Speeding up numerical computationsvia conformal maps

Nick Trefethen, Oxford University

Thanks to Nick Hale,Nick Higham and Wynn Tee

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SIAM 1997 SIAM 2000 Cambridge 2003

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Princeton 2005 Bornemann et al., SIAM 2004

Suppose f is analytic, bounded, and 2-periodicin the strip Sa = {z: -a < Im z < a} .

Sample f in equally spaced points

x

a

PERIODIC STRIPS, INFINITE STRIPS, AND ELLIPSES

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Sample f in equally spaced points

Error in trigonometric interpolation: O(ea/x)

Error in trapezoid rule quadrature: O(e2a/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)

Now suppose f is analytic and bounded in the ellipse Eρwith foci ±1, ρ = semimajor + semiminor axis lengths > 1.

cosh(a)

sinh(a)

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ρ = exp(a)

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cosh(a)

Error in polynomial interpolation inChebyshev or Gauss-Legendre points: O(n)

Error in Gauss quadrature: O(2n)

(Bernstein 1919)

1. New formulas for quadrature on [−1,1]

2. Evaluating f(A), A = matrix or operator

3. Tee’s adaptive spectral method

PLAN OF THE TALK:

WE’LL APPLY THESE RESULTS TO THREE PROBLEMS,EACH INVOLVING A CONFORMAL CHANGE OF VARIABLES

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4. Double Exponential quadrature

5. Analytic continuation

6. Inverse Laplace transforms

RELATED TOPICS WE WON’T HAVE TIME FOR:

1. New formulas for quadrature on [−1,1]

JOINT WORK WITH NICK HALE, OXFORD U.

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SIAM J. Numer. Anal., to appear

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 middlethan an equispaced grid.

1 1

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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 rulefor periodic integrands.”

“Chebyshev spectral methods need /2 times as many grid pointsas Fourier spectral methods — or in 3D, (/2)34 times as many.”

1 1

Q: Where do ellipses come from?

A: From using polynomials to derive the quadrature formula.

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Q: Why do we have to use polynomials?

A: We don’t!

Our solution: conformally map the ρ-ellipse to a region withstraighter sides. For example, map it to an infinite strip:

1 11 1

gsx

Gauss quadrature here… …gives us a non-polynomialtransplanted quadrature rule here

strip is π/2 times narrower than ellipse

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Transplanted integral: f(x) dx = f(g(s)) g’(s) ds1

1

1

1

THM: If f is analytic in the strip, the transplantedGauss formula has error O( ρ 2n ) for any ρ < ρ .

transplanted quadrature rule here

~ ~

Conformal map from ellipse to infinite strip

sin−1 tanh−1

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sin−1

sn

tanh

GAUSS vs. TRANSPLANTED GAUSS quadrature points

(for a typical choice of parameter ρ )

N=16

1 1

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N=32

N=64

1 1

1 1

Convergence for f(x) = 1/(cosh(1)cos(16x))

(analytic in the strip of half-width a = 1/16)

error

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Gauss quadrature

TransplantedGauss quadrature

n

error

Nine more examples (strip map with ρ=1.4)

Gauss

transplantedGauss

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Standard theorems for Gauss quadrature New theorems for transplanted Gauss quadrature.

E.G.: Suppose f is analytic and bounded in the ε-nbhdof [−1,1] for any ε < 0.05, and we use the ρ=1.1 strip map.

THEOREMS

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THM: Gauss quadrature: error O( (1+ε)−2n )

Transplanted Gauss: error O( (1+ε)−3n )

Gauss

A wilder example

integrand quadrature error

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transplantedGauss

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

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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.

2. Evaluating f(A), A = matrix or operator

JOINT WORK WITH NICK HALE AGAIN AND ALSO NICK HIGHAM, U. OF MANCHESTER

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SIAM J. Numer. Anal., submitted

Aim: compute f(A) , A = operator or large matrix(e.g. of dimension 106)

or f(A)b for various vectors b

Examples: A , A , log(A) , exp(A) , . . .

