Auxiliary Problem Research Articles

Sabit Katsayılı İki Boyutlu Lineer Hiperbolik Denklemler İçin Cauchy Probleminin Süreksiz Fonksiyonlar Sınıfında Yüksek Mertebeden Hassas Sonlu Farklar Şeması

In this study we develop a finite difference schema for practical calculation of the Cauchy problem for the 2D scalar advection equation with a higher accuracy oder constant coefficient, encountered in different areas of hydrodynamics. For this aim to develop an auxiliary problem having some advantages over the main problem is introduced. The proposed auxiliary problem permits us construct a higher order sensitive finite differences schema.

Kernel-Based Approximation of the Koopman Generator and Schrödinger Operator.

Many dimensionality and model reduction techniques rely on estimating dominant eigenfunctions of associated dynamical operators from data. Important examples include the Koopman operator and its generator, but also the Schrödinger operator. We propose a kernel-based method for the approximation of differential operators in reproducing kernel Hilbert spaces and show how eigenfunctions can be estimated by solving auxiliary matrix eigenvalue problems. The resulting algorithms are applied to molecular dynamics and quantum chemistry examples. Furthermore, we exploit that, under certain conditions, the Schrödinger operator can be transformed into a Kolmogorov backward operator corresponding to a drift-diffusion process and vice versa. This allows us to apply methods developed for the analysis of high-dimensional stochastic differential equations to quantum mechanical systems.

ON THE ALGORITHM FOR SOLVING OF A LINEAR BOUNDARY VALUE PROBLEM FOR ORDINARY DIFFERENTIAL EQUATION WITH PARAMETER

A linear boundary value problem for a system of ordinary differential equations containing a parameter is considered on a bounded segment. For a fixed parameter value, the Cauchy problem for an ordinary differential equation is solved. Using the fundamental matrix of differential part and assuming uniqueness solvability of the Cauchy problem an origin boundary value problem is reduced to the system of linear algebraic equation with respect to unknown parameter. The existence of a solution to this system ensures the existence of a solution to the boundary value problem under study. The algorithm of finding of solution for initial problem is offered based on a construction and solving of the linear algebraic equation. The basic auxiliary problem of algorithm is: the Cauchy problem for ordinary differential equations. The numerical implementation of algorithm offered in the article uses the method of Runge-Kutta of fourth order to solve the Cauchy problem for ordinary differential equations.

EQUATION DECOMPOSITION METHOD FOR SOLVING OF PROBLEMS OF STATICS, VIBRATIONS AND STABILITY OF THIN-WALLED CONSTRUCTIONS

The work suggests the effective equation decomposition method (EDM) for solving of statics, vibrations and stability problems of thin-walled constructions. This method is based on the partition of the initial problem on the consideration of more simple auxiliary problems. The additional unknown functions are introduced for definition of the sought solutions. The paper shows the method’s advantages on the examples of the boundary value problems for rectangular areas. The problem of anisotropic plate resting on an elastic subgrade and subjected to an action of expanding forces acting in the middle surface and to transverse loads is under study. The plate’s edges are elastically supported. Also free vibrations of the rectangular plates of variable thickness with different boundary conditions were under consideration. The approximate analytical solutions with high exactness are obtained.

On an inverse boundary-value problem for a second-order elliptic equation with non-classical boundary conditions

An inverse boundary value problem for a second-order elliptic equation with periodic and integral condition is investigated. The problem is considered in a rectangular domain. To investigate the solvability of the inverse problem, we perform a conversion from the original problem to some auxiliary inverse problem with trivial boundary conditions. By the contraction mapping principle we prove the existence and uniqueness of solutions of the auxiliary problem. Then we make a conversion to the stated problem again and, as a result, we obtain the solvability of the inverse problem.

Leaky waves in a nonlinear metamaterial rod

We consider propagation of leaky TE waves in a rod filled with nonlinear metamaterial medium. The problem is reduced to the analysis of a nonlinear integral equation with a kernel in the form of the Green function of an auxiliary boundary value problem on an interval. The existence of propagating nonlinear leaky TE waves for the chosen nonlinearity (Kerr law) is proved using the method of contraction. For the numerical solution, a method based on solving an auxiliary Cauchy problem (a version of the shooting method) is proposed. New propagation regimes are discovered.

Building a certified reduced basis for coupled thermo-hydro-mechanical systems with goal-oriented error estimation

A goal-oriented a-posteriori error estimator is developed for transient coupled thermo-hydro-mechanical (THM) parametric problems solved with a reduced basis approximation. The estimator assesses the error in some specific Quantity of Interest (QoI). The goal-oriented error estimate is derived based on explicitly-computed weak residual of the primal problem and implicitly-computed adjoint solution associated with the QoI. The time-dependence of the coupled THM system poses an additional complexity as the auxiliary adjoint problem evolves backwards in time. The error estimator guides a greedy adaptive procedure that constructs progressively an optimal reduced basis by smartly selecting snapshot points over a given parametric training sample. The reduced basis obtained is used to drastically reduce the coupled system spatial degrees of freedom by several orders of magnitude. The computational gain obtained from the developed methodology is demonstrated through applications in 2D and 3D parametrized problems simulating the evolution of coupled THM processes in rock masses.

