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This article is cited in 1 scientific paper (total in 1 paper)
MATHEMATICAL MODELING AND NUMERICAL SIMULATION
Application of gradient optimization methods to solve the Cauchy problem for the Helmholtz equation
N. V. Pletneva, P. E. Dvurechenskiib, A. V. Gasnikovacd a Moscow Institute of Physics and Technology,
9 Institutskiy per., Dolgoprudny, Moscow Region, 141701, Russia
b Weierstrass Institute for Applied Analysis and Stochastics,
39 Mohrenstraße, Berlin, 10117, Germany
c Caucasus Mathematical Center, Adyghe State University,
208 Pervomaysk st., Maikop, Adyghe, 385000, Russia
d Institute for Information Transmission Problems of Russian Academy of Sciences,
19/1 Bol’shoy Karetnyy per., Moscow, 212705, Russia
Abstract:
The article is devoted to studying the application of convex optimization methods to solve the Cauchy problem for the Helmholtz equation, which is ill-posed since the equation belongs to the elliptic type. The Cauchy problem is formulated as an inverse problem and is reduced to a convex optimization problem in a Hilbert space. The functional to be optimized and its gradient are calculated using the solution of boundary value problems, which, in turn, are well-posed and can be approximately solved by standard numerical methods, such as finite-difference schemes and Fourier series expansions. The convergence of the applied fast gradient method and the quality of the solution obtained in this way are experimentally investigated. The experiment shows that the accelerated gradient method — the Similar Triangle Method — converges faster than the non-accelerated method. Theorems on the computational complexity of the resulting algorithms are formulated and proved. It is found that Fourier's series expansions are better than finite-difference schemes in terms of the speed of calculations and improve the quality of the solution obtained. An attempt was made to use restarts of the Similar Triangle Method after halving the residual of the functional. In this case, the convergence does not improve, which confirms the absence of strong convexity. The experiments show that the inaccuracy of the calculations is more adequately described by the additive concept of the noise in the first-order oracle. This factor limits the achievable quality of the solution, but the error does not accumulate. According to the results obtained, the use of accelerated gradient optimization methods can be the way to solve inverse problems effectively.
Keywords:
inverse problems, convex optimization, optimization in a Hilbert space, first-order methods, fast gradient method, inexact oracle.
Received: 13.02.2022 Accepted: 13.02.2022
Citation:
N. V. Pletnev, P. E. Dvurechenskii, A. V. Gasnikov, “Application of gradient optimization methods to solve the Cauchy problem for the Helmholtz equation”, Computer Research and Modeling, 14:2 (2022), 417–444
Linking options:
https://www.mathnet.ru/eng/crm976 https://www.mathnet.ru/eng/crm/v14/i2/p417
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