Conformal Accelerations Method and Efficient Evaluation of Stable Distributions
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Conformal Accelerations Method and Efficient Evaluation of Stable Distributions Svetlana Boyarchenko1 · Sergei Levendorski˘i2
Received: 28 September 2018 / Accepted: 24 February 2020 © Springer Nature B.V. 2020
Abstract We suggest 3 families of conformal deformations and changes of variables for evaluation of integrals arising in applications of the Fourier analysis to fractional partial differential equations and evaluation of special functions, probability distribution functions, cumulative probability distribution functions and quantiles of stable distributions. For the error tolerance E-15, hypergeometric functions can be calculated much faster (in Matlab implementation) than using SFT in Matlab, Python and Mathematica; even when the index α of the stable distribution is small or close to 1, the same error tolerance can be satisfied in 0.005–0.1 msec. For the calculation of quantiles in wide regions in the tails using the Newton or bisection method, it suffices to precompute several hundred values of the characteristic exponent at points of an appropriate grid (conformal principal components) and use these values in formulas for cpdf and pdf. The same three families can be used to evaluate more general distributions and solutions of boundary problems for fractional partial differential equations more general than the ones related to stable distributions. The methods of the paper are applicable to other classes of integrals, highly oscillatory ones especially. Keywords Stable Lévy processes · Fractional partial differential equations · Special functions · Signal processing · Spectral methods · Conformal acceleration · Sinh-acceleration · Conformal principal components · Fourier transforms Mathematics Subject Classification (2010) 26A33 · 35R11 · 65M70 · 65T99 · 60G52 · 65G70 · 33C48 · 42A38
B S. Boyarchenko
[email protected] S. Levendorski˘i [email protected]
1
Department of Economics, The University of Texas at Austin, 2225 Speedway Stop C3100, Austin, TX 78712, USA
2
Calico Science Consulting, Austin, TX, USA
S. Boyarchenko, S. Levendorski˘i
1 Introduction The Fourier-Laplace transform and Wiener-Hopf factorization are ubiquitous in mathematics, physics, mathematical biology, engineering, statistics, and finance. The oscillation and/or slow decay at infinity of integrands in the corresponding formulas makes accurate and fast calculations difficult, especially if the integrands are not smooth. The prominent examples are probability distribution functions (pdf), cumulative probability distribution functions (cpdf) of stable Lévy processes and solutions of boundary problems for the fractional Laplacian and other fractional partial differential equations. As a fairly general example, consider integrals of the form 1 0 eixξ −t (−iμξ +ψ (ξ )) g(ξ ˆ )dξ, (1.1) V (t, x) = 2π R where μ, x ∈ R, t > 0, and (i) ψ 0 and gˆ admit analytic continuation to a cone of the form Cγ − ,γ + := Cγ+− ,γ + ∪ Cγ−− ,γ + , 0
0
0
0
0
0
where Cγ+− ,γ + := {ξ ∈ C | arg ξ ∈ (γ0− , γ0+ )}, Cγ−− ,γ + := {ξ ∈ C | a
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