Alex Townsend Cornell University

Transcription

1 Continuous analogues of Krylov methods for differential operators Alex Townsend Cornell University Joint work with: Marc Aurèle Gilles Based on Continuous analogues of Krylov methods for differential operator, submitted last week.

2 Computing trends Punchcards DOS Windows95 Device oblivious interaction Tabulations IEEE 754 Software Discretization oblivious software

3 Discretize-then-solve Discretize-then-solve Finite diff. and pseudospectral Linear, elliptic, 2nd order, PDE Matrix Discrete solution Lu = f on = g Ax = b x Solve-then-discretize L is an unbounded operator A is a bounded operator Want A to be well-conditioned Want A to capture L 2nd order diff matrices Approx. solution û (see [Weideman & Trefethen, 1988] and [Fornberg & Flyer, 2015])

4 Matrix Krylov subspace methods Task: Solve Ax = b for x A is large fast mat-vecs Krylov subspace methods are iterative solvers, computing iterates from: K k (A, b) = Span b, Ab, A 2 b,..., A k 1 b Preconditioning: P Ax = Pb Popular strategies for spectral methods: 1. Low-order methods (same DE, crude disc.) e.g. [Canuto & Quarteroni, 1985], [Orszag, 1980] [Shewchuk,1994] and [Liesen & Strakoš, 2013] CG + Legendre-Galerkin disc. 2. Equivalent preconditioning (crude DE, same disc.) e,g, [Axelsson & Karatson, 2009], [Hiptmair, 2006] For details, see [Shen, Tang, and Wang, 2011].

5 Operator Krylov methods Task: Solve Lu = f with =0 L = 2nd order di op fast op-funs Qu: What is a continuous analogue of Krylov subspaces for diff ops? A discussion in Sept 2012, involving 120 numerical analysts: For Krylov methods you apply the operator in forward mode, so imposing bcs does not make sense. [Daniel, 1965] Marcus Webb One needs to consider the weak formulation and define the right spaces for the domain and range. Nick Trefethen I find it remarkable that on these fundamental issues we are so much in the fog. The first SIAM spotlight book is on this topic. [Malèk & Strakoš, 2014] Folkmar Bornemann

6 Continuous analogues Task: Solve Lu = f with =0 L : H 2 0 7! L 2, B[v, w] = Lu = dx i,j=1 Z dx i,j=1 (a ij (x)u xi ) xj + a ij v xi w xj + dx i=1 Z L = 2nd order di op fast op-funs dx i=1 b i v xi w + b i (x)u xi + c(x)u Z cvw [Evans, 2010] Krylov method GMRES [Saad & Schultz, 1986] MINRES [Paige & Saunders, 1975] conjugate gradient [Hestenes & Stiefel, 1952] Matrix properties A = invertible A = symm. A = pos. def. Operator properties Ellipticity & some regularity L is self-adjoint L has pos. eigs

7 Four problems Task: Solve Lu = f with =0 L =di op fast op-funs Naive attempt: Example: K k (L,f) = Span f,lf,...,l k 1 f u xx =1 x 2 u(±1) = 0 K k (L,f) = Span 1 x 2, 2 Doesn't contain sol. Problem 1 L destroys bcs Repeated op-funs? Problem 2 L destroys smoothness Repeated op-funs? Problem 3 L is an unbounded op Convergence? Problem 4 6=0 Imposing bc in K k? Ideas abound: Josef Málek Zdenek Strakoš Sheehan Olver Jared Aurentz Ralf Hiptmair Jie Shen

8 Idea 1: Orthogonal projections Orthogonal projections allow us to do repeated op-fun products. Pick an approximation space V 0 Assume Set V0 L(Lf) f 2 V 0 for now. = arg min q2v 0 k qk L 2 Without projections *If repeated op-funs are possible. Lf f V 0 H 2 0( ) for u in Lu = f. (Need V 0 H0( ) 2 closed L 2 -subspace.) L( V0 Lf) With projections V 0 H 2 0( ) Related to projected Krylov methods in quadratic programming. [Gould, Orban, & Rees, 2014] Lf ( V0 L) 2 f V0 Lf f

9 Idea 1: Orthogonal projections f 2 V0 Why orthogonal projection? L V ( L V f ) L V0 f V0 0 0 f Self-adjoint(!): V0 = V0 L f V V 0 0 Respects bilinear form: 2 V0 H0 ( ) B[ V0 v, V0 w] = B[v, w] 2 ( V0 L V0 ) f for v, w 2 V0 Krylov subspace: Kk ( V0 L V0, f ) = Span{f, V0 L V0 f,..., ( V0 L V0 )k 1 f} 2 uxx = 1 x u(±1) = 0 Example, again: 4 X j V0 = {p(x) = j x : p(±1) = 0} V0 = arg min k j=0 K3 ( L, f ) = Span 1 q2v0 x2, ( 21x4 + 14x2 + 7)/4, 147x x2 qkl2 14

10 Operator CG method Matrix CG Operator CG A = symm. pos. def. matrix L = self-adj., pos. eigvals., f 2 V 0 The convergence of this operator CG method is poor! This op. CG method is equiv. to matrix CG on specific Galerkin disc.

