# Showing papers in "Computational Mathematics and Mathematical Physics in 2009"

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TL;DR: A parallel method is implemented in C-MPI for the global minimization of functions whose gradient satisfies the Lipschitz condition and the performance of the algorithm is demonstrated using the calculation of the structure of a protein molecule as an example.

Abstract: On the basis of the method of nonuniform coverings, a parallel method for the global optimization of Lipschitzian functions is developed. This method is implemented in C-MPI for the global minimization of functions whose gradient satisfies the Lipschitz condition. The performance of the algorithm is demonstrated using the calculation of the structure of a protein molecule as an example.

41 citations

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TL;DR: In this paper, a high-order accurate method for analyzing two-dimensional rarefied gas flows is proposed on the basis of a nonstationary kinetic equation in arbitrarily shaped regions.

Abstract: A high-order accurate method for analyzing two-dimensional rarefied gas flows is proposed on the basis of a nonstationary kinetic equation in arbitrarily shaped regions. The basic idea behind the method is the use of hybrid unstructured meshes in physical space. Special attention is given to the performance of the method in a wide range of Knudsen numbers and to accurate approximations of boundary conditions. Examples calculations are provided.

33 citations

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TL;DR: In this paper, the authors studied a bilevel facility location problem in which the clients choose suppliers based on their own preferences and showed that the coopertative and anticooperative statements can be reduced to a particular case in which every client has a linear preference order on the set of facilities to be opened.

Abstract: A bilevel facility location problem in which the clients choose suppliers based on their own preferences is studied. It is shown that the coopertative and anticooperative statements can be reduced to a particular case in which every client has a linear preference order on the set of facilities to be opened. For this case, various reductions of the bilevel problem to integer linear programs are considered. A new statement of the problem is proposed that is based on a family of valid inequalities that are related to the problem on a pair of matrices and the set packing problem. It is shown that this formulation is stronger than the other known formulations from the viewpoint of the linear relaxation and the integrality gap.

30 citations

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TL;DR: The discrete extremal problems to which certain problems of searching for subsets of vectors and cluster analysis are reduced are proved to be NP-complete as discussed by the authors, which is the case for the problem of clustering.

Abstract: The discrete extremal problems to which certain problems of searching for subsets of vectors and cluster analysis are reduced are proved to be NP-complete.

27 citations

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TL;DR: In this article, two classes of competitive facility location models are considered, in which several persons (players) sequentially or simultaneously open facilities for serving clients, and the tight PLS-completeness of the problem of finding Nash equilibriums is proved.

Abstract: Two classes of competitive facility location models are considered, in which several persons (players) sequentially or simultaneously open facilities for serving clients. The first class consists of discrete two-level programming models. The second class consists of game models with several independent players pursuing selfish goals. For the first class, its relationship with pseudo-Boolean functions is established and a novel method for constructing a family of upper and lower bounds on the optimum is proposed. For the second class, the tight PLS-completeness of the problem of finding Nash equilibriums is proved.

25 citations

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TL;DR: In this paper, a fast algorithm for solving the N-body problem arising in flow simulation when the flow is represented as a set of many interacting vortex elements is proposed, which is used to compute the flow over a circular cylinder at high Reynolds numbers.

Abstract: A fast algorithm is proposed for solving the N-body problem arising in flow simulation when the flow is represented as a set of many interacting vortex elements. The algorithm is used to compute the flow over a circular cylinder at high Reynolds numbers.

25 citations

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TL;DR: In this paper, a harmonic mapping of a parametric domain with a given non-degenerate grid onto a physical domain is used to generate structured difference grids in two-dimensional nonconvex domains.

