Speeding up numerical computations via conformal maps

Transcription

1 Speeding up numerical computations via conformal maps Nick Trefethen, Oxford University Thanks to Nick Hale, Nick Higham and Wynn Tee 1/35

2 SIAM 1997 SIAM 2000 Cambridge /35 Princeton 2005 Bornemann et al., SIAM 2004

3 PERIODIC STRIPS, INFINITE STRIPS, AND ELLIPSES Suppose f is analytic, bounded, and 2 -periodic in the strip S a = {z: -a < Im z < a}. a x Sample f in equally spaced points Error in trigonometric interpolation: O(e a/ x ) Error in trapezoid rule quadrature: O(e 2 a/ x ) (Poisson 1826, Davis 1959) If f is nonperiodic on the whole real line (but integrable): Same results under mild assumptions (sinc interpolation) (Turing 1943, Goodwin 1949, Milne 1953, Martensen 1968, Stenger 1970s) 3/35

4 Now suppose f is analytic and bounded in the ellipse E ρ with foci ±1, ρ = semimajor + semiminor axis lengths > sinh(a) ρ = exp(a) cosh(a) Error in polynomial interpolation in Chebyshev or Gauss-Legendre points: O( n ) Error in Gauss quadrature: O( 2n ) (Bernstein 1919) 4/35

5 PLAN OF THE TALK: WE LL APPLY THESE RESULTS TO THREE PROBLEMS, EACH INVOLVING A CONFORMAL CHANGE OF VARIABLES 1. New formulas for quadrature on [ 1,1] 2. Evaluating f(a), A = matrix or operator 3. Tee s adaptive spectral method RELATED TOPICS WE WON T HAVE TIME FOR: 4. Double Exponential quadrature 5. Analytic continuation 6. Inverse Laplace transforms 5/35

6 1. New formulas for quadrature on [ 1,1] JOINT WORK WITH NICK HALE, OXFORD U. SIAM J. Numer. Anal., to appear 6/35

7 1 1 Analyticity in an ellipse is a strange condition. - It entails more smoothness in the middle than near the ends. - A Gauss or Chebyshev grid is /2 times coarser in the middle than an equispaced grid. - pts per wavelength are needed in total to resolve a sine wave. Gauss quad. is /2 times less efficient than the trapezoid rule for periodic integrands. Chebyshev spectral methods need /2 times as many grid points as Fourier spectral methods or in 3D, ( /2) 3 4 times as many. 7/35

8 Q: Where do ellipses come from? 1 1 A: From using polynomials to derive the quadrature formula. Q: Why do we have to use polynomials? A: We don t! 8/35

9 Our solution: conformally map the ρ-ellipse to a region with straighter sides. For example, map it to an infinite strip: s 1 1 Gauss quadrature here g strip is π/2 times narrower than ellipse x 1 1 gives us a non-polynomial transplanted quadrature rule here 1 1 Transplanted integral: f(x) dx = f(g(s)) g (s) ds 1 1 THM: If f is analytic in the strip, the transplanted Gauss formula has error O( ρ~ 2n ) for any ρ ~ < ρ. 9/35

10 Conformal map from ellipse to infinite strip sin 1 tanh 1 sn 10/35

11 GAUSS vs. TRANSPLANTED GAUSS quadrature points (for a typical choice of parameter ρ ) 1 1 N= N= N=64 11/35

12 Convergence for f(x) = 1/(cosh(1) cos(16x)) (analytic in the strip of half-width a = 1/16) error Gauss quadrature Transplanted Gauss quadrature n 12/35

13 Nine more examples (strip map with ρ=1.4) Gauss transplanted Gauss 13/35

14 THEOREMS Standard theorems for Gauss quadrature New theorems for transplanted Gauss quadrature. E.G.: Suppose f is analytic and bounded in the ε-nbhd of [ 1,1] for any ε < 0.05, and we use the ρ=1.1 strip map. THM: Gauss quadrature: error O( (1+ε) 2n ) Transplanted Gauss: error O( (1+ε) 3n ) 14/35

15 A wilder example integrand quadrature error Gauss transplanted Gauss 15/35

16 RELATED WORK Gregory formulas : trapezoid rule with endpoint corrections Bakhvalov 1967: theoretical results on conformal maps & quadrature Kosloff & Tal-Ezer 1993: arcsine change of vars. for spectral methods Beylkin, Boyd, Rokhlin & others: prolate spheroidal wave functions Alpert 1999: hybrid trapezoid/gauss quadrature formulas The last three seem roughly as effective as our method in practice. But they come with no thms about geometric convergence for analytic f. 16/35

17 2. Evaluating f(a), A = matrix or operator JOINT WORK WITH NICK HALE AGAIN AND ALSO NICK HIGHAM, U. OF MANCHESTER SIAM J. Numer. Anal., submitted 17/35

