusion equation Convection-diusion equation: ɛ u + w u = f, in Ω R d with d = 2, 3, ɛ > 0 (viscosity parameter), w R d (convection vector) Dominant Convection w ɛ Incompressible ow div(w) = 0 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
4 The convection-diusion equation The convection-diusion equation Convection-diusion equation: ɛ u + w u = f, in Ω R d with d = 2, 3, ɛ > 0 (viscosity parameter), w R d (convection vector) Dominant Convection w ɛ Incompressible ow div(w) = 0 Further assumptions: w separable coecients, e.g., in 2D w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) Ω is a rectangle (parallelepipedal) domain Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
5 The convection-diusion equation Discretization phase Discretizing by FEM or FD, we get the nonsymmetric linear system Au = f, where A R N N Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
6 The convection-diusion equation Discretization phase Discretizing by FEM or FD, we get the nonsymmetric linear system Au = f, where A R N N REMARK: The discretization step is crucial to avoid spurious oscillations in the numerical solution: Rene the meshsize h huge increase of N Dierent strategies: Articial diusion SUPG... Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
7 A matrix-equation-based strategy A matrix-equation-based strategy AIM: Preconditioning the very large and sparse nonsymmetric linear system Au = f, where A R N N with a simplied matrix version of the discretized dierential operator Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
8 A matrix-equation-based strategy A matrix oriented formulation Poisson equation u xx u yy = f, in Ω = (0, 1) 2 (w/ zero Dirichlet b.c.) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
9 A matrix-equation-based strategy A matrix oriented formulation Poisson equation u xx u yy = f, in Ω = (0, 1) 2 (w/ zero Dirichlet b.c.) Discretize Ω with a uniform mesh Ω h with nodes (x i, y j ), i, j = 1,..., n 1. Dene T, F R (n 1) (n 1) such that T = 1 h 2 tridiag( 1, 2, 1) and F i,j = f (x i, y j ) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
10 A matrix-equation-based strategy A matrix oriented formulation Poisson equation u xx u yy = f, in Ω = (0, 1) 2 (w/ zero Dirichlet b.c.) Discretize Ω with a uniform mesh Ω h with nodes (x i, y j ), i, j = 1,..., n 1. Dene T, F R (n 1) (n 1) such that T = 1 h 2 tridiag( 1, 2, 1) and F i,j = f (x i, y j ) Centered FD leads to the symmetric linear system Au = f, f = vec(f ) where A = T I n 1 + I n 1 T R (n 1)2 (n 1) 2 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
11 A matrix-equation-based strategy A matrix oriented formulation U i,j u(x i, y j ) at interior node (x i, y j ) For each i, j = 1,..., n 1, u xx (x i, y j ) U i 1,j 2U i,j + U i+1,j = 1 [1, 2, 1] h 2 h2 U i 1,j U i,j U i+1,j and u yy (x i, y j ) U i,j 1 2U i,j + U i,j+1 h 2 = 1 1 h 2 [U i,j 1, U i,j, U i,j+1] 2 1 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
12 A matrix-equation-based strategy A matrix oriented formulation U i,j u(x i, y j ) at interior node (x i, y j ) For each i, j = 1,..., n 1, u xx (x i, y j ) U i 1,j 2U i,j + U i+1,j = 1 [1, 2, 1] h 2 h2 U i 1,j U i,j U i+1,j and u yy (x i, y j ) U i,j 1 2U i,j + U i,j+1 h 2 = 1 1 h 2 [U i,j 1, U i,j, U i,j+1] 2 1 TU + UT = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
