Papers by Vladimir Rokhlin
ArXiv, 2020
In the present paper we describe a method for solving inverse problems for the Helmholtz equation... more In the present paper we describe a method for solving inverse problems for the Helmholtz equation in radially-symmetric domains given multi-frequency data. Our approach is based on the construction of suitable trace formulas which relate the impedance of the total field at multiple frequencies to derivatives of the potential. Using this trace formula we obtain a system of coupled differential equations which can be solved to obtain the potential in a stable manner. Finally, the performance of the reconstruction algorithm is illustrated with several numerical examples.
ArXiv, 2020
In the present paper we describe a method for solving inverse problems for the Helmholtz equation... more In the present paper we describe a method for solving inverse problems for the Helmholtz equation in radially-symmetric domains given multi-frequency data. Our approach is based on the construction of suitable trace formulas which relate the impedance of the total field at multiple frequencies to derivatives of the potential. Using this trace formula we obtain a system of coupled differential equations which can be solved to obtain the potential in a stable manner. Finally, the performance of the reconstruction algorithm is illustrated with several numerical examples.
In the present paper we describe a simple black box algorithm for efficiently and accurately solv... more In the present paper we describe a simple black box algorithm for efficiently and accurately solving scattering problems related to the scattering of time-harmonic waves from radially-symmetric potentials in two dimensions. The method uses FFTs to convert the problem into a set of decoupled second-kind Fredholm integral equations for the Fourier coefficients of the scattered field. Each of these integral equations are solved using scattering matrices, which exploit certain low-rank properties of the integral operators associated with the integral equations. The performance of the algorithm is illustrated with several numerical examples including scattering from singular and discontinuous potentials. Finally, the above approach can be easily extended to time-dependent problems. After outlining the necessary modifications we show numerical experiments illustrating the performance of the algorithm in this setting.
Applied and Computational Harmonic Analysis, 2020
n, where α i are real numbers, and x i are points in a compact interval of R. This expression can... more n, where α i are real numbers, and x i are points in a compact interval of R. This expression can be viewed as representing the electrostatic potential generated by charges on a line in R 3. While fast algorithms for computing the electrostatic potential of general distributions of charges in R 3 exist, in a number of situations in computational physics it is useful to have a simple and extremely fast method for evaluating the potential of charges on a line; we present such a method in this paper, and report numerical results for several examples. 2010 Mathematics Subject Classification. 31C20 (primary) and 41A55, 41A50 (secondary). Key words and phrases. Fast multipole method, Chebyshev system, generalized Gaussian quadrature. N.F.M. was supported in part by NSF DMS-1903015. V.R. was supported in part by AFOSR FA9550-16-1-0175 and ONR N00014-14-1-0797.
Journal of Mathematical Physics, 2017
We present a fast summation method for lattice sums of the type which arise when solving wave sca... more We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.
SIAM Journal on Scientific and Statistical Computing
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Given an m×n matrix A and a positive integer k, we describe a randomized procedure for the approx... more Given an m×n matrix A and a positive integer k, we describe a randomized procedure for the approximation of A with a matrix Z of rank k. The procedure relies on applying A T to a collection of l random vectors, where l is an integer equal to or slightly greater than k; the scheme is efficient whenever A and A T can be applied rapidly to arbitrary vectors. The discrepancy between A and Z is of the same order as √ lm times the (k + 1) st greatest singular value σ k+1 of A, with negligible probability of even moderately large deviations. The actual estimates derived in the paper are fairly complicated, but are simpler when l − k is a fixed small nonnegative integer. For example, according to one of our estimates for l − k = 20, the probability that the spectral norm A − Z is greater than 10 p (k + 20) m σ k+1 is less than 10 −17 . The paper contains a number of estimates for A − Z , including several that are stronger (but more detailed) than the preceding example; some of the estimates are effectively independent of m. Thus, given a matrix A of limited numerical rank, such that both A and A T can be applied rapidly to arbitrary vectors, the scheme provides a simple, efficient means for constructing an accurate approximation to a singular value decomposition of A. Furthermore, the algorithm presented here operates reliably independently of the structure of the matrix A. The results are illustrated via several numerical examples.
We introduce a new approach to the rapid numerical application to arbitrary vectors of certain ty... more We introduce a new approach to the rapid numerical application to arbitrary vectors of certain types of linear operators. Inter alia, our scheme is applicable to many classical integral transforms, and to the expansions associated with most families of classical special functions; among the latter are Bessel functions, Legendre, Hermite, and Laguerre polynomials, Spherical Harmonics, Prolate Spheroidal Wave functions, and a number of others. In all these cases, the CPU time requirements of our algorithm are of the order O(n log n), where n is the size of the matrix to be applied. The performance of our algorithm is illustrated via a number of numerical examples.
I Remark 1.2 The Lagrange interpolation formula has traditionally been less favored for practical... more I Remark 1.2 The Lagrange interpolation formula has traditionally been less favored for practical calculations than other classical methods (see, for example, ). However, the algorithms of this paper are numerically stable and very efficient, as demonstrated by our numerical experiments, thus affording the Lagrange formula a substantial advantage over other techniques for the manipulation of polynomials.
We introduce a new class of methods for the Cauchy problem for ordinary differential equations (O... more We introduce a new class of methods for the Cauchy problem for ordinary differential equations (ODEs). We begin by converting the original ODE into the corresponding Picard equation and apply a deferred correction procedure in the integral formulation, driven by either the explicit or the implicit Euler marching scheme. The approach results in algorithms of essentially arbitrary order accuracy for both non-stiff and stiff problems; their performance is illustrated with several numerical examples. For nonstiff problems, the stability behavior of the obtained explicit schemes is very satisfactory and algorithms with orders between 8 and 20 should be competitive with the best existing ones. In our preliminary experiments with stiff problems, a simple adaptive implementation of the method demonstrates performance comparable to that of a stateof-the-art extrapolation code (at least, at moderate to high precision).
Seg Technical Program Expanded Abstracts, 1999
Nasa Sti Recon Technical Report N, Sep 30, 1994
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Papers by Vladimir Rokhlin