Nonlocal inverse boundary-value problem for a 2D parabolic equation with integral overdetermination condition

This article studies a nonlocal inverse boundary-value problem for a two-dimensional second-order parabolic equation in a rectangular domain. The purpose of the article is to determine the unknown coefficient and the solution of the considered problem. To investigate the solvability of the inverse problem, we transform the original problem into some auxiliary problem with trivial boundary conditions. Using the contraction mappings principle, existence and uniqueness of the solution of an equivalent problem are proved. Further, using the equivalency, the existence and uniqueness theorem of the classical solution of the original problem is obtained.

Wave propagation in immersed waveguide with radial inhomogeneity

In this paper, we investigate wave propagation in an inhomogeneous cylindrical waveguide immersed in a medium. To model external medium influence, we use the impedance boundary conditions on the waveguide outer wall. Some structural properties of the dispersion set are studied. To treat the problem of the waveguide forced oscillations, we solve boundary value problems for the first order vector differential equation sequentially in the Fourier transform space and then build the actual space on the basis of the residue theory. The residues for the numerically given function with the first order poles were found by considering auxiliary boundary value problems. The wave fields on the waveguide internal wall are obtained for different inhomogeneity laws and boundary conditions.

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.

Minimizers of curl prescribed full trace

This paper concerns the minimization problem of L2 norm of curl of vector fields prescribed full trace on the boundary of a multiconnected bounded domain. The existence of the minimizers in H1 are shown by orthogonal decompositions of vector function spaces and a constructed auxiliary variational problem. And the H2 estimate of the type II divergence-free part of the minimizers is established by div-curl-gradient type estimates of vector fields.

The elliptic homoeoid inclusion in plane elasticity

The transformation problem of an elliptical homoeioid inclusion with a uniform eigenstrain embedded in an unbounded homogeneous isotropic medium is studied in the context of plane elasticity. The term homoeoid is used to name a region of a plane medium bounded by two concentric, similar and similarly-oriented elliptic contours. The solution to the problem is achieved by solving first an auxiliary problem corresponding to the case in which the region of the medium (core) surrounded by the inclusion is replaced by a hole. A particular feature of the elastic field of the auxiliary problem is the unmoving of the hole boundary. This result suggests that the solution to the auxiliary problem is, also, the solution to the problem under consideration; additionally, it is the solution whatever is the mechanical property of the core and its bonding conditions with the inclusion. The solution to the problem is obtained in closed form, in terms of the complex potentials of the inclusion and its surrounding (matrix). Based on the complex potentials obtained, a simple expression for the total elastic energy stored in the unbounded medium is derived. It is shown that the total area change of the unbounded medium is that of the inclusion, which is determined in a simple form.

Social Optima in Robust Mean Field LQG Control: From Finite to Infinite Horizon

This article studies social optimal control of mean field linear-quadratic-Gaussian models with uncertainty. Specially, the uncertainty is represented by an uncertain drift, which is common for all agents. A robust optimization approach is applied by assuming all agents treat the uncertain drift as an adversarial player. In our model, both dynamics and costs of agents are coupled by mean field terms, and both finite- and infinite-time horizon cases are considered. By examining social functional variation and exploiting person-by-person optimality principle, we construct an auxiliary control problem for the generic agent via a class of forward-backward stochastic differential equation system. By solving the auxiliary problem and constructing consistent mean field approximation, a set of decentralized control strategies is designed and shown to be asymptotically optimal.

On a multiple credit rating migration model with stochastic interest rate

In this paper, we explore a pricing model for corporate bond accompanied with multiple credit rating migration risk and stochastic interest rate. The bond price volatility strongly depends on potentially multiple credit rating migration and stochastic change of interest rate. A free boundary problem of partial differential equation is presented, which is the equivalent transformation of the pricing model. The existence, uniqueness, and regularity for the free boundary problem are established to guarantee the rationality of the pricing model. Due to the stochastic change of interest rate, the discontinuous coefficient in the free boundary problem depends explicitly on the time variable but is convergent as time tends to infinity. Accordingly, an auxiliary free boundary problem is constructed, whose coefficient is the convergent limit of the coefficient in the original free boundary problem. With some constraint on the risk discount rate satisfied, we prove that a unique traveling wave exists in the auxiliary free boundary problem. The inductive method is adopted to fit the multiplicity of credit rating. Then we show that the solution of the original free boundary problem converges to the traveling wave in the auxiliary free boundary problem. Returning to the pricing model with multiple credit rating migration and stochastic interest rate, we conclude that the bond price profile can be captured by a traveling wave pattern coupling with a guaranteed bond price with face value equal to one at the maturity.

