Solution Of Auxiliary Problem Research Articles

Sensitivity of Seawater Temperature and Salinity to their Restoring Times, Appending in the Boundary Conditions for these Variables at the Free Surface of the Laptev Sea in the No-Ice Period

In order to study sensitivity of seawater temperature and salinity in the no-ice Laptev Sea to their restoring times, appending in the boundary conditions for these variables at the free surface, the 3D finite-element hydrostatic model QUODDY-4 and the indirect means of describing internal tidal waves have been applied. The latter is based on the use of the corrected eddy diffusivity representing the sum of the same uncorrected diffusivity and the diapycnal diffusivity. The first of them characterizes the influence of non-tidal factors, computed using 2.5-level eddy closure scheme, the second, determined by the ratio of internal tidal waves induced baroclinic tidal energy dissipation to the buoyancy frequency in square, the influence of purely tidal factor. The estimate of this dissipation obtained from a solution of auxiliary problem on the internal tidal waves dynamics and energetics is attached. Seawater temperature and salinity in the subsurface and near-bottom layers of the Sea and also their vertical distributions along the 120E meridional transection obtained for the strong, moderate, and mixed restoring times are discussed. As a result, it is clarified that seawater temperature and salinity are weakly sensitive to changes in the restoring time. The saying follows from a comparison of model averaged (over a chosen period and by the Sea area) values of seawater temperature and salinity, and from their local vertical profiles, computed with regard to accepted estimates of the restoring time.

Optimal control in the boundary value problem for elliptic equations with degeneration

The problem of optimal control of the system described by the oblique derivative problem forthe elliptic equation of the second order is studied. Cases of internal and boundary managementare considered. The quality criterion is given by the sum of volume and surface integrals.The coefficients of the equation and the boundary condition allow power singularities of arbitraryorder in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special H\"{o}lder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will bethe solution to the given problem.
 The necessary and sufficient conditions for the existence of the optimal solution of the systemdescribed by the boundary value problem for the elliptic equation with degeneracy have been established.

OPTIMAL CONTROL IN THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH DEGENERATION

The theory of optimal control of systems, which is described by partial differential equations, is rich in results and is actively developing nowadays. The popularity of this kind of research is connected with its active use in solving problems of natural science, in particular hydro and gas dynamics, heat physics, diffusion, and the theory of biological populations. The problem of optimal control of the system described by the Dirichlet problem for the elliptic equation of the second order is studied. Cases of internal control are considered. The quality criterion is given by the volumetric integral. The coefficients of the equation admit power singularities of arbitrary order in any variables at some set of points. Solutions of auxiliary problems with smooth coefficients are studied to solve the given problem. Using a priori estimates, inequalities are established for solving problems and their derivatives in special Hölder spaces. Using the theorems of Archel and Riess, a convergent sequence is distinguished from a compact sequence of solutions to auxiliary problems, the limiting value of which will be the solution to the given problem. The necessary and sufficient conditions for the existence of the optimal solution of the system described by the Dirichlet problem for the elliptic equation with degeneracy have been established.

Some methods for solving equations with an operator function and applications for problems with a fractional power of an operator

Several applied problems are characterized by the need to numerically solve equations with an operator function (matrix function). In particular, in the last decade, mathematical models with a fractional power of an elliptic operator and numerical methods for their study have been actively discussed. Computational algorithms for such non-standard problems are based on approximations by the operator function. In this case, an approximate solution is determined by solving auxiliary standard problems. This paper discusses the main directions for constructing acceptable approximations of operator functions when solving equations.The most widespread are the approaches using various options for rational approximation. Also, we note the methods that relate to approximation by exponential sums. We propose to use a new approach, which is based on the application of approximation by exponential products. The solution of an equation with an operator function is based on the transition to standard stationary or evolutionary problems. Estimates of the accuracy of the approximate solution of the operator equation at known absolute or relative errors of approximation functions are obtained. The influence of the error in the solution of auxiliary problems is investigated separately.General approaches are illustrated by a problem with a fractional power of the operator. The first class of methods is based on the integral representation of the operator function under rational approximation, approximation by exponential sums, and approximation by exponential products. The second class of methods is associated with solving an auxiliary Cauchy problem for some evolutionary equation. In this case, we can distinguish a method based on approximation by exponential products.