Abstract: Generation of structured difference grids in two-dimensional nonconvex domains is considered using a mapping of a parametric domain with a given nondegenerate grid onto a physical domain. For that purpose, a harmonic mapping is first used, which is a diffeomorphism under certain conditions due to Rado’s theorem. Although the harmonic mapping is a diffeomorphism, its discrete implementation can produce degenerate grids in nonconvex domains with highly curved boundaries. It is shown that the degeneration occurs due to approximation errors. To control the coordinate lines of the grid, an additional mapping is used and universal elliptic differential equations are solved. This makes it possible to generate a nondegenerate grid with cells of a prescribed shape.

23 citations

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TL;DR: In this article, a minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows is described, which is explicit, conservative, and second-order accurate in space and time.

Abstract: A minimum-stencil difference scheme for computing two-dimensional axisymmetric gas flows is described. The scheme is explicit, conservative, and second-order accurate in space and time. The numerical results obtained for pulsating flows and contact discontinuity instabilities are discussed. The mechanisms of flow pulsation and instability generation are described.

22 citations

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TL;DR: In this paper, a nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered, and a necessary optimality condition is formulated in the form of a smooth maximum principle.

Abstract: A nonlinear optimal impulsive control problem with trajectories of bounded variation subject to intermediate state constraints at a finite number on nonfixed instants of time is considered. Features of this problem are discussed from the viewpoint of the extension of the classical optimal control problem with the corresponding state constraints. A necessary optimality condition is formulated in the form of a smooth maximum principle; thorough comments are given, a short proof is presented, and examples are discussed.

21 citations

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TL;DR: In this paper, a nonlinear eigenvalue problem related to determining the stress and strain fields near the tip of a transverse crack in a power-law material is studied, where the eigenvalues are found by a perturbation method based on representations of an eigen value, the corresponding eigen function, and the material nonlinearity parameter in the form of series expansions in powers of a small parameter.

Abstract: A nonlinear eigenvalue problem related to determining the stress and strain fields near the tip of a transverse crack in a power-law material is studied. The eigenvalues are found by a perturbation method based on representations of an eigenvalue, the corresponding eigenfunction, and the material nonlinearity parameter in the form of series expansions in powers of a small parameter equal to the difference between the eigenvalues in the linear and nonlinear problems. The resulting eigenvalues are compared with the accurate numerical solution of the nonlinear eigenvalue problem.

21 citations

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TL;DR: In this article, the optimal design of topologies for mechanical structures can be reduced to problems of vanishing constraints, where some constraints must hold in certain regions of the corresponding space rather than everywhere.

Abstract: A new class of optimization problems is discussed in which some constraints must hold in certain regions of the corresponding space rather than everywhere. In particular, the optimal design of topologies for mechanical structures can be reduced to problems of this kind. Problems in this class are difficult to analyze and solve numerically because their constraints are usually irregular. Some known first- and second-order necessary conditions for local optimality are refined for problems with vanishing constraints, and special Newton-type methods are developed for solving such problems.

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TL;DR: In this paper, a relationship between weighted pseudoinverses and weighted normal pseudosolutions is found, and iterative methods for calculating both pseudo-invariant and pseudosolution are constructed.

Abstract: Weighted pseudoinverses with singular weights can be defined by a system of matrix equations. For one of such definitions, necessary and sufficient conditions are given for the corresponding system to have a unique solution. Representations of the pseudoinverses in terms of the characteristic polynomials of symmetrizable and symmetric matrices, as well as their expansions in matrix power series or power products, are obtained. A relationship is found between the weighted pseudoinverses and the weighted normal pseudosolutions, and iterative methods for calculating both pseudoinverses and pseudosolutions are constructed. The properties of the weighted pseudoinverses with singular weights are shown to extend the corresponding properties of weighted pseudoinverses with positive definite weights.

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TL;DR: In this paper, a unified numerical algorithm based on the quasigasdynamic system of equations was proposed to solve ten well-known one-dimensional test problems reflecting the characteristic features of unsteady inviscid gas flows.

Abstract: Ten well-known one-dimensional test problems reflecting the characteristic features of unsteady inviscid gas flows are successfully solved by a unified numerical algorithm based on the quasigasdynamic system of equations. In all the cases, the numerical solution converges to a self-similar one as the spatial grid is refined.