13 A matrix-equation-based strategy A matrix oriented formulation, ɛ u + w u = f Proposition. (2D case) Assume w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)). Let (x i, y j ) Ω h, i, j = 1,..., n 1, and set, for k = 1, 2, Φ k = diag(φ k (x 1 ),..., φ k (x n 1 )) Ψ k = diag(ψ k (y 1 ),..., ψ k (y n 1 )) Then, the centered FD discretization of the continuous operator leads to the following operator: L : u ɛ u + φ 1 (x)ψ 1 (y)u x + φ 2 (x)ψ 2 (y)u y L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) where B = 1 tridiag( 1, 0, 1) R(n 1) (n 1) 2h Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
14 A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
15 A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Ψ 1 = Φ 2 = I Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
16 A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Ψ 1 = Φ 2 = I ɛt U + ɛut + (Φ 1 B)U + U(B T Ψ 2 ) = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
17 A matrix-equation-based strategy The easy case: ɛ u + φ 1 (x)u x + ψ 2 (y)u y = f w = (φ 1 (x), ψ 2 (y)) (common in academic examples!) Ψ 1 = Φ 2 = I ɛt U + ɛut + (Φ 1 B)U + U(B T Ψ 2 ) = F (ɛt + Φ 1 B)U + U(ɛT + B T Ψ 2 ) = F This is a Sylvester equation that can be explicitly and eciently solved! Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
18 The new preconditioner The general case: preconditioning strategy w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
19 The new preconditioner The general case: preconditioning strategy w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) Approximating Ψ 1 ψ 1 I, Φ 2 φ 2 I, where, e.g., ψ 1, φ 2 mean values Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
20 The new preconditioner The general case: preconditioning strategy w = (w 1, w 2 ) = (φ 1 (x)ψ 1 (y), φ 2 (x)ψ 2 (y)) L h : U ɛt U + ɛut + (Φ 1 B)UΨ 1 + Φ 2 U(B T Ψ 2 ) Approximating Ψ 1 ψ 1 I, Φ 2 φ 2 I, where, e.g., ψ 1, φ 2 mean values P : Y (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + φ 2 B T Ψ 2 ) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
21 Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
22 Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. At each iteration k: ṽ k = P 1 v k R (n 1)2 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
23 Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. At each iteration k: ṽ k = P 1 v k R (n 1)2 How to compute it? Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
24 Implementation details Implementation details Nonsymmetric linear system: Au = f, A R (n 1)2 (n 1) 2 Solver: GMRES with right preconditioning. At each iteration k: ṽ k = P 1 v k R (n 1)2 How to compute it? 1 Compute G k R (n 1) (n 1) s.t. v k = vec(g k ) 2 Solve (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ2 Ψ 2 ) = G k 3 Compute ṽ k = vec(y) Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
25 Implementation details Implementation details Sylvester equation: (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ 2 Ψ 2 ) = G k Solver: KPIK [V. Simoncini, 2007], [T. Breiten, V. Simoncini, M. Stoll, 2014]. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
26 Implementation details Implementation details Sylvester equation: (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ 2 Ψ 2 ) = G k Solver: KPIK [V. Simoncini, 2007], [T. Breiten, V. Simoncini, M. Stoll, 2014]. Observations: Inner: Inexact method Outer: FGMRES rhs must be low rank: Truncated SVD of G k Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
