Exponential integration of large systems of ODEs

Transcription

1 Exponential integration of large systems of ODEs Jitse Niesen (University of Leeds) in collaboration with Will Wright (Melbourne University) 23rd Biennial Conference on Numerical Analysis, June 2009

2 Plan Aim: To solve the ODE u = Lu + N(u) where L is a big matrix and the stiffness is concentrated in L. Motivation: Such equations arise when semi-discretizing semi-linear PDEs. Method: Some exponential integrator with Krylov methods.

3 Outline Exponential integrators Definition of the ϕ-functions Connection ϕ-functions and ODEs Exponential Euler method An exponential predictor-corrector method Computing ϕ-function with matrix argument Reducing to the exponential Krylov methods Timestepping Numerical examples Uni Florida sparse matrices Heston PDE from finance (linear) Allen Cahn PDE in 2d (nonlinear) Conclusions

4 The solution of inhomogeneous linear DEs The solution of u = au, u(0) = u 0 is u(t) = e at u 0 = ϕ 0 (at)u 0. The solution of u = au + b, u(0) = 0 is u(t) = t 0 e as b ds = 1 a eas b where ϕ 1 (x) = (e x 1)/x. t s=0 = eat 1 b = tϕ 1 (at)b, a The solution of u = au + ct, u(0) = 0 is u(t) = t 2 ϕ 2 (at)c, where ϕ 2 (x) = (e x 1 x)/x 2. These solutions are also valid for vector ODEs.

5 The solution of inhomogeneous linear DEs II The solution of u (t) = Lu(t) + v ! tv ! t2 v 2 +, u(0) = u 0, where u(t), u 0, v 0, v 1,... are vectors and L is a matrix, is u(t) = ϕ 0 (tl)u 0 + tϕ 1 (tl)v 0 + t 2 ϕ 2 (tl)v 1 + t 3 ϕ 3 (tl)v 2 + where the ϕ-functions are ϕ 0 (x) = e x = 1 + x x ! x 3 + ϕ 1 (x) = ex 1 x = x + 1 3! x ! x 3 + ϕ 2 (x) = ex 1 x x 2 = ! x + 1 4! x ! x 3 + ϕ 3 (x) = ex 1 x 1 2 x 2 x 3 = 1 3! + 1 4! x + 1 5! x ! x 3 +

6 Exponential Euler method Back to u = Lu + N(u) (L = linear, N = nonlinear) Replace the nonlinear term with the constant N(u(0)) N(u(t)) (for small t) and use the results from the previous slide: u(t) = e tl u(0) + tϕ 1 (tl) N(u(0)). This leads to the exponential Euler method u n+1 = e hl u n + hϕ 1 (hl)n(u n ). This method has order 1 (like the normal Euler method). If DN(u n ) = 0 then it has order 2. (Certaine 1960) Exponential integrators are exact for linear equations. Thus, they can solve stiff equations if the stiffness is in the linear part.

7 An exponential predictor corrector method Higher-order exponential integrators are built from higher-order normal integrators with the higher ϕ-functions. We like exponential integrators based on AB/AM predictor corrector methods. E.g., for order 2, base method: u n+1 = u n hf (u n) 1 2 hf (u n 1) u n+1 = u n h( f (u n+1) 2f (u n ) + f (u n 1 ) ) Exponential integrator: u n+1 = e hl u n + hϕ 1 (hl)n(u n ) + h 2 ϕ 2 (hl) ( N(u n ) N(u n 1 ) ) u n+1 = u n+1 + h 2 ϕ 2 (hl) ( N(u n+1) 2N(u n ) + N(u n 1 ) ) Other exponential methods also exist (without corrector step, based on Runge Kutta, using exact Jacobian,... ). All need to evaluate ϕ-function of matrix times vector.

8 Outline Exponential integrators Definition of the ϕ-functions Connection ϕ-functions and ODEs Exponential Euler method An exponential predictor-corrector method Computing ϕ-function with matrix argument Reducing to the exponential Krylov methods Timestepping Numerical examples Uni Florida sparse matrices Heston PDE from finance (linear) Allen Cahn PDE in 2d (nonlinear) Conclusions

9 Evaluating the ϕ-functions The biggest problem for EIs is the evaluation of (linear combinations of) ϕ-functions with a matrix argument. Sometimes (e.g., spectral methods) you can diagonalize the matrix easily and use L = V 1 DV = ϕ k (hl) = hv 1 ϕ k (D)V. General case: Reduce problem to evaluation of matrix exponential (this step enlarges the matrix slightly, by two rows and columns): A v 0 e A ϕ 1 (A)v ϕ 2 (A)v exp = (Saad 1992; Sidje 1998) Now use well-known method to compute matrix exponential (scaling-and-squaring with Padé).