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TL;DR: Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed in this paper.

Abstract: Parallel versions of a method based on reducing a linear program (LP) to an unconstrained maximization of a concave differentiable piecewise quadratic function are proposed. The maximization problem is solved using the generalized Newton method. The parallel method is implemented in C using the MPI library for interprocessor data exchange. Computations were performed on the parallel cluster MVC-6000IM. Large-scale LPs with several millions of variables and several hundreds of thousands of constraints were solved. Results of uniprocessor and multiprocessor computations are presented.

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TL;DR: In this paper, the third boundary value problem for a loaded heat equation in a p-dimensional parallelepiped is considered, and an a priori estimate for the solution to a locally one-dimensional scheme is derived, and the convergence of the scheme is proved.

Abstract: The third boundary value problem for a loaded heat equation in a p-dimensional parallelepiped is considered. An a priori estimate for the solution to a locally one-dimensional scheme is derived, and the convergence of the scheme is proved.

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TL;DR: In this article, the Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied.

Abstract: The Cauchy problem for a fourth-order pseudoparabolic equation describing liquid filtration problems in fissured media, moisture transfer in soil, etc., is studied. Under certain summability and boundedness conditions imposed on the coefficients, the operator of this problem and its adjoint operator are proved to be homeomorphism between certain pairs of Banach spaces. Introduced under the same conditions, the concept of a θ-fundamental solution is introduced, which naturally generalizes the concept of the Riemann function to the equations with discontinuous coefficients; the new concept makes it possible to find an integral form of the solution to a nonhomogeneous problem.

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TL;DR: In this article, the weighted least squares problem is considered and error estimates for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix.

Abstract: The weighted least squares problem is considered. Given a generally inconsistent system of linear algebraic equations, error estimates are obtained for its weighted minimum-norm least squares solution under perturbations of the matrix and the right-hand side, including the case of rank modifications of the perturbed matrix.

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TL;DR: In this article, the gradient of the cost functional in the discrete optimal control problem of metal solidification in casting is exactly evaluated and the mathematical model describing the solidification process is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type.

Abstract: The gradient of the cost functional in the discrete optimal control problem of metal solidification in casting is exactly evaluated. The mathematical model describing the solidification process is based on a three-dimensional two-phase initial-boundary value problem of the Stefan type. Formulas determining exact gradient determination are derived using the fast automatic differentiation technique.

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TL;DR: In this paper, a mathematical model for determining the degree of roughness and other parameters of multilayer nanostructures from the angular spectrum of the intensity of the reflected X-rays is studied.

Abstract: Mathematical models and methods for determining the degree of roughness and other parameters of multilayer nanostructures from the angular spectrum of the intensity of the reflected X-rays are studied. The proposed mathematical model for solving the direct problem of x-ray propagation and the distribution of their electromagnetic field within a multilayer nanostructure takes into account the refraction effect due to the inclusion of the second derivative with respect to the structure depth. A numerical method for solving the resulting problem is developed, and the numerical results are analyzed. The approximation-combinatorial method of the decomposition and composition of systems is used to solve the inverse problem.

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TL;DR: In this article, the inverse problem of coupled thermoelasticity is considered in the static, quasi-static, and dynamic cases, and uniqueness theorems are proved in all the cases under study.

Abstract: The inverse problem of coupled thermoelasticity is considered in the static, quasi-static, and dynamic cases. The goal is to recover the thermal stress state inside a body from the displacements and temperature given on a portion of its boundary. The inverse thermoelasticity problem finds applications in structural stability analysis in operational modes, when measurements can generally be conducted only on a surface portion. For a simply connected body consisting of a mechanically and thermally isotropic linear elastic material, uniqueness theorems are proved in all the cases under study.

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TL;DR: The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.

