The automatic solution of PDEs using a global spectral method

Transcription

1 The automatic solution of PDEs using a global spectral method 2014 CBMS-NSF Conference, 29th June 2014 Alex Townsend PhD student University of Oxford (with Sheehan Olver) Supervised by Nick Trefethen, University of Oxford. Supported by EPSRC grant EP/P505666/1.

2 Introduction Chebfun Piecewise smooth (2010) Endpoint singularities (2010*) Reducing regularity Blow up functions (2011) Chebfun (2004) Linear ODEs (2008) Nonlinear ODEs (2012) Chebgui (2011*) Ordinary differential equations Chebfun2 (2013) Piecewise smooth Linear PDEs Two dimensions Arbitrary domains Nonlinear PDEs Alex Oxford 1/21

3 Introduction Chebop: Spectral collocation for ODEs In 2008: Overload the MATLAB backslash command \ for operators [Driscoll, Bornemann, & Trefethen 2008]. L = chebop(@(x,u) diff(u,2)-x.*u,[-30 30]); % Airy equation L.lbc = 1; L.rbc = 0; % Set boundary conditions u = L \ 0; plot(u) % Solve and plot Alex Oxford 2/21

4 Introduction Spectral collocation basics Given values on a grid, what are the values of the derivative on that same grid?: u u 1 2 u 5 u 9 u 1 u 2 u 5 u 9 x 1 x 2 x 5 x 9 D n u 1 u n u 1 u n. =., D n = diffmat(n). For example, u (x) + cos(x)u(x) is represented as L n = D n + diag (cos(x 1 ),..., cos(x n )) R n n. Alex Oxford 3/21

5 Introduction Why do spectral methods get a bad press? 1. Dense matrices. 2. Ill-conditioned matrices. 3. When has it converged? Tricky. See, for example: [Canuto et al. 07],, [Fornberg 98], [Trefethen 00]. Alex Oxford 4/21

6 Introduction Why do spectral methods get a bad press? 1. Dense matrices. 2. Ill-conditioned matrices. 3. When has it converged? Tricky. See, for example: [Canuto et al. 07],, [Fornberg 98], [Trefethen 00]. Alex Oxford 4/21

7 Introduction Why do spectral methods get a bad press? 1. Dense matrices. 2. Ill-conditioned matrices. 3. When has it converged? Tricky Condition number See, for example: [Canuto et al. 07],, [Fornberg 98], [Trefethen 00]. Alex Oxford 4/21

8 Introduction Why do spectral methods get a bad press? 1. Dense matrices. 2. Ill-conditioned matrices. 3. When has it converged? Tricky Condition number 10 0 Error in solution See, for example: [Canuto et al. 07],, [Fornberg 98], [Trefethen 00]. Alex Oxford 4/21

9 A fast and well-conditioned spectral method Differentiation operator Work with coefficients: Spectral methods do not have to result in dense, illconditioned matrices. (Just don t discretize the differentiation operator faithfully.) The idea is to use simple relations between Chebyshev polynomials: 1 2 dt k dx = ku k 1, k 1, (U k U k 2 ), k 2, 1 T k = 2 0, k = 0, U 1, k = 1, U 0, k = 0. D = , S = Olver & T., A fast and well-conditioned spectral method, SIAM Review, Alex Oxford 5/21

10 A fast and well-conditioned spectral method Multiplication operator T j T k = 1 2 T j k + 2a 0 a 1 a 2 a 3... a 1 2a 0 a 1 a T j+k a 1 a 2 a 3 a 4... M[a] = 1 a 2 a 1 2a 0 a a 2 a 3 a 4 a a 3 a 2 a 1 2a 0... a 3 a 4 a 5 a } {{ } Toeplitz } {{ } Hankel + rank-1 Multiplication is not a dense operator in finite precision. It is m-banded: m a(x) = a k T k (x) = ã k T k (x) + O(ɛ), k=0 Alex Oxford 6/21 k=0

