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On numerical integration of isotropic singularities in $\mathbb R^d$
Keywords: numerical integration; algebraic singularity; approximation on the semiaxis
Thu, 11:10--11:35
  • Chernov, Alexey (Hausdorff Center for Mathematics; Institute for Numerical Simulation, University of Bonn, Germany)

We consider the integrals of the type \begin{equation*} I_1(R) = \int_{\mathbb R^d} \frac{F(x)}{\|x-R\|^\alpha} \, dx,\qquad I_2 = \int_{\mathbb R^d}\int_{\mathbb R^d} \frac{G(x,y)}{\|x-y\|^\alpha} \, dy dx \end{equation*} where $F$ and $G$ are smooth and exponentially decaying at infinity and $\alpha<d$, and introduce a family of quadrature rules for approximate computation of $I_1(R)$, $I_2$. Important special cases of integrals of this type are e.g. one- and two-electron-like integrals with application in Quantum Chemistry \begin{equation*} J_1(R) = \int_{\mathbb R^3} \frac{e^{-a\|x-R_a\|^2}}{\|x-R\|}f(x) ,\quad J_2 = \int_{\mathbb R^3}\int_{\mathbb R^3} \frac{e^{-a\|x-R_a\|^2}e^{-b\|x-R_b\|^2}}{\|x-y\|} f(x)g(y). \end{equation*} In this case, analytic integration is possible if $f=1=g$; if $f$, $g$ are polynomials the evaluation can be performed via recursive formulae [2]. Similar integrals appear in pricing of options on Lévy driven assets: \begin{equation*} \mathcal A_J[\varphi](x) = -\int_{\mathbb R^d}(\varphi(x+z) - \varphi(x)-z\cdot\nabla \varphi(x)) k(z) \, dz, \quad k(z) = \frac{e^{-M\|z\|}}{\|z\|^\alpha}. \end{equation*} Here the integral kernel $k(z)$ corresponds to a tempered process, which can be viewed as an isotropic generalization of the CGMY process to $d \geq 2$ [3] and $\alpha$ is not necessarily an integer.

For numerical computation of $I_1(R), I_2$ we introduce a family of product-type quadrature rules and study its convergence properties. The construction is based on a change of coordinates moving the singularity to the origin and involves suitable tensor-product rules in the radial and angular directions. This approach is related to [1], where the numerical integration over two regular simplices has been studied, in contrast to the present case of unbounded domains.

We illustrate the performance of the suggested quadrature rules on several numerical examples and give preliminary convergence estimates for the quadrature error under suitable smoothness and decay assumptions on the integrands.

[1] Alexey Chernov, Tobias von Petersdorff, and Christoph Schwab. Exponential convergence of $hp$ quadrature for integral operators with Gevrey kernels. M2AN Math. Model. Numer. Anal., 45:387–422, 2011.
[2] Larry E. McMurchie and Ernest R. Davidson. One- and two-electron integrals over cartesian gaussian functions. J. Computational Phys., 26:218–231, 1978.
[3] Ricardo H. Nochetto, Tobias von Petersdorff, and Chen-Song Zhang. A posteriori error analysis for a class of integral equations and variational inequalities. Numer. Math., 116(3):519–552, 2010.