ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2018, Vol. 39, No. 4, pp. 486–493. c  Pleiades Publishing, Ltd., 2018. Parallel CPU- and GPU-Algorithms for Inverse Problems in Nondestructive Testing E. G. Bazulin 1,2* , A. V. Goncharsky 3** , S. Y. Romanov 3*** , and S. Y. Seryozhnikov 3**** (Submitted by V. V. Voevodin) 1 Moscow State University, Moscow, 119991 Russia 2 ECHO+ Ltd, Technopark Strogino, Moscow, 123458 Russia 3 Lomonosov Moscow State University. Vorobyevy Gory 1, 119991 Moscow, Russia Received January 15, 2018 Abstract—This paper is concerned with developing efficient methods for solving inverse problems of ultrasonic nondestructive imaging in the framework of a scalar wave model, which describes the propagation, diffraction and refraction of longitudinal ultrasonic waves. The problem of recovering the velocity of a longitudinal wave in a solid is formulated as a coefficient inverse problem, which in this formulation is nonlinear. The proposed scalable numerical algorithms can be efficiently parallelized both on CPU- and GPU-equipped supercomputers. The efficiency of the algorithms is illustrated by applying them to model problems. The computations were performed on the “Lomonosov” supercomputer at Lomonosov Moscow State University. DOI: 10.1134/S1995080218040030 Keywords and phrases: Nondestructive imaging, inverse problem, tomography, supercomputer, GPU. 1. INTRODUCTION Currently, intensive works are being carried out to develop ultrasonic tomography methods and devices for medical imaging [1–4]. These developments are currently at the stage of prototype devices. The most interesting application of ultrasound tomography in medicine is the differential diagnosis of breast cancer. A characteristic feature of medical ultrasonic imaging is that the speed of sound in soft tissues differs from that in water by less than 10%. This study is dedicated to developing ultrasonic tomographic methods of nondestructive testing (NDT) of solids. Typical NDT tasks include ultrasonic inspection of welds, non-destructive testing of concrete, products made of plastics and composite materials [5,6]. The presence of several types of waves—longitudinal, transverse, and surface waves, is a distinctive feature of ultrasonic waves in solids. Therefore, the most accurate mathematical models of wave propagation are vector models [7]. Attempts have been made to formulate inverse problems of ultrasonic imaging of solids in terms of tensor models [8]. Developing algorithms to solve inverse problems using a tensor formulation appears to be an extremely complicated task. This work is dedicated to developing tomographic methods of nondestructive imaging of solids within the framework of the simplest wave propagation model—the scalar wave model. This approach is applicable if the velocities of longitudinal and transverse waves in the studied material differ significantly from one another, and the longitudinal wave can be separated from the transverse wave using pulse arrival times. * E-mail: bazulin@echoplus.ru ** E-mail: gonchar@srcc.msu.ru *** E-mail: romanov60@gmail.com **** E-mail: s2110sj@gmail.com 486