Optimization of numerical algorithms for solving inverse problems of ultrasonic tomography on a supercomputer

Abstract

The paper is dedicated to optimizing numerical algorithms to solve wave tomography problems by using supercomputers. The problem is formulated as a non-linear coefficient inverse problem for the wave equation. Due to the huge amount of computations required, solving such problems is impossible without the use of high-performance supercomputers. Gradient iterative methods are employed to solve the problem. The gradient of the residual functional is calculated from the solutions of the direct and the “conjugate” wave-propagation problems with transparent boundary conditions. Two formulations of the transparency condition are compared. We show that fourth-order finite-difference schemes allow us to reduce the size of the grid by a factor of 1.5–2 in each coordinate compared to second-order schemes. This makes it possible to significantly reduce the amount of computations and memory required, which is especially important for 3D problems of wave tomography. The primary application of the method is medical ultrasonic tomography.