11 A fast and well-conditioned spectral method What about this new spectral method? 1. Almost banded matrices. 2. Well-conditioned matrices. 3. When has it converged? Trivial. Other approaches: [Clenshaw 57], [Greengard 91], [Shen 03]. Alex Oxford 7/21

12 A fast and well-conditioned spectral method What about this new spectral method? 1. Almost banded matrices. 2. Well-conditioned matrices. 3. When has it converged? Trivial. Other approaches: [Clenshaw 57], [Greengard 91], [Shen 03]. Alex Oxford 7/21

13 A fast and well-conditioned spectral method What about this new spectral method? 1. Almost banded matrices. 2. Well-conditioned matrices. 3. When has it converged? Trivial Condition number Other approaches: [Clenshaw 57], [Greengard 91], [Shen 03]. Alex Oxford 7/21

14 A fast and well-conditioned spectral method What about this new spectral method? 1. Almost banded matrices. 2. Well-conditioned matrices. 3. When has it converged? Trivial Condition number 10 0 Error in solution Other approaches: [Clenshaw 57], [Greengard 91], [Shen 03]. Alex Oxford 7/21

15 A fast and well-conditioned spectral method First example u (x) + x 3 u(x) = 100 sin(20,000x 2 ), u( 1) = 0. The exact solution is ( x ) u(x) = e x e t4 4 sin(20,000t 2 )dt. 1 u(x) degree(u) = time = 15.5s x N = chebop(@(x,u) ); N.lbc = 0; u = N \ f; Adaptively selects the discretisation size. Forms a chebfun object [Chebfun V4.2]. ũ u = Alex Oxford 8/21

16 A fast and well-conditioned spectral method Another example u 1 (x) + u(x) = 0, u( 1) = ,000x2 The exact solution with a = 50,000 is ( u(x) = exp tan 1 ( ax) + tan 1 ( ) a). a u(x) degree(u) = x Absolute error Old chebop Preconditioned collocation US method x Alex Oxford 9/21

17 A fast and well-conditioned spectral method A high-order example u (10) (x) + cosh(x)u (8) (x) + cos(x)u (2) (x) + x 2 u(x) = 0 u(±1) = 0, u (±1) = 1, u (k) (±1) = 0, k = 2, 3, u(x) u n+1 u n ( 1 1 (ũ(x) + ũ( x)) 2 ) 1 2 = x n Alex Oxford 10/21

18 Chebop and Chebop2 Convenience for the user L = chebop(@(x,u) diff(u,2)-x.*u,[-30 30]); % Airy equation L.lbc = 1; L.rbc = 0; % Set boundary conditions u = L \ 0; % u = chebfun Convert handle into discretisation instructions Construct discretisation A R n n Impose bcs and solve Ãx = b Converged to the solution? yes Construct a chebfun no, increase n L = chebop2(@(x,y,u) laplacian(u)+(1000+y)*u);% Helmholtz with gravity L.lbc = 1; L.rbc = 1; L.ubc = 1; L.dbc = 1;% Set boundary conditions u = L \ 0; % u = chebfun2 Convert handle into discretisation instructions Construct a matrix equation with an n y n x solution matrix Impose bcs and solve matrix equation Converged to the solution? yes Construct a chebfun2 no, increase n x or n y or both Alex Oxford 11/21

19 Chebop and Chebop2 Convenience for the user L = chebop(@(x,u) diff(u,2)-x.*u,[-30 30]); % Airy equation L.lbc = 1; L.rbc = 0; % Set boundary conditions u = L \ 0; % u = chebfun Convert handle into discretisation instructions Construct discretisation A R n n Impose bcs and solve Ãx = b Converged to the solution? yes Construct a chebfun no, increase n L = chebop2(@(x,y,u) laplacian(u)+(1000+y)*u);% Helmholtz with gravity L.lbc = 1; L.rbc = 1; L.ubc = 1; L.dbc = 1;% Set boundary conditions u = L \ 0; % u = chebfun2 Convert handle into discretisation instructions Construct a matrix equation with an n y n x solution matrix Impose bcs and solve matrix equation Converged to the solution? yes Construct a chebfun2 no, increase n x or n y or both Alex Oxford 11/21