27 Implementation details Implementation details Sylvester equation: (ɛt + ψ 1 Φ 1 B)Y + Y(ɛT + B T φ 2 Ψ 2 ) = G k Solver: KPIK [V. Simoncini, 2007], [T. Breiten, V. Simoncini, M. Stoll, 2014]. Observations: Inner: Inexact method Outer: FGMRES rhs must be low rank: Truncated SVD of G k Three parameters: 1 inner tolerance: tol_inner= truncation tolerance: tol_truncation= maximum rank allowed: r_max= 10 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
28 3D case 3D case To x ideas: Ω = (0, 1) 3. U (k) i,j u(x i, y j, z k ), and dene the tall matrix U := where e j = I (:, j). U (1). U (n 1) n 1 = k=1 (e k U (k) ) R (n 1)2 (n 1) Other orderings are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
29 3D case 3D case: L : u ɛ u + w 1 u x + w 2 u y + w 3 u z Proposition. (3D case) Assume w = (w1, w2, w3) s.t. w1 = φ 1(x)ψ 1(y)υ 1(z) w2 = φ 2(x)ψ 2(y)υ 2(z) w3 = φ 3(x)ψ 3(y)υ 3(z) Let (x i, y j, z k ) Ω h, i, j, k = 1,..., n 1, and set, for l = 1, 2, 3, Φ l = diag(φ l (x1),..., φ l (x n 1)) Ψ l = diag(ψ l (y1),..., ψ l (y n 1)) Υ l = diag(υ l (z1),..., υ l (z n 1)) Then, the centered FD discretization leads to the following operator: L h : U (I ɛt )U + ɛut + (ɛt I )U + (Υ 1 Φ 1 B)UΨ 1 + (Υ 2 Φ 2 )UB T Ψ 2 + [(Υ 3 B) Φ 3 ]UΨ 3 Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
30 Numerical experiments Numerical experiments Competitors on Au = f: GMRES+MI20 FGMRES+AGMG REMARK: our preconditioner/solver is implemented in interpreted Matlab functions while both MI20 and AGMG are fortran90 compiled codes provided with mex les control.one_pass_coarsen= 1, [J. Boyle, M.D. Mihajlovi c, J.A. Scott, 2010] all default parameters, [Y. Notay, 2010] Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
31 Numerical experiments Test 1. Explicit matrix solution (2D) Consider ɛ u+w u = 0, in Ω = (0, 1) 2 ( ) w = (x + 1)2, 0 Dirichlet b.c.: { u(x, 0) = 1 x [0, 1], u(x, 1) = 0 x [0, 1]. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
![Numerical experiments Test 1. Explicit matrix solution (2D) Consider ɛ u+w u = 0, in Ω = (0, 1) 2 ( ) w = 1 + 1 4 (x + 1)2, 0 Dirichlet b.c.: { u(x, 0) = 1 x [0, 1], u(x, 1) = 0 x [0, 1].](/docs-images/79/79474396/images/32-4.jpg "(ɛt + Φ 1 B)U + ɛut = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/2015 15 / 23")
32 Numerical experiments Test 1. Explicit matrix solution (2D) Consider ɛ u+w u = 0, in Ω = (0, 1) 2 ( ) w = (x + 1)2, 0 Dirichlet b.c.: { u(x, 0) = 1 x [0, 1], u(x, 1) = 0 x [0, 1]. (ɛt + Φ 1 B)U + ɛut = F Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
33 Numerical experiments Test 1. Explicit matrix solution (2D) ɛ n x FGMRES+AGMG GMRES+MI20 KPIK time (# its) time (# its) time (# its) (13) ( 7) (24) (15) ( 8) (32) (20) ( 8) (44) (17) ( 7) (46) (14) ( 7) (22) (14) ( 9) (32) (15) ( 9) (38) (18) ( 8) (64) (18) ( 8) (52) (14) (10) (22) (15) ( 8) (34) (14) (10) (42) (18) (11) (66) (18) ( 9) (58) Table : Test 1 (2D). Performance achieved as the viscosity and mesh parameter vary. tol_outer= Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
![except for the side y = 0: { u(x, 0) = 1 + tanh[10 + 20(2x 1)], 0](/docs-images/79/79474396/images/34-4.jpg "x 0.5 u(x, 0) = 2, 0.")