11 Not yet a continuous analogue But, we want a discretization oblivious algorithm V 0 = H 2 0( ) so need to take!

12 Idea 2: Operator preconditioners Symmetric operator preconditioners improve convergence. Symm. preconditioning: R T ARy = R T b Symm. operator preconditioning: =0 R LRv = R f, x = Ry u = Rv Less art in the science: (Preconditioner only depends on L.) B[R, R ] h, i Example: BVP on ( 1, 1) Z x Pick R = (s)ds, 1 W 0 = {v 2 L 2 ( ) : Z 1 1 R = For this example, the preconditioner is similar to integral reformulation. [Zebib, 1981], [Greengard, 1991], & [Driscoll, 2014] Z 1 x v(s)ds =0} R is easy [Hiptmair, 2006] and [Axelsson & Karátson, 2009] (s)ds = 1 2 Z 1 1 (s)ds

13 Operator PCG method L = self-adj., pos. eigvals., R f 2 W 0 Combine ortho. projections with op. preconditioners: T = R LR K k (T, R f) = Span{R f,tr f,...,t k 1 R f} Convergence of op. PCG method Operator PCG method E1: E2: E3: ((2 + cos( x))u 0 ) 0 =(1+x 2 ) 1, u(±1) = 0 ((1 + x 2 )u 0 ) 0 +( 4 cos( x))2 u =(1+x 2 ) 1, u(±1) = 0 u ( 4 )2 u =(1+x 2 ) 1, u(±1) = 0

14 PCG theorems Matrix CG Operator PCG CG iterates: x 0 =0,x 1,..., CG iterates: u 0 =0,u 1,..., Fact 1: rj T r k =0, j 6= k Fact 1: hr j,r k i =0, j 6= k r k = kb Ax k k 2 Fact 2: p T j Ap k =0, j 6= k p k = search direction r k = kr f R Lu k k L 2 Fact 2: B[Rp j, Rp k ]=0, p k = search function j 6= k Fact 3: x k = arg min y2k k (A,b) kx yk A Fact 3: u k = arg min q2x k ku qk L X k = {p 2 H 1 0( ) :p = Rq, q 2 K k (T, R f)} Fact 4: kx x k k A apple 2 p! k Fact 4: apple(a) 1 p kxk A ku u k k L apple 2 apple(a)+1 p apple(t ) 1 p apple(t )+1! k kuk L (see [Kirby, 2010] for a def. of apple(t ).)

15 Idea 3: An ancillary problem for righthand side The ancillary problem allows us to deal with general righthand sides. We solve R LRv = R f, using the Krylov subspace =0 K k (T, R f) = Span{R f,tr f,...,t k 1 R f} T = R LR We need R(R =0to apply the operator PCG method. Modify RHS before solving: 1. Find a v 1 so that RR LRv = RR 2. Solve R LRv 2 = R g for v 2 with g = f R LRv 1 3. Set v = v 1 + v 2

16 Practical realization of the PCG method for BVPs We let Chebfun deal with the discretization of functions: ApproxFun

17 Numerical experiment: smooth functions Family of BVPs: d (2 + cos(! 1 x)) du dx dx = f(x) on ( 1, 1) u(±1) = 0 f(x) chosen so that the exact solution is u(x) =sin(! 2 x) Oscillatory sol., simple var. coeffs Simple sol., oscillatory var. coeffs! 1 = 10! 2 = 10

18 Numerical experiment: piecewise smooth BVP with piecewise smooth functions: ((1 + 2 cos( x) )u 0 ) 0 = sign(cos(30 x)) on ( 1, 1) u(±1) = 0 chebfunpref.setdefaults( splitting,true) L = chebop(@(x,u) -diff((1+2*abs(cos(pi*x))).*diff(u))); f = chebfun(@(x) sign(cos(3*pi*x))); L.lbc = 0; L.rbc = 0; u = pcg(l, f) u(x) u 0 (x) Similar to [Lee & Greengard, 1997]

19 Operator GMRES Matrix GMRES solves Ax = b via the following approx. problems: Operator GMRES solves R LRv = R f via where K k (T, R f) = Span{R f,tr f,...,t k 1 R f} T = R LR We hope that u k = Rv k converges to the solution of Lu = f

20 Numerical experiment: operator GMRES BVP with smooth functions: (e x u 0 ) 0 + u 0 10u = sin(30 x) on ( 1, 1) u(±1) = 0 L = chebop(@(x,u) -diff(exp(x).*diff(u)) + diff(u) - 10*u); f = chebfun(@(x) sin(30*pi*x)); L.lbc = 0; L.rbc = 0; GMRES with restarts u = gmres(l, f); plot(u) u(x) From experience, the behavior of operator GMRES is similar to matrix GMRES.

21 Thank you Idea 1: Orthogonal projections Idea 2: Operator preconditioners Idea 3: Ancillary problem for rhs Matrix Krylov methods u = pcg(a, b) u = minres(a, b) u = gmres(a, b) Operator Krylov methods u = pcg(l, f) u = minres(l, f) u = gmres(l, f) What s next: Elliptic PDEs with inv. sqrt Laplacian preconditioners AND Dr. Gilles