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Vol 211, No 4 (2020)
- Year: 2020
- Articles: 6
- URL: https://ogarev-online.ru/0368-8666/issue/view/7464
The gradient projection algorithm for a proximally smooth set and a function with Lipschitz continuous gradient
Abstract
We consider the minimization problem for a nonconvex function with Lipschitz continuous gradient on a proximally smooth (possibly nonconvex) subset of a finite-dimensional Euclidean space. We introduce the error bound condition with exponent $\alpha\in(0,1]$ for the gradient mapping. Under this condition, it is shown that the standard gradient projection algorithm converges to a solution of the problem linearly or sublinearly, depending on the value of the exponent $\alpha$. This paper is theoretical. Bibliography: 23 titles.
Matematicheskii Sbornik. 2020;211(4):3-26
3-26
Optimal boundary control of nonlinear-viscous fluid flows
Abstract
The optimal control problem for a stationary model of a nonlinear-viscous incompressible fluid flowing through a bounded domain is considered under the wall slip condition. As a control parameter, the dynamic pressure at the in-flow and out-flow parts of the boundary is used. Using methods of the theory of pseudomonotone mappings, the existence of a weak solution (a velocity–dynamic pressure pair) minimizing a given cost functional is proved. The behaviour of solutions and optimal values of the cost functional are studied when the set of admissible controls varies. In particular, it is shown that the marginal function of this control system is lower semicontinuous. Bibliography: 23 titles.
Matematicheskii Sbornik. 2020;211(4):27-43
27-43
The wave model of a metric space with measure and an application
Abstract
Let $(\Omega,d)$ be a complete metric space and let $\mu$ be a Borel measure on $\Omega$. Under certain fairly general assumptions about the metric and the measure, we use lattice theory to construct an isometric copy $(\widetilde\Omega,\widetilde d)$ of the space $(\Omega,d)$, which is called its wave model. The construction is motivated by applications to inverse problems of mathematical physics. We show how the wave model solves the problem of reconstructing a Riemannian manifold with boundary from its spectral data. Bibliography: 13 titles.
Matematicheskii Sbornik. 2020;211(4):44-62
44-62
Boundary behaviour of open discrete mappings on Riemannian manifolds. II
Abstract
The boundary behaviour of classes of ring mappings, which generalize quasiconformal mappings in the sense of Gehring, is under investigation. Theorems proving that they have continuous boundary extensions are established in terms of prime ends of regular domains. Results on the equicontinuity of mappings in these classes in the closure of a fixed domain are also established in these terms. Bibliography: 45 titles.
Matematicheskii Sbornik. 2020;211(4):63-111
63-111
112-122
123-144