34 Numerical experiments Test 2. Preconditioning strategy (2D) Consider ɛ u + w u = 0, in Ω = (0, 1) 2 w = (y(1 (2x + 1) 2 ), 2(2x + 1)(1 y 2 )) Zero Dirichlet b.c. except for the side y = 0: { u(x, 0) = 1 + tanh[ (2x 1)], 0 x 0.5 u(x, 0) = 2, 0.5 < x 1 slightly modication of Example V in [H.C. Elman, A. Ramage, 2002]. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
35 Numerical experiments Test 2. Preconditioning strategy (2D) ɛ n x FGMRES+AGMG GMRES+MI20 FGMRES+KPIK time (# its) time (# its) time (# its) (11) ( 4) ( 8) (14) ( 4) ( 7) (11) ( 4) ( 7) (17) ( 4) ( 6) (15) ( 4) ( 5) ( 9) ( 4) (10) (12) ( 4) ( 9) (11) ( 4) ( 8) (18) ( 4) ( 7) (17) ( 4) ( 6) ( 8) ( 4) (11) ( 9) ( 4) (11) (14) ( 4) ( 9) (12) ( 4) ( 8) (23) ( 4) ( 7) Table : Test 2 (2D). Performance achieved as the viscosity and mesh parameter vary. tol_outer= Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
36 Numerical experiments Test 3. Explicit matrix solution (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (x sin x, y cos y, e z2 1 ) Zero Dirichlet b.c. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
37 Numerical experiments Test 3. Explicit matrix solution (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (x sin x, y cos y, e z2 1 ) Zero Dirichlet b.c. [I (ɛt + Φ 1 B) + (ɛt + Ψ 2 B) T I ] U + U (ɛt + BΥ 3 ) = 11 T where 1 is the vector of all ones REMARK: low rank rhs in the matrix equation is crucial for explicit solution! Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
38 Numerical experiments Test 3. Explicit matrix solution (3D) ɛ n x FGMRES+AGMG GMRES+MI20 KPIK time (# its) time (# its) time (# its) (14) ( 7) (18) (14) ( 7) (20) (14) ( 7) (20) (15) ( 7) (22) (15) ( 7) (22) (14) ( 7) (18) (14) ( 7) (20) (14) ( 7) (20) (14) ( 7) (22) (14) ( 7) (22) (14) ( 6) (18) (14) ( 7) (20) (14) ( 7) (22) (14) ( 7) (22) (14) ( 7) (24) Table : Test 3 (3D). Performance achieved as the viscosity and mesh parameter vary. tol_outer= Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
39 Numerical experiments Test 4. Preconditioning strategy (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (yz(1 x 2 ), 0, e z ) Zero Dirichlet b.c. Other variable aggregations are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
40 Numerical experiments Test 4. Preconditioning strategy (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (yz(1 x 2 ), 0, e z ) Zero Dirichlet b.c. The discretization yields ɛt U + U(I ɛt + ɛt I ) + Υ 3 BU + Υ 1 U(Ψ 1 Φ 1 B) = 11 T where 1 is the vector of all ones Other variable aggregations are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
41 Numerical experiments Test 4. Preconditioning strategy (3D) Consider ɛ u + w u = 1, in Ω = (0, 1) 3 w = (yz(1 x 2 ), 0, e z ) Zero Dirichlet b.c. The discretization yields ɛt U + U(I ɛt + ɛt I ) + Υ 3 BU + Υ 1 U(Ψ 1 Φ 1 B) = 11 T where 1 is the vector of all ones Υ 1 ῡ 1 I and obtain the preconditioning operator P : Y (ɛt + Υ 3 B)Y + Y(I ɛt + ɛt I + Ψ 1 ῡ 1 Φ 1 B) Other variable aggregations are possible. Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
42 Numerical experiments Test 4. Preconditioning strategy (3D) ɛ n x FGMRES+AGMG GMRES+MI20 FGMRES+KPIK (15) ( 7) ( 6) (15) ( 7) ( 6) (16) ( 7) ( 6) (16) ( 8) ( 6) (16) ( 8) ( 6) (18) - ( -) ( 8) (18) - ( -) ( 8) (19) - ( -) ( 9) (19) - ( -) ( 9) (19) - ( -) ( 9) (18) - ( -) (10) (19) - ( -) (10) (20) - ( -) (10) (20) - ( -) (10) (21) - ( -) (10) Table : Test 4 (3D). Performance achieved as the viscosity and mesh parameter vary. - stands for excessive time in building the preconditioner. tol_outer= Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
43 Conclusions and outlook Conclusions and outlook Preliminary numerical experiments show that the new approach performs comparably well with respect to state-of-the-art approaches. Generalize the approach to more general settings overcoming the limitation to the use of uniform mesh on rectangle (parallelepipedal) domains. Modify the approach to handle dierent discretization strategies, e.g., SUPG (early attempts in [D.P., 2014, Master's thesis]). Reference: Matrix-equation-based strategies for convection-diusion equations D. Palitta and V. Simoncini ArXiv: Davide Palitta (UniBo) Matrix equations & conv-di. 23/06/ / 23
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