20 Interpreting user-defined input Automatic differentiation Implemented by forward-mode operator overloading u xx + u yy + 50u + yu + Interpret anonymous function as a sequence of elementary operations + + * y.* u Can also calculate Fréchet derivatives u xx u yy u Key people: Ásgeir Birkisson and Toby Driscoll u u Alex Oxford 12/21

21 Low rank approximation Numerical rank For A C m n, SVD gives best rank k wrt 2-norm [Eckart & Young 1936] A = min(m,n) σ j u j v j k σ j u j v j, σ k+1 < tol. For Lipschitz smooth bivariate functions [Schmidt 1909, Smithies 1937] f(x, y) = σ j u j (y)v j (x) k σ j u j (y)v j (x). For compact linear operators acting on functions of two variables, L! = σ j L y L x j j k σ j L y L x j j. Alex Oxford 13/21

22 Low rank approximation Numerical rank For A C m n, SVD gives best rank k wrt 2-norm [Eckart & Young 1936] A = min(m,n) σ j u j v j k σ j u j v j, σ k+1 < tol. For Lipschitz smooth bivariate functions [Schmidt 1909, Smithies 1937] f(x, y) = σ j u j (y)v j (x) k σ j u j (y)v j (x). For compact linear operators acting on functions of two variables, L! = σ j L y L x j j k σ j L y L x j j. Alex Oxford 13/21

23 Low rank approximation Numerical rank For A C m n, SVD gives best rank k wrt 2-norm [Eckart & Young 1936] A = min(m,n) σ j u j v j k σ j u j v j, σ k+1 < tol. For Lipschitz smooth bivariate functions [Schmidt 1909, Smithies 1937] f(x, y) = σ j u j (y)v j (x) k σ j u j (y)v j (x). For compact linear operators acting on functions of two variables, L! = σ j L y L x j j k σ j L y L x j j. Alex Oxford 13/21

24 Low rank approximation Do the low rank stuff before discretization Low rank-then-discretize: Instead of low rank techniques after discretization, do them before. For example, Helmholtz is of rank 2 2 u + K 2 u = (u xx + K 2 2 u) + (u yy + K 2 2 u) = (D2 + K 2 2 I) I + I (D2 + K 2 Let A be your favourite ODE discretization of D 2 + K 2 2 I, then (typically) In general, if L is of rank k we have AXI + IXA T. 2 I). k A j XB T j = F Alex Oxford 14/21

25 Low rank approximation Do the low rank stuff before discretization Low rank-then-discretize: Instead of low rank techniques after discretization, do them before. For example, Helmholtz is of rank 2 2 u + K 2 u = (u xx + K 2 2 u) + (u yy + K 2 2 u) = (D2 + K 2 2 I) I + I (D2 + K 2 Let A be your favourite ODE discretization of D 2 + K 2 2 I, then (typically) In general, if L is of rank k we have AXI + IXA T. 2 I). k A j XB T j = F Alex Oxford 14/21

26 Low rank approximation Do the low rank stuff before discretization Low rank-then-discretize: Instead of low rank techniques after discretization, do them before. For example, Helmholtz is of rank 2 2 u + K 2 u = (u xx + K 2 2 u) + (u yy + K 2 2 u) = (D2 + K 2 2 I) I + I (D2 + K 2 Let A be your favourite ODE discretization of D 2 + K 2 2 I, then (typically) In general, if L is of rank k we have AXI + IXA T. 2 I). k A j XB T j = F Alex Oxford 14/21