A general boundary layer corrector for the asymptotic homogenization of elastic linear composite structures

Asymptotic homogenization method is often used in multiscale analysis of periodic structures instead of conducting a full field heterogeneous analysis, in order to achieve computational feasibility and efficiency. When completed with a relocalization process, this method may provide relevant estimates to microscale fields within the material. Nevertheless, the construction of a solution near the boundaries remains beyond the capabilities of classical relocalization schemes due to the loss of periodicity in the vicinity of the boundaries. This paper proposes a post-processing scheme in order to conduct the relocalization step within a finite element framework for periodic linear elastic composite materials. It also assesses the boundary layer effect and a new general method, effective for various boundary conditions (Dirichlet, Neumann or mixed), is proposed based on the idea of computing corrective terms as solution of auxiliary problems on the unit-cell. These terms are finally added to the usual fields obtained from the relocalization process to obtain the corrected solution near the boundaries. The efficiency, accuracy and limitation of the proposed approach are studied on various numerical examples.

Application of the penalty method to limit quasi-equilibrium problems

We solve a general quasi-equilibrium problem in a finite-dimensional setting, where only approximation sequences are used instead of exact values of the bi-function and feasible map. We suggest to utilize a sequence of solutions of auxiliary problems based on a penalty method. Its convergence is attained without concordance of penalty and approximation parameters under mild coercivity type conditions.

A Goal-Oriented Error Estimator for a Class of Homogenization Problems

We present a goal-oriented a posteriori error estimator for finite element approximations of a class of homogenization problems. As a rule, homogenization problems are defined through the coupling of a macroscopic solution and the solution of auxiliary problems. In this work we assume that the homogenized problem is known and that it depends on a finite number of auxiliary problems. The accuracy in the goal functional depends therefore on the discretization error of the macroscopic and the auxiliary solutions. We show that it is possible to compute the error contributions of all solution components separately and use this information to balance the different discretization errors. Additionally, we steer a local mesh refinement for both the macroscopic problem and the auxiliary problems. The high efficiency of this approach is shown by numerical examples. These include the upscaling of a periodic diffusion tensor, the case of a Stokes flow over a porous bed, and the homogenization of a fuel cell model which includes the flow in a gas channel over a porous substrate coupled with a multispecies nonlinear transport equation.

Reciprocal theorem for convective heat and mass transfer from a particle in Stokes and potential flows

A reciprocal theorem for convective heat and mass transfer offers an alternative approach for calculating integrated quantities such as an average flux from a particle. The results for nonuniform boundary conditions are obtained via solutions of auxiliary problems with constant surface conditions.

Решение вспомогательных задач диагностики утечек для обеспечения безопасности функционирующих трубопроводных систем

The article presents modern approaches to the solution of auxiliary problems of leakage diagnosis for heating oil and gas pipelines. Peculiarities of solving problems: the definition of a rational presentation of the hydraulic characteristics of the subscriber subsystems; establishing the optimal density of the manometer shooting functioning facility; establishing the date of leakage

An Inexact Penalty Method for Non Stationary Generalized Variational Inequalities

We consider a set-valued (generalized) variational inequality problem in a finite-dimensional setting, where only approximation sequences are known instead of exact values of the cost mapping and feasible set. We suggest to apply a sequence of inexact solutions of auxiliary problems involving general penalty functions. Its convergence is attained without concordance of penalty, accuracy, and approximation parameters under certain coercivity type conditions.

Stable difference schemes for certain parabolic equations

In some applications, boundary value problems for second-order parabolic equations with a special nonself-adjoint operator have to be solved approximately. The operator of such a problem is a weighted sum of self-adjoint elliptic operators. Unconditionally stable two-level schemes are constructed taking into account that the operator of the problem is not self-adjoint. The possibilities of using explicit-implicit approximations in time and introducing a new sought variable are discussed. Splitting schemes are constructed whose numerical implementation involves the solution of auxiliary problems with self-adjoint operators.