18 Aim: compute f(a), A = operator or large matrix (e.g. of dimension 10 6 ) or f(a)b for various vectors b Examples: A, A, log(a), exp(a),... Applications: anomalous diffusion, finance, semigroups,... Higham has written a book about f(a) problems. 18/35

19 For a scalar a, Cauchy integrals where C encloses a and lies in the region of analyticity of f. For a matrix or operator A, where C encloses spec(a ). If C is a circular contour, equally spaced points should be perfect periodic trapezoid rule! 19/35

20 ASSUMPTIONS f is analytic in the complex plane except (-, 0]. A has spectrum in [m,m], M» m > 0. E.G.: A A log(a) tanh( A ) (A)... singularities of f spectrum of A 0 m M 20/35

21 A BAD IDEA Take the contour C to be a circle surrounding the spectrum. For this you ll need a very large number of sample points:» M/m. Reason: annulus of analyticity is narrow. singularities of f 0 spectrum of A m M Instead we want to map a much thicker annulus onto the WHOLE LIGHT GRAY REGION. 21/35

22 MAP FROM THE ANNULUS (equivalently could use periodic strip) g As always we use a change of variables: f(z) (z-a) -1 dz = f(g(s)) (g(s)-a) -1 g (s) ds 22/35

23 CONFORMAL MAP FROM ANNULUS (plots show the upper half) log sn (Jacobi elliptic function again) Möbius 23/35

24 MATLAB TEST CODE FOR MAP 1, f = % method1.m - evaluate f(a) by contour integral. The functions % ellipkkp and ellipjc are from Driscoll's SC Toolbox. f % change this for another function f A = pascal(6); % change this for another matrix A X = sqrtm(a); % change this if f is not sqrt I = eye(size(a)); e = eig(a); m = min(e); M = max(e); k = (sqrt(m/m)-1)/(sqrt(m/m)+1); L = -log(k)/pi; RESULTS [K,Kp] = ellipkkp(l); for N = 5:5:50 >> method1 t =.5i*Kp - K + (.5:N)*2*K/N; [u,cn,dn] = ellipjc(t,l); z = sqrt(m*m)*((1/k+u)./(1/k-u)); snp = cn.*dn./(1/k-u).^2; S = zeros(size(a)); for j = 1:N S = S + f(z(j))*inv(z(j)*i-a)*snp(j); end S = -4*K*sqrt(m*M)*imag(S)/(k*pi*N); error = norm(s-x)/norm(x); fprintf('%4d %16.12f\n', N, error) end 24/35

25 A more practical example A = negative of discrete Laplacian (sparse, dimension 2500) b = random vector of same dimension Compute A 1/2 b : Contour integral & conformal map: 0.76 secs. on this laptop Matlab sqrtm : 4 min. 48 secs. 25/35

26 Comments about conformal mapping methods for f(a) Further improvements get a further factor of 2 speedup We have reduced f(a)b to a dozen or two backslashes Competitor for small A: Schur reduction, Padé approx. Competitor for large A: Krylov subspace compressions This technique is very general, applicable to many f and A Deeper understanding: link with rational approximation 26/35

27 3. Tee s adaptive spectral method JOINT WORK WITH WYNN TEE, OXFORD DPhil 2007 SIAM J. Sci. Comp., /35

28 This final topic is the most complex. I told Wynn it would never work. But it did! The aim: adaptive spectral method for PDEs for problems with spikes, fronts, rapid variation RELATED WORK Bayliss, Matkowsky and others `87,`89,`90,`92,`95 Guillard and Peyret `88 Augenbaum `89 Kosloff and Tal-Ezer `93 Mulholland, Huang, Sloan, Qiu `97,`98 Weideman `99 Berrut, Baltesnsperger, Mittelmann `00,`01,`02,`04,`05 Good ideas here. But no method that can handle extreme cases. Why not? None of them thought in terms of conformal maps. 28/35

29 Tee s new method combines: 1. Padé/Chebyshev-Padé location of complex singularities 2. Conformal mapping onto domains with slits 3. Spectral differentiation by rational barycentric formulas 29/35

30 At each time step we construct conformal map from ellipse to plane minus slits ending at estimated singularities 30/35

31 Examples of adaptively constructed irregular grids For these computations we achieve 10-digit accuracy with grids of <100 points (spectral in x, 9 th or 13 th order in t) 31/35

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34 demonstrations to 10-digit accuracy with <100 grid points in x burgers allencahn blowup 34/35

35 RECAP OF OUR PROBLEMS 1. New formulas for quadrature on [-1,1] 2. Evaluating f(a), A = matrix or operator 3. Tee s adaptive spectral method MORAL OF THE STORY It s not enough for a grid to look good. It must correspond to a transplantation with a wide region of analyticity. And if it does, you get exponential convergence. 35/35

36 Speeding up numerical computations via conformal maps Nick Trefethen, Oxford University Thanks to Nick Hale, Nick Higham and Wynn Tee 36/35