Indefinite and physics-based preconditioning

Transcription

1 Indefinite and physics-based preconditioning Jed Brown VAW, ETH Zürich

2 Newton iteration Standard form of a nonlinear system F (u) 0 Iteration Solve: Update: J(ũ)u F (ũ) ũ + ũ + u Example (p-bratu) Suppose F is a discretization of (η u ) λe u f 0 η(γ) ( ɛ2 + γ ) p 2 1 2, γ γ 0 2 u 2. Then J(ũ)u is a discretization of (η u + η ( ũ u) ũ ) λeũu.

3 Matrices and Preconditioners Definition (Matrix) A matrix is a linear transformation between finite dimensional vector spaces. Definition (Forming a matrix) Forming or assembling a matrix means defining it s action in terms of entries (usually stored in a sparse format). Left preconditioning in a Krylov iteration (P 1 A)x P 1 b {P 1 b, (P 1 A)P 1 b, (P 1 A) 2 P 1 b,... } Definition (Preconditioner) A preconditioner P is a method for constructing a matrix (just a linear function, not assembled!) P 1 P(A, A p ) using a matrix A and extra information A p, such that the spectrum of P 1 A (or AP 1 ) is well-behaved.

4 Domain decomposition Domain size L, subdomain size H, element size h Overlapping/Schwarz Solve Dirichlet problems on overlapping subdomains No overlap: its O ( ) ( L Hh, Overlap: its O L ) H BDDC and FETI-DP Neumann problems on subdomains with coarse grid correction its O ( 1 + log H ) h

5 Normal preconditioners fail for indefinite problems

6 Model problem: Stokes system Strong form: Find (u, p) V P such that η 2 u + p f u 0 Minimization form: Find u V which minimizes I(u) η u: u f u subject to u 0 Lagrangian: L(u, p) Ω Ω η u: u p u f u Weak form: Find (u, p) V P such that η v : u q u p v f v 0 Ω for all (v, q) V P.

7 Stokes Weak form Find (u, p) V P such that η v : u q u p v f v 0 Ω for all (v, q) V P. Matrix Block factorization [ A B T Jx B ] ( ) u p ( ) f 0 [ ] A B T B [ 1 BA 1 1 where the Schur complement is ] [ ] A B T S S BA 1 B T. [ A B ] [ 1 A 1 B T ] S 1

8 Block factorization [ ] A B T B where [ 1 BA 1 1 ] [ ] A B T S S BA 1 B T. [ A B ] [ 1 A 1 B T ] S 1 S is symmetric negative definite if A is SPD and B has full rank. S is dense We only need to multiply B, B T with vectors. We need a preconditioner for A and S. Any definite preconditioner (from last time) can be used for A. It s not obvious how to precondition S, more on that later.

9 Reduced factorizations are sufficient Theorem (GMRES convergence) GMRES applied to T x b converges in n steps for all right hand sides if there exists a polynomial of degree n such that p n (A) 0 and p n (0) 1. That is, if the minimum polynomial of A has degree n. A lower-triangular preconditioner Left precondition J: [ ] 1 [ ] T P 1 A A B T J B S B [ A 1 ] [ ] [ A B T 1 A S 1 BA 1 S 1 1 B T ] B 1 Since (T 1) 2 0, GMRES converges in at most 2 steps.

10 Preserving symmetry, for CG or MinRes Either P 1 A must be symmetric or both P 1 and A must be symmetric [ ] 1 P 1 A S [ ] [ ] [ T P 1 A 1 A B T 1 A J S 1 1 B T ] B S 1 B ( T 2) 1 2 [ 1 4 A 1 B T S 1 ] B ( T 1 ) 2 1 [ A B T S 1 B Now Q A 1 B T S 1 B is a projector (Q 2 Q) so [ ( T 1 ) ] ( T 1 ) Rearranging, T (T 1)(T 2 T 1) 0. GMRES converges in at most 3 iterations ]

11 Preconitioning the Schur complement S BA 1 B T is dense so we can t form it, but we need S 1. Least squares commutator Suppose B is square and nonsingular. Then S 1 B T AB 1. B is not square, replace B 1 with Moore-Penrose pseudoinverse B B T (BB T ) 1, (B T ) (BB T ) 1 B. Then Ŝ 1 (BB T ) 1 BAB T (BB T ) 1. Navier-Stokes: BB T ( ) is essentially a homogeneous Poisson operator in the pressure space Multigrid on BB T is an effective preconditioner. Requires 2 Poisson preconditioners per iteration

12 Preconditioning the Schur complement Physics-based commutators Unsteady Navier-Stokes with Picard linearization: ( ) 1 A α η 2 + w B ( ) Suppose we have formal commutativity ( ) 1 1 ( ) 1 1 α η 2 + w ( 2 ) α η 2 + w Discrete form S BA 1 B T ( BM 1 u B T )A 1 p M p Ŝ where M u, M p are mass matrices, L BMu 1 B T a homogeneous Laplacian in the pressure space. Then Ŝ 1 Mp 1 A p L 1 Stokes: S ( 2 )( η 2 ) 1 1 η, mass matrix scaled by 1 η.

13 Physics-based preconditioners Shallow water with stiff gravity wave h is hydrostatic pressure, u is velocity, gh is fast wave speed h t (uh) x 0 (uh) t + (u 2 h) x + ghh x 0 Semi-implicit method Suppress spatial discretization, discretize in time, implicitly for the terms contributing to the gravity wave h n+1 h n t (uh) n+1 (uh) n t Rearrange, eliminating (uh) n+1 h n+1 h n t + (uh) n+1 x 0 + (u 2 h) n x + gh n h n+1 x 0 t(gh n h n+1 x ) x Sx n

14 Delta form Preconditioner should work like the Newton step F (x) δx Recast semi-implicit method in delta form δh t + (δuh) x F 0 δuh t + ghn (δh) x F 1 Eliminate δuh δh t t(ghn (δh) x ) x F 0 + ( tf 1 ) x Solve for δh, then evaluate δuh t [ gh n ] (δh) x F 1 Fully implicit solver Is nonlinearly consistent (no splitting error) Can be high-order in time Leverages existing code (the semi-implicit method)