Abstract: A mathematical model of tumor cell population dynamics is considered. The tumor is assumed to consist of cells of two types: amenable and resistant to chemotherapeutic treatment. It is assumed that the growth of the cell populations of both types is governed by logistic equations. The effect of a chemotherapeutic drug on the tumor is specified by a therapy function. Two types of therapy functions are considered: a monotonically increasing function and a nonmonotone one with a threshold. In the former case, the effect of a drug on the tumor is stronger at a higher drug concentration. In the latter case, a threshold drug concentration exists above which the effect of the therapy reduces. The case when the total drug amount is subject to an integral constraint is also studied. A similar problem was previously studied in the case of a linear therapy function with no constraint imposed on the drug amount. By applying the Pontryagin maximum principle, necessary optimality conditions are found, which are used to draw important conclusions about the character of the optimal therapy strategy. The optimal control problem of minimizing the total number of tumor cells is solved numerically in the case of a monotone or threshold therapy function with allowance for the integral constraint on the drug amount.

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TL;DR: In this paper, a modification of the classical needle variation, namely the so-called two-parameter variation of controls is proposed, which can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions.

Abstract: A modification of the classical needle variation, namely, the so-called two-parameter variation of controls is proposed. The first variation of a functional is understood as a repeated limit. It is shown that the modified needle variation can be effectively used to derive necessary optimality conditions for a rather wide class of optimal control problems involving partial differential equations with weak solutions. Specifically, the two-parameter variation is used to obtain necessary optimality conditions in the form of a maximum principle for the optimal control of divergent hyperbolic equations.

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TL;DR: In this article, a simulation of pulsating flows with contact discontinuity instabilities and generated regions of turbulent-like gas parameter fluctuations were numerically simulated for an energy source located symmetrically or asymmetrically relative to the body.

Abstract: Pulsating flows with contact discontinuity instabilities and generated regions of turbulent-like gas parameter fluctuations were numerically simulated. The flows were driven by the interaction of an infinite heated rarefied channel with a shock layer. Pulsating flow regimes were analyzed for an energy source located symmetrically or asymmetrically relative to the body. Averaged steady flows characterized by stagnation point oscillations about a new position at the end face were examined, and steady-state flow structures with gas parameter nonmonotonicities near the body end face were studied. The simulation was based on minimum-stencil modified schemes.

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TL;DR: Low-velocity inviscid and viscous flows are simulated using the compressible Euler and Navier-Stokes equations with finite-volume discretizations on unstructured grids to speed up the convergence of the iterative process.

Abstract: Low-velocity inviscid and viscous flows are simulated using the compressible Euler and Navier-Stokes equations with finite-volume discretizations on unstructured grids. Block preconditioning is used to speed up the convergence of the iterative process. The structure of the preconditioning matrix for schemes of various orders is discussed, and a method for taking into account boundary conditions is described. The capabilities of the approach are demonstrated by computing the low-velocity inviscid flow over an airfoil.

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TL;DR: In this paper, a KP1 acceleration scheme for inner iterations that is consistent with the weighted diamond differencing (WDD) scheme is constructed for the transport equation in three-dimensional (r, ϑ, z) geometry, including those with an important role of scattering anisotropy.

Abstract: For the transport equation in three-dimensional (r, ϑ, z) geometry, a KP1 acceleration scheme for inner iterations that is consistent with the weighted diamond differencing (WDD) scheme is constructed. The P1 system for accelerating corrections is solved by an algorithm based on the cyclic splitting method (SM) combined with Gaussian elimination as applied to auxiliary systems of two-point equations. No constraints are imposed on the choice of the weights in the WDD scheme, and the algorithm can be used, for example, in combination with an adaptive WDD scheme. For problems with periodic boundary conditions, the two-point systems of equations are solved by the cyclic through-computations method elimination. The influence exerted by the cycle step choice and the convergence criterion for SM iterations on the efficiency of the algorithm is analyzed. The algorithm is modified to threedimensional (x, y, z) geometry. Numerical examples are presented featuring the KP1 scheme as applied to typical radiation transport problems in three-dimensional geometry, including those with an important role of scattering anisotropy. A reduction in the efficiency of the consistent KP1 scheme in highly heterogeneous problems with dominant scattering in non-one-dimensional geometry is discussed. An approach is proposed for coping with this difficulty. It is based on improving the monotonicity of the difference scheme used to approximate the transport equation.

