Bruno, O. P., & Kunyansky, L. A. (2001). A Fast, High-Order Algorithm for the Solution of Surface Scattering Problems: Basic Implementation, Tests, and Applications. Journal of Computational Physics, 169(1), 80-110.
Abstract:
We present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three-dimensional space. This algorithm evaluates scattered fields through fast, high-order solution of the corresponding boundary integral equation. The high-order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities. The acceleration, in turn, results from use of a novel approach which, based on high-order "two-face" equivalent source approximations, reduces the evaluation of far interactions to evaluation of 3-D fast Fourier transforms (FFTs). This approach is faster and substantially more accurate, and it runs on dramatically smaller memories than other FFT and k-space methods. The present algorithm computes one matrix-vector multiplication in O(N6/5logN) to O(N4/3logN) operations, where N is the number of surface discretization points. The latter estimate applies to smooth surfaces, for which our high-order algorithm provides accurate solutions with small values of N; the former, more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling. Further, our approach exhibits super-algebraic convergence; it can be applied to smooth and nonsmooth scatterers, and it does not suffer from accuracy breakdowns of any kind. In this paper we introduce the main algorithmic components in our approach, and we demonstrate its performance with a variety of numerical results. In particular, we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds. © 2001 Academic Press.
Kuchment, P., Kuchment, P., Kunyansky, L., & Kunyansky, L. (2010). Synthetic focusing in ultrasound modulated tomography. Inverse Problems and Imaging, 4(4), 665-673.
Abstract:
Several hybrid tomographic methods utilizing ultrasound modulation have been introduced lately. Success of these methods hinges on the feasibility of focusing ultrasound waves at an arbitrary point of interest. Such focusing, however, is difficult to achieve in practice. We thus propose a way to avoid the use of focused waves through what we call synthetic focusing, i.e. by reconstructing the would-be response to the focused modulation from the measurements corresponding to realistic unfocused waves. Examples of reconstructions from simulated data are provided. This non-technical paper describes only the general concept, while technical details will appear elsewhere. © 2010 American Institute of Mathematical Sciences.
Kunyansky, L. (2015). Inversion of the spherical means transform in corner-like domains by reduction to the classical Radon transform. Inverse Problems, 31(9).
Kuchment, P., & Kunyansky, L. (2008). A survey in mathematics for industry: Mathematics of thermoacoustic tomography. European Journal of Applied Mathematics, 19(2), 191-224.
Abstract:
The article presents a survey of mathematical problems, techniques and challenges arising in thermoacoustic tomography and its sibling photoacoustic tomography. © 2008 Cambridge University Press.
Kunyansky, L. A. (1992). Generalized and attenuated radon transforms: Restorative approach to the numerical inversion. Inverse Problems, 8(5), 809-819.
Abstract:
The problem of the function reconstruction on its line integrals with known weight function is considered. The approach studied consists of treating the attenuated projections by the radon transform inversion formula and considering the result of the inversion as a distorted image. A helpful formula describing the distortion is obtained. The norm of the distortion operator is estimated and several iterative restoration algorithms based on the integral transfers are investigated. The results of the numerical inversion of the attenuated radon transform are presented to demonstrate the features of the algorithms.