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Examples: A , A , log(A) , exp(A) , . . .

Applications: anomalous diffusion, finance, semigroups, . . .

Higham has written a book about f(A) problems.

where C encloses a and liesin the region of analyticity of f .

For a matrix or operator A ,

For a scalar a ,

Cauchy integrals

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where C encloses spec(A ).

For a matrix or operator A ,

If C is a circular contour,equally spaced pointsshould be perfect —periodic trapezoid rule!

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)...

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0 m M

singularities of f spectrum of A

A BAD IDEA

Take the contour C to be a circle surrounding the spectrum.

For this you’ll need a very large numberof sample points: » M/m .

Reason: annulus of analyticity is narrow.Insteadwe wantto mapa much

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0

singularities of f spectrum of A

m M

a muchthickerannulusonto theWHOLELIGHTGRAYREGION.

MAP FROM THE ANNULUS(equivalently could use periodic strip)

g

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f(z) (z-A)-1 dz = f(g(s)) (g(s)-A)-1 g’(s) ds

As always we use a change of variables:

CONFORMAL MAP FROM ANNULUS(plots show the upper half)

log

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sn (Jacobi elliptic function again)

Möbius

% method1.m - evaluate f(A) by contour integral. The functions% ellipkkp and ellipjc are from Driscoll's SC Toolbox.

f = @sqrt; % change this for another function fA = pascal(6); % change this for another matrix AX = sqrtm(A); % change this if f is not sqrtI = 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;[K,Kp] = ellipkkp(L);for N = 5:5:50

MATLAB TEST CODE FOR MAP 1 , f =

>> method1

RESULTS

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for N = 5:5:50t = .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:NS = S + f(z(j))*inv(z(j)*I-A)*snp(j);

endS = -4*K*sqrt(m*M)*imag(S)/(k*pi*N);error = norm(S-X)/norm(X);fprintf('%4d %16.12f\n', N, error)

end

>> method15 5.983430140320

10 0.37194156608715 0.01748713246020 0.00074193428025 0.00002971644430 0.00000114669035 0.00000004310840 0.00000000159045 0.00000000005850 0.000000000002

A more practical example

A = negative of 5050 discrete Laplacian (sparse, dimension 2500)

b = random vector of same dimension

Compute A1/2 b :

Contour integral & conformal map: 0.76 secs. on this laptop

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Matlab “sqrtm”: 4 min. 48 secs.

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

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• Competitor for large A: Krylov subspace compressions

• This technique is very general, applicable to many f and A

• Deeper understanding: link with rational approximation

3. Tee’s adaptive spectral method

JOINT WORK WITH WYNN TEE, OXFORD DPhil 2007

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SIAM J. Sci. Comp., 2006

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 WORKBayliss, Matkowsky and others `87,`89,`90,`92,`95Guillard and Peyret `88Augenbaum `89

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Augenbaum `89Kosloff and Tal-Ezer `93Mulholland, Huang, Sloan, Qiu `97,`98Weideman `99Berrut, 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.

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

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At each time step we construct conformal map

from ellipse… …to plane minus slits endingat estimated singularities

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Examples of adaptively constructed irregular grids

For these computations we achieve 10-digit accuracy with gridsof <100 points (spectral in x, 9th or 13th order in t)

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demonstrations to 10-digit accuracywith <100 grid points in x

burgersallencahnblowup

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blowup

1. New formulas for quadrature on [-1,1]

2. Evaluating f(A), A = matrix or operator

3. Tee’s adaptive spectral method

RECAP OF OUR PROBLEMS

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MORAL OF THE STORY

It’s not enough for a grid to “look good”.

It must correspond to a transplantationwith a wide region of analyticity. And if it

does, you get exponential convergence.

Speeding up numerical computationsvia conformal maps

Nick Trefethen, Oxford University

Thanks to Nick Hale,Nick Higham and Wynn Tee

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