General problems of regularizability

A sequential minimisation algorithm for the numerical solution of inverse problems of frequency sounding is presented. The algorithm is based on the concept of convexification of a multiextremal objective function proposed recently by the authors. A key point in the sequential minimisation algorithm is that unlike conventional layerstripping algorithms, it provides the stable approximate solution via minmisation of a finite sequence of strictly convex objective functions resulted from applying the nonlinear weighet least squares method with Carleman's weight functions. Another advantage of the proposed algorithm is that the starting vectors for the descent methods of minimisation are directly determined from the data eliminating the uncertainty inherent to the local methods, such as the gradient or Newton-like methods. The 1-D inverse model of frequency sounding is selected to demonstrate its computational feasibility. Based on the localising property of Carleman's weight functions, it is proven that the distance between the approximate and "exact" solutions is small if the approximation error is small. The computational experiments with several realistic and synthetic marine shallow water configurations are presented to demonstrate the computational feasibility of the proposed algorithm.

Section: Inversion Of Incomplete Datamentioning

confidence: 99%

A sequential minimization algorithm based on the convexification approach

Klibanov¹,

Timonov²

2003

“…The inverse problems considered are ill-posed, and therefore regularization terms can be added to functionals (4) and (8). However, we can do without such terms in these functionals because the iterative methods used to solve the corresponding problems have, in a sense, certain regularizing properties [15]. We now write out the formulas for the gradients of residual functionals (4) and (8).…”

Section: Base Model Of Wave Propagation In Attenuating Mediamentioning

confidence: 99%

“…The first approach is based on the use of Green functions. In this case, inverse problems of ultrasound tomography can be reduced to a set of nonlinear Fredholm integral equations of the first kind [15]. The resulting problem is ill-posed.…”

Section: Introductionmentioning

confidence: 99%

“…The resulting problem is ill-posed. Methods of solution of linear and nonlinear inverse problems represented by operator equations are well known and efficient procedures have been developed to solve them [15,16]. This representation is naturally suitable for studying the capabilities of various linear approximations to the nonlinear problem of ultrasound tomography.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Supercomputer technologies in inverse problems of ultrasound tomography

Goncharsky

Romanov

2013

This study focuses on the development of efficient methods for solving inverse problems of ultrasound tomography as coefficient inverse problems for the wave equation. The inverse problem consists in finding the unknown wave propagation velocity as a function of coordinates in three-dimensional space. Efficient iterative methods are proposed for solving the inverse problem based on a direct computation of the residual functional. One of the most promising directions of ultrasound tomography is the development of ultrasound tomographs for medical research, and primarily for the differential diagnosis of breast cancer. From a medical viewpoint, diagnostic facilities for the differential cancer diagnosis should have a resolution of 3 mm or better. Because of this requirement, inverse problems of ultrasound tomography have to be solved on dense grids with sizes of up to 1000 × 1000 on cross sections of three-dimensional objects studied. Supercomputers are needed to address such inverse problems in terms of the wave model described by second-order hyperbolic equations. The algorithms developed in this study are easily scalable on supercomputers running up to several tens of thousands of processes. The problem of choosing the initial approximation for iterative algorithms when solving the inverse problem has been studied.

“…In accordance with the theory of ill-posed problems (see, e.g. [1]), the regularization consists of selecting an element of a minimizing sequence, so that the norm of the residual does not exceed the prescribed level of perturbation. The feasibility of the proposed method is demonstrated in numerical experiments.…”

Section: Introductionmentioning

confidence: 99%

A regularized weighted least gradient problem for conductivity imaging

Tamasan¹,

Timonov²

2019

We propose and study a regularization method for recovering an approximate electrical conductivity solely from the magnitude of one interior current density field. Without some minimal knowledge of the boundary voltage potential, the problem has been recently shown to have nonunique solutions, thus recovering the exact conductivity is impossible. The method is based on solving a weighted least gradient problem in the subspace of functions of bounded variations with square integrable traces. The computational effectiveness of this method is demonstrated in numerical experiments.