Skip to main content Accessibility help


  • Flavia Lanzara (a1) and Gunther Schmidt (a2)


We study a fast method for computing potentials of advection–diffusion operators $-{\rm\Delta}+2\mathbf{b}\boldsymbol{\cdot }{\rm\nabla}+c$ with $\mathbf{b}\in \mathbb{C}^{n}$ and $c\in \mathbb{C}$ over rectangular boxes in $\mathbb{R}^{n}$ . By combining high-order cubature formulas with modern methods of structured tensor product approximations, we derive an approximation of the potentials which is accurate and provides approximation formulas of high order. The cubature formulas have been obtained by using the basis functions introduced in the theory of approximate approximations. The action of volume potentials on the basis functions allows one-dimensional integral representations with separable integrands, i.e. a product of functions depending on only one of the variables. Then a separated representation of the density, combined with a suitable quadrature rule, leads to a tensor product representation of the integral operator. Since only one-dimensional operations are used, the resulting method is effective also in the high-dimensional case.



Hide All
1.Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions, Dover (1968).
2.Beylkin, G. and Mohlenkamp, M. J., Numerical operator calculus in higher dimensions. Proc. Natl. Acad. Sci. USA 99 2002, 1024610251.
3.Beylkin, G. and Mohlenkamp, M. J., Algorithms for numerical analysis in high dimensions. SIAM J. Sci. Comput. 26 2005, 21332159.
4.Hestenes, M. R., Extension of the range of differentiable functions. Duke Math. J. 8 1941, 183192.
5.John, F., Partial Differential Equations, 4th edn., Springer (1982).
6.Khoromskij, B. N., Fast and accurate tensor approximation of a multivariate convolution with linear scaling in dimension. J. Comput. Appl. Math. 234 2010, 31223139.
7.Lanzara, F., Maz’ya, V. and Schmidt, G., On the fast computation of high dimensional volume potentials. Math. Comp. 80 2011, 887904.
8.Lanzara, F., Maz’ya, V. and Schmidt, G., Accuracy cubature of volume potentials over high-dimensional half-spaces. J. Math. Sci. 173 2011, 683700.
9.Lanzara, F., Maz’ya, V. and Schmidt, G., Fast cubature of volume potentials over rectangular domains by approximate approximations. Appl. Comput. Harmon. Anal. 36 2014, 167182.
10.Maz’ya, V., Approximate Approximations (The Mathematics of Finite Elements and Applications. Highlights 1993) (ed. Whiteman, J. R.), Wiley (Chichester, 1994), 77104.
11.Maz’ya, V., Sobolev Spaces with Applications to Elliptic Partial Differential Equations, Vol. 342, Springer (2011).
12.Maz’ya, V. and Schmidt, G., Approximate Approximations (Mathematical Surveys and Monographs 141), American Mathematical Society (Providence, RI, 2007).
13.Takahasi, H. and Mori, M., Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci. Kyoto Univ. 9 1974, 721741.
14.Waldvogel, J., Towards a general error theory of the trapezoidal rule. Springer Optim. Appl. 42 2011, 267282.
MathJax is a JavaScript display engine for mathematics. For more information see

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed