speeding up numerical computations via conformal...
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![Page 1: Speeding up numerical computations via conformal mapspeople.maths.ox.ac.uk/trefethen/mappings.pdf · Speeding up numerical computations via conformal maps Nick Trefethen, Oxford University](https://reader036.vdocuments.site/reader036/viewer/2022062317/5ae76b377f8b9a8b2b8e96e3/html5/thumbnails/1.jpg)
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
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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)
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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)
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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:
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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
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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.”
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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!
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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
~ ~
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Conformal map from ellipse to infinite strip
sin−1 tanh−1
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sin−1
sn
tanh
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GAUSS vs. TRANSPLANTED GAUSS quadrature points
(for a typical choice of parameter ρ )
N=16
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N=32
N=64
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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
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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 )
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Gauss
A wilder example
integrand quadrature error
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transplantedGauss
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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.
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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
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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.
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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!
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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
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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.
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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:
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CONFORMAL MAP FROM ANNULUS(plots show the upper half)
log
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sn (Jacobi elliptic function again)
Möbius
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% 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
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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.
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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
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3. Tee’s adaptive spectral method
JOINT WORK WITH WYNN TEE, OXFORD DPhil 2007
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SIAM J. Sci. Comp., 2006
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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.
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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
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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.
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Speeding up numerical computationsvia conformal maps
Nick Trefethen, Oxford University
Thanks to Nick Hale,Nick Higham and Wynn Tee
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