Indefinite Preconditioners for PDE-constrained optimization problems. V. Simoncini

Transcription

1 Indefinite Preconditioners for PDE-constrained optimization problems V. Simoncini Dipartimento di Matematica, Università di Bologna, Italy Partly joint work with Debora Sesana, Università del Piemonte Orientale 1

2 A BT B C The problem u = f v g Ax = b Hypotheses: A R n n symmetric B T R n m tall, m n C symmetric positive (semi)definite More hypotheses later on specific problems... Computational Algebraic Aspects: Elman, Silvester, Wathen 2005 (book) Benzi, Golub and Liesen, Acta Num

3 P = Constraint (Indefinite) Preconditioner à B T = I 0 à 0 I à 1 B B C Bà 1 I 0 S 0 I with C = S Bà 1 B T for some S. Assume B = B. For particular choices of Ã, C, all eigs of AP 1 are real and positive (under certain conditions, variants of the CG method can be used) Many contributions ( Bai, Bergamaschi, Cao, Dollar, Durazzi, Ewing, Gondzio, Gould, Herzog, Keller, Lazarov, Lu, Lukšan, Ng, Perugia, Rozložník, Ruggiero, Sachs, Schilders, Schöberl, Vassilevski, Venturin, Vlček, Wang, Wathen, Zilli, Zulehner,...) 3

4 The Magnetostatic problem (3D) Maxwell equations: divb = 0 curlh = J Constitutive law: B = µh (B displ. field; H magn. field; µ magn. perm.; J current dens.) Constrained quadratic programming formulation: min 1 µ 1 B µh 2 dx 2 Ω with B n = f B on Γ B and H n = f H on Γ H divb = 0 curlh = J 4

5 Magnetostatic problem: Algebraic Saddle-Point problem A BT B C x y = f g 2D: A pos.def. on Ker(B) B full row rank, C = 0 3D: A pos.def. on Ker(B), B rank deficient C semidefinite matrix Range(C), Range(B) complementary spaces BB T +C sym. positive definite A zero-order operator, B first-order operator 5

6 Magnetostatic problem: Indefinite Preconditioning C = 0. After scaling, Exact preconditioner: P = I BT = I 0 I 0 I BT B 0 B I 0 H 0 I - H = BB T Weyr canonical form ( B T = B T H 1 2) AP 1 X = X I n m +Θ I m I m I m, X = X B T (A I) 1 BT 0 0 m B(A I) 1 BT where (A I)(I B T B) X = XΘ partial eigenvalue decomposition, associated with its nonzero eigenvalues All real and positive eigenvalues: {1} {1+θ i } 6

7 Magnetostatic problem: Indefinite Preconditioning Inexact Indefinite preconditioning: P inex = I n 0 I n 0 I n B T, BB T +C H inex spd B I m 0 H inex 0 I m with E A AP 1 inex = AP 1 +E, BT I m H 1 2 E rank-m max λ i(hh 1 i=1,...,m inex ) 1 7

8 Inexact Indefinite Preconditioning. On the choice of H inex If H inex > 0 is such that H H inex has k m zero eigenvalues, then AP 1 inex retains 2k unit eigenvalues with geometric multiplicity k. First order perturbation of (multiple) unit eigenvalue: λ(ap 1 inex ) λ(ap 1 )+ξ 1 2 Assume A I < 0. Then ξ real. If H H inex 0 then ξ 0 1) Spectral approximation matters 2) Sign of approximation matters ξ independent of meshsize 8

9 Inexact Indefinite Preconditioning. On the choice of H inex Incomplete Choleski (tol=1e-3) Spectrum of AP 1 inex AMG preconditioning imaginary part of eigenvalues imaginary part of eigenvalues real part of eigenvalues real part of eigenvalues 9

10 The Stokes problem Minimize J(u) = 1 2 subject to u = 0 in Ω Ω u 2 dx Ω f udx Lagrangian: L(u,p) = J(u)+ Ω p udx Optimality condition on discretized Lagrangian leads to: A BT B C x y = f 0 A second-order operator, B first-order operator, C zero-order operator Thanks to Walter Zulehner 10

11 The Stokes problem. Inexact contraint preconditioning P inex = I n 0 à 0 I n à 1 B T Bà 1 I m 0 H inex 0 I m with H = Bà 1 B T +C H inex spd First order spectral perturbation of simple eigenvalues: λ(ap inex ) λ(ap 1 ) cκ(ã 1 A I) 1 2 max λ j(hh 1 j=1,...,m inex ) 1 (for à 1 A I definite) Spectrum independent of mesh parameter (for judicious choices of Ã,H inex) 11

12 The Stokes problem. Inexact contraint preconditioning Selection of Ã, H inex: Ã = amg(a), H inex = Q (pressure mass matrix) IFISS 3.1 (Elman, Ramage, Silvester): Flow over a backward facing step Stable Q2-Q1 approximation (C = 0) stopping tolerance: 10 6 n m # it

13 Constrained Optimal Control Problem. A toy problem. Let Ω R d, d = 2,3. Given û (desired state) in ˆΩ Ω, find u: min u,f s.t. 1 2 u û 2 L 2 (ˆΩ) +β f 2 L 2 (Ω) 2 u = f in Ω with u = û on Ω. Lagrangian of discretized problem: L(f,u,λ) = 1 2 ut Mu u T Mû+ 1 2 û 2 +βf T Mf +λ T (Ku Mf d) K stiffness matrix. First order optimality condition yields: 2βM 0 M f 0 0 M K T u = b M K 0 λ d M could be singular (depending on where û is defined) 13

14 Dimension reduction 2βM 0 M f 0 M K T u = M K 0 λ that is, 2βf = λ. Therefore 0 b d M K T u K 1 2β M = λ b d with M = M T 0, K = K T square, M = M T > 0 14

15 Indefinite Preconditioning strategy P = 0 K, K 1 2β M P 1 = K 1 K 1 K 1 C K 1 0, K K If K = K, then λ i (AP 1 ) = 1+η, 0 η c β (independent of meshsize) If K spectrally equivalent to K, still independence of meshsize 15

16 Numerical results: 2D and 3D Ω Ω D: û(x,y) = 2 in Ω 0 and û(x,y) = 0 on Ω (undefined elsewhere) Data thanks to Sue H. Thorne, RAL, UK 16

17 Numerical results M singular, K =amg(k) 2D: β = 10 5 β = 10 2 n # it. # it D: β = 10 5 β = 10 2 n # it. # it

18 Final considerations Plain use of Indefinite (constraint) preconditioning should not be discouraged Interplay between Solvers and Preconditioners is crucial Preconditioning strategies for Saddle Point systems largely expanding topic (also: block diagonal/triangular, augmented, projected CG, etc...) References for this talk: V.Simoncini, Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems, to appear in COAP. D. Sesana and V. Simoncini, Spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices, Submitted, Jan