10 Krylov methods Padé approximation with squaring and scaling is too costly for large matrices (N 10 3 ). Idea of Krylov-subspace methods is to project the action of A on the Krylov subspace with dimension m N: span{v, Av, A 2 v,..., A m 1 v}. The Lanczos (if A symmetric) or Arnoldi algorithm produces an orthogonal N-by-m matrix V m and an m-by-m matrix H m such that A V m H m V T m = ϕ k (A) V m ϕ k (H m )V T m = ϕ k (A)v v V m ϕ k (H m )e 1 H m is small so we can compute ϕ k (H m )e 1 via Padé. (Friesner, Tuckerman & Dornblaser 1989)

11 Dimension of Krylov subspace The error in the Krylov approximation may be estimated with error v h m+1,m e T mϕ k+1 (H m )e 1 v m+1. This can be computed very cheaply. (Saad 1992; Sidje 1998) We use this error estimate to choose the dimension m of the Krylov subspace adaptively. That is, we compute the error estimate and if it is too big, we increase m and try again. (Hochbruck, Lubich & Selhofer 1998) We also use the error estimate to correct our initial approximation.

12 Timestepping The error in the Krylov approximation is bounded by error 2 v (ρ(a))m (1 + o(1)) m! for m large, where ρ(a) is the spectral radius. (Saad 1992) This suggest that m has to be large if ρ(a) is large. An alternative is to use timestepping. Remember that the ϕ-functions solve ODEs: The solution of u = Au + tv, u(0) = 0 is u(t) = t 2 ϕ 2 (ta)v. To compute ϕ 2 (A)v, we can split the interval [0, 1] up in several sub-intervals and solve the ODE on each sub-interval by computing a ϕ-function. (Sidje 1998; Sofroniou & Spaletta 2007) The length of the sub-intervals is chosen adaptively with standard techniques for ODE solvers.

13 Outline Exponential integrators Definition of the ϕ-functions Connection ϕ-functions and ODEs Exponential Euler method An exponential predictor-corrector method Computing ϕ-function with matrix argument Reducing to the exponential Krylov methods Timestepping Numerical examples Uni Florida sparse matrices Heston PDE from finance (linear) Allen Cahn PDE in 2d (nonlinear) Conclusions

14 Uni Florida sparse matrices We compare our routine with expokit, using the same experiments as in Sidge (1998). Compute ϕ 1 (A)v for: orani678: N = 2529, nnz = 90158, TOL = bcspwr10: N = 5300, nnz = 21842, TOL = 10 5, symm. gr 30 30: N = 900, nnz = 7744, TOL = 10 14, symm. helm2d03: N 400k, nnz 2.7M, TOL = 10 8, symm. Estimate error by repeating computation with tight tolerance; for gr 30 30, by integrating forward and backward. expokit our code time error time error orani bcspwr gr helm2d

15 Heston PDE The Heston PDE is used in mathematical finance to prize European call options with stochastic volatility: u t = 1 2 xy 2 u yy + ρλxyu xy λ2 xu xx + ryu y + κ(η x)u x ru. Use standard second-order finite differences and incorporate boundary conditions to get ODE of the form u = Au + v 1, u(0) = v 0. The matrix A has size 5100 and non-zero elements. The ODE can be solved by evaluating ϕ-functions. In t Hout (2007) advocates the use of ADI. We compare his ADI schemes with the standard matlab routine ode15s and our code.

16 Error vs time maximum error Phipm Crank Nicolson Douglas Craig Hundsdorfer Ode15s cpu time

17 Krylov dimension m and step size τ 55 m x τ t

18 Error estimate (blue) and actual error (red) 9 x error t

19 Allen Cahn in two dimensions u t = α 2 u + u u 3, x, y [0, 1], t [0, 1] Finite differences, Neumann BCs, α = 0.1, grid D Allen Cahn α = 0.1 odeexp5 oderos4 ode15s radau5 exp4 global error cpu time Implicit: ode15s (Matlab), radau5 (Hairer & Wanner) Exponential: odeexp5 (us), oderos4 (Caliari & Ostermann), exp4 (Hochbruck, Lubich & Selhofer)

20 Conclusion EIs are meant for ODEs with stiffness concentrated in the linear part, e.g., semi-discretized semi-linear PDEs. EIs work very well with spectral discretization. Otherwise need to calculate ϕ(a)v challenging if A large. EIs with Krylov subspace methods are competitive with state-of-the-art for mildly stiff problems. Performance degrades for very stiff problems. Current work / future plans: How to combine with finite elements Mu = Lu + N(u)? Krylov subspace may not be best, so look at: Leja point interpolation (Caliari & Ostermann) Rational Krylov (Grimm & Hochbruck) RD-rational approximations (Moret & Novati) Using contour integrals (Schmelzer & Trefethen) Using inverse Laplace transform (López-Fernández)