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TL;DR: A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients and is used to design the schemes proposed and to obtain fifth-order accurate conditions.

Abstract: New two-stage Rosenbrock schemes with complex coefficients are proposed for stiff systems of differential equations. The schemes are fourth-order accurate and satisfy enhanced stability requirements. A one-parameter family of L1-stable schemes with coefficients explicitly calculated by formulas involving only fractions and radicals is constructed. A single L2-stable scheme is found in this family. The coefficients of the fourth-order accurate L4-stable scheme previously obtained by P.D Shirkov are refined. Several fourth-order schemes are constructed that are high-order accurate for linear problems and possess the limiting order of L-decay. The schemes proposed are proved to converge. A symbolic computation algorithm is developed that constructs order conditions for multistage Rosenbrock schemes with complex coefficients. This algorithm is used to design the schemes proposed and to obtain fifth-order accurate conditions.

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TL;DR: In this paper, an analytical solution of the skin effect problem in a metal with specular-diffuse boundary conditions is obtained, which makes it possible to obtain a solution up to an arbitrary degree of accuracy.

Abstract: An analytical solution of the skin effect problem in a metal with specular-diffuse boundary conditions is obtained. A new analytical method is developed that makes it possible to obtain a solution up to an arbitrary degree of accuracy. The method is based on the idea of representing not only the boundary condition on the field in the form of a source (which is conventional) but also the boundary condition on the distribution function. The solution is obtained in the form of a von Neumann series.

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TL;DR: In this article, the homotopy perturbation method is employed to compute an approximation to the solution of the Davey-Stewartson equations, and the absolute errors between the exact solutions of the DSP equations and the HPM solutions are presented.

Abstract: In this paper, we extend the homotopy perturbation method to solve the Davey-Stewartson equations. The homotopy perturbation method is employed to compute an approximation to the solution of the equations. Computation the absolute errors between the exact solutions of the Davey-Stewartson equations and the HPM solutions are presented. Some plots are given to show the simplicity the method.

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TL;DR: In this paper, the Fourier-Bessel integral transform in L 2 (ℝ+) was shown to have a generalized modulus of continuity, and two estimates useful in applications were proved.

Abstract: Two estimates useful in applications are proved for the Fourier-Bessel integral transform in L2(ℝ+) as applied to some classes of functions characterized by a generalized modulus of continuity.

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TL;DR: In this article, a system of two nonlinear Schrodinger equations is considered that governs the frequency doubling of femtosecond pulses propagating in an axially symmetric medium with quadratic and cubic nonlinearity.

Abstract: A system of two nonlinear Schrodinger equations is considered that governs the frequency doubling of femtosecond pulses propagating in an axially symmetric medium with quadratic and cubic nonlinearity. A numerical method is proposed to find soliton solutions of the problem, which is previously reformulated as an eigenvalue problem. The practically important special case of a single Schrodinger equation is discussed. Since three-dimensional solitons in the case of cubic nonlinearity are unstable with respect to small perturbations in their shape, a stabilization method is proposed based on weak modulations of the cubic nonlinearity coefficient and variations in the length of the focalizing layers. It should be emphasized that, according to the literature, stabilization was previously achieved by alternating layers with oppositely signed nonlinearities or by using nonlinear layers with strongly varying nonlinearities (of the same sign). In the case under study, it is shown that weak modulation leads to an increase in the length of the medium by more than 4 times without light wave collapse. To find the eigenfunctions and eigenvalues of the nonlinear problem, an efficient iterative process is constructed that produces three-dimensional solitons on large grids.