27 Low rank approximation Computing the rank of a partial differential operator Recast differential operators as polynomials: Once you have polynomials computing the rank is easy. The rank of N y N x L = i=0 j=0 a ij (x, y) i y i j x j equals a TT-rank [Oseledets 2011] (between {x, s} and {y, t}) of N y N x k h(x, s, y, t) = a ij (s, t)y i x j = c j (t, y)r j (s, x). i=0 j=0 Rank 1: ODEs Trivial PDEs Rank 2: Laplace, Helmholtz Transport, Heat, Wave Black-Scholes Rank 3: Biharmonic Lots here. Alex Oxford 15/21

28 Low rank approximation Computing the rank of a partial differential operator Recast differential operators as polynomials: Once you have polynomials computing the rank is easy. The rank of N y N x L = i=0 j=0 a ij (x, y) i y i j x j equals a TT-rank [Oseledets 2011] (between {x, s} and {y, t}) of N y N x k h(x, s, y, t) = a ij (s, t)y i x j = c j (t, y)r j (s, x). i=0 j=0 Rank 1: ODEs Trivial PDEs Rank 2: Laplace, Helmholtz Transport, Heat, Wave Black-Scholes Rank 3: Biharmonic Lots here. Alex Oxford 15/21

29 Low rank approximation Computing the rank of a partial differential operator Recast differential operators as polynomials: Once you have polynomials computing the rank is easy. The rank of N y N x L = i=0 j=0 a ij (x, y) i y i j x j equals a TT-rank [Oseledets 2011] (between {x, s} and {y, t}) of N y N x k h(x, s, y, t) = a ij (s, t)y i x j = c j (t, y)r j (s, x). i=0 j=0 Rank 1: ODEs Trivial PDEs Rank 2: Laplace, Helmholtz Transport, Heat, Wave Black-Scholes Rank 3: Biharmonic Lots here. Alex Oxford 15/21

30 Low rank approximation Construct a n x by n y generalised Sylvester matrix equation If the PDE is Lu = f, where L is of rank-k then we solve for X C n y n x in, k X = solution s coefficients σ j A j XB T j = F, A j C n y n y, B j C n x n x. A j = A j, B j = 1D spectral discretization of L y j, Lx j Alex Oxford 16/21

31 Low rank approximation Matrix equation solvers Rank 1: A 1 XB T 1 = F. Solve A 1Y = F, then B 1 X T = Y T. Rank 2: A 1 XB T 1 + A 2XB T = F. Generalised Sylvester solver (RECSY) 2 [Jonsson & Kågström, 2002]. Rank k, k 3: Solve N N system using almost banded structure Execution time O(N 2 ) O(N 3/2 ) O(N) blue = rank 1 green = rank 2 red = rank N Alex Oxford 17/21

32 Examples Helmholtz equation 2 u + 2ω2 u = 0, u(x, ±1) = f (x, ±1), u(±1, y) = f (±1, y), where f (x, y) = cos(ωx) cos(ωy). Wave numbers vs. polynomial degree ω = n y Alex Oxford x ω 18/21

33 Examples Variable helmholtz equation N = chebop2(@(x,y,u) laplacian(u) (1/2+sin(x)ˆ2).*cos(y)ˆ2.*u); N.lbc = 1; N.rbc = 1; N.ubc = 1; N.dbc = 1; u = N \ chebfun2(@(x,y) cos(x.*y)); 10 0 Absolute maximum error N N = 1,050,625, error , time = 44.2s. Alex Oxford 19/21

34 Examples Wave and Klein Gordon equation N = chebop2(@(u) diff(u,2,1) - diff(u,2,2) + 5*u); % u_tt - u_xx + 5u N.dbc [u-exp(-10*x) diff(u)]; N.lbc = 0; N.rbc = 0; u = N \ 0; Alex Oxford 20/21

35 Conclusion Spectral methods do not have to be ill-conditioned. (Don t discretize differentiation faithfully.) Spectral methods are extremely convenient and flexible. As of 2014, global spectral methods are heavily restricted to a few geometries. Thank you for listening Alex Oxford 21/21