A generalization of an extensible beam equation with critical growth in [formula omitted

In this work we obtain an existence result for a generalized extensible beam equation with critical growth in RN of the type Δ2u−M(∫RN|∇u|2dx)Δu+u=λf(u)+|u|2∗∗−2uinRN, where N≥5 and λ>0. The functions M:[0,+∞)→R and f:R→R are continuous. Since there is a competition between the function M and the critical exponent given by 2∗∗=2NN−4, we need to make a truncation on function M. Using the size of λ, we show that each solution of auxiliary problem is a solution of original problem. Our approach is variational and uses minimax point critical theorems.

A modification of the method of combinatorial truncation in optimization problems over vertex-located sets

A modification of the combinatorial truncation method for optimization over vertex-located sets is considered. The modification allows dealing with degenerate solutions of auxiliary problems. The form of the truncating inequality is substantiated. An example is given to illustrate the application of the method.

A numerical approximation of the thermal coupling of fluids and solids

AbstractIn this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context. Copyright © 2008 John Wiley & Sons, Ltd.

Viscosity Solutions of a Level-Set Method for Anisotropic Geometric Diffusion in Image Processing

We discuss the existence of viscosity solutions for a class of anisotropic level-set methods which can be seen as an extension of the mean-curvature motion with a nonlinear anisotropic diffusion tensor. In an earlier work (Mikula et al. in Comput. Vis. Sci. 6(4):197---209, [2004]; Preusser and Rumpf in SIAM J. Appl. Math. 62(5):1772---1793, [2002]) we have applied such methods for the denoising and enhancement of static images and image sequences. The models are characterized by the fact that--unlike the mean-curvature motion--they are capable of retaining important geometric structures like edges and corners of the level-sets. The article reviews the definition of the model and discusses its geometric behavior. The proof of the existence of viscosity solutions for these models is based on a fixed point argument which utilizes a compactness property of the diffusion tensor. For the application to image processing suitable regularizations of the diffusion tensor are presented for which the compactness assumptions of the existence proof hold. Finally, we consider the half relaxed limits of the solutions of auxiliary problems to show the compactness of the solution operator and thus the existence of a solution to the original problem.

Probability distribution in varying-structure queueing networks

The paper is devoted to the computation of stationary distributions of queueing systems in random media. Results obtained for considered models follow from a theorem proved in the paper. As an application of the theorem, we consider Jackson networks whose structure (the set of working nodes, service and input flow intensities, routing matrix, state set) and type (open/closed network) varies as the state of another network or of the environment changes. Product-form formulas for the computation of stationary distributions of the considered networks are obtained, and algorithms for the solution of auxiliary problems are developed.

Metal-Hydride Systems for Processing Hydrogen Isotopes for Power Plants

We consider the main approaches and designs of using metal hydrides in supply systems of vacuum power plants with hydrogen isotopes as a working medium. By analyzing the results of investigations and developments in this field, we show that the most promising method is connected with the creation of multifunctional metal-hydride elements, which are a part of the structural units of the working zone of a vacuum chamber (e.g., the electrodes of a device supporting plasma) and are made of hydride-forming materials that are reversible absorbers of low-pressure hydrogen. In this case, in addition to the complex solution of auxiliary problems of compact storage, purification, and programmed bleeding-in of hydrogen isotopes into the vacuum chamber of a power plant, efficient pumping-out and improved operation of the plant as a whole can be ensured in certain cases.

Augmented Lagrangians with Adaptive Precision Control forQuadratic Programming with Equality Constraints

In this paper we introduce an augmented Lagrangian type algorithm for strictly convex quadratic programming problems with equality constraints. The new feature of the proposed algorithm is the adaptive precision control of the solution of auxiliary problems in the inner loop of the basic algorithm. Global convergence and boundedness of the penalty parameter are proved and an error estimate is given that does not have any term that accounts for the inexact solution of the auxiliary problems. Numerical experiments illustrate efficiency of the algorithm presented

Die methode der sukzessiven stützhalbräume zur lösung konvcxer optimierungsaufgaben

Basing on the close relation between solutions of auxiliary problems and perturbated primal problems a new method for handling constrained optimization problems via unconstrained ones is derived. I...

Quasi-Newton methods with approximate solution of auxiliary problems

Quasi-Newton methods with approximate solution of auxiliary problems