Model predictive control for nonlinear systems in Takagi-Sugeno’s form under round-robin protocol

In this paper, we focus on the investigation of the fuzzy model predictive control (FMPC) problem for nonlinear systems characterized by Takagi-Sugeno (T-S) fuzzy form with hard constraints on the control input and the state. To reduce the data transmission burden and improve the network utilization, a so-called Round-Robin (RR) communication protocol is employed to orchestrate data transmission orders via the shared communication network, which is located from controller nodes to the actuator node. In light of the token-dependent quadratic function approach, a switching system is constructed based on the data transmission period regarding the adopted RR protocol. Then, with respect to system nonlinearities in the case of the T-S fuzzy form, a “min-max” problem is formulated by resorting to the objective function over the infinite horizon. Our purpose is to find a series of desired controllers in terms of FMPC approach such that the underlying T-S fuzzy system is asymptotically stable. With taking the T-S fuzzy nonlinearities and the RR protocol fully into consideration, sufficient conditions are provided through an online auxiliary optimization problem. Finally, two simulation examples are used to demonstrate the effectiveness and the validity of the proposed methods.

Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions

<p style='text-indent:20px;'>The paper deals with the optimal control problem described by second order evolution differential inclusions; to this end first we use an auxiliary problem with second order discrete and discrete-approximate inclusions. Then applying infimal convolution concept of convex functions, step by step we construct the dual problems for discrete, discrete-approximate and differential inclusions and prove duality results. It seems that the Euler-Lagrange type inclusions are "duality relations" for both primary and dual problems and that the dual problem for discrete-approximate problem make a bridge between them. At the end of the paper duality in problems with second order linear discrete and continuous models and model of control problem with polyhedral DFIs are considered.

Temporal Decomposition-Based Stochastic Economic Dispatch for Smart Grid Energy Management

This paper presents a temporal decomposition strategy to decompose security-constrained economic dispatch (SCED) over the scheduling horizon with the goal of reducing its computational burden and enhancing its scalability. A set of subproblems, each with respect to demand response, normal constraints, and $N-1$ contingency corrective actions at a subhorizon, is formulated. The proposed decomposition deals with computational complexities originated from intertemporal interdependencies of system equipment, i.e., generators’ ramp constraints and state of charge of storage devices. The concept of overlapping intervals is introduced to make SCED subproblems solvable in parallel. Intertemporal connectivity related to energy storage is also modeled in the context of temporal decomposition. Besides, reserve up and down requirements are formulated as data-driven nonparametric chance constraints to account for wind generation uncertainties. The concept of $\phi -$ divergence is used to convert nonparametric chance constraints to more conservative parametric constraints. A reduced risk level is calculated with respect to wind generation prediction errors to ensure the satisfaction of system constraints with a confidence level after the true realization of uncertainty. Auxiliary problem principle is applied to coordinate SCED subproblems in parallel. Numerical results on three test systems show the effectiveness of the proposed algorithm.

Optimal archgrids: a variational setting

The paper deals with the variational setting of the optimal archgrid construction. The archgrids, discovered by William Prager and George Rozvany in 1970s, are viewed here as tension-free and bending-free, uniformly stressed grid-shells forming vaults unevenly supported along the closed contour of the basis domain. The optimal archgrids are characterized by the least volume. The optimization problem of volume minimization is reduced to the pair of two auxiliary mutually dual problems, having mathematical structure similar to that known from the theory of optimal layout: the integrand of the auxiliary minimization problem is of linear growth, while the auxiliary maximization problem involves test functions subjected to mean-square slope conditions. The noted features of the variational setting governs the main properties of the archgrid shapes: they are vaults over a subregion of the basis domain being the effective domain of the minimizer of the auxiliary problem. Thus, the method is capable of cutting out the material domain from the design domain; this process is built in within the theory. Moreover, the present paper puts forward new methods of numerical construction of optimal archgrids and discusses their applicability ranges.

Optimization Problems Arising from the Design of Pipes Made of Composite Materials

The work is devoted to the construction of analytical solutions for the stress-strain state of a cylindrical hollow elastic rod with a layered structure along the radius. Earlier, the problem of finding the stress-strain state of a rod of composite material fixed at one end with the applied forces and moments of forces at the other end was considered. An approximate representation of the solutions was given, which included auxiliary problems on one fragment of the cylinder, consisting of periodically repeating similar fragments. Such auxiliary problems in the general case do not have an analytical solution. In this paper it is shown that in the presence of radial symmetry of the rod section, it is possible to construct a stress-strain state in an analytical form. In addition, tensile and bending stiffness can be presented in an analytical form. The latter circumstance allows us to set a problem of optimizing the stiffness characteristics of a rod with its fixed weight. Optimization is carried out by varying the thickness of the layers of the same materials.

Numerical Solution of Linear Differential Equations with Nonlocal Nonlinear Conditions

The numerical solution of systems of linear ordinary differential equations with nonlocal nonlinear conditions depending on the values of the desired function at intermediate points is investigated. Conditions for the existence of a solution to the problem under consideration are given. For the numerical solution, an approach is proposed that reduces the problem to two auxiliary linear systems of differential equations with linear conditions and to a single nonlinear algebraic system with a dimension depending only on the number of given intermediate points. The proposed approach is illustrated by solving two problems. The auxiliary problems are solved analytically in one of them and numerically in the other.