Auxiliary Problem Research Articles

Axisymmetric residual stresses in a solid cylinder of finite length.

A solution technique is presented for the determination of residual stresses in a finite-length solid cylinder subject to non-uniform axisymmetric distributions of incompatible residual strains. The problem is reduced to the sequential solving of three individual problems: a problem on the determination of residual stresses in an infinitely long cylinder (the basic state) and two auxiliary problems for evaluating the stresses induced by the end-face effects (the disturbed states). The variational method of homogeneous solutions is implemented in order to determine the disturbed states within the framework of two latter problems. The solution technique is verified numerically for typical distribution profiles of incompatible strains depending on both the radial and axial coordinates. The approach can be used to evaluate residual stresses in a solid cylinder of finite length due to the intense thermal treatment.

A path-following inexact Newton method for PDE-constrained optimal control in BV

We study a PDE-constrained optimal control problem that involves functions of bounded variation as controls and includes the TV seminorm of the control in the objective. We apply a path-following inexact Newton method to the problems that arise from smoothing the TV seminorm and adding an H^1 regularization. We prove in an infinite-dimensional setting that, first, the solutions of these auxiliary problems converge to the solution of the original problem and, second, that an inexact Newton method enjoys fast local convergence when applied to a reformulation of the auxiliary optimality systems in which the control appears as implicit function of the adjoint state. We show convergence of a Finite Element approximation, provide a globalized preconditioned inexact Newton method as solver for the discretized auxiliary problems, and embed it into an inexact path-following scheme. We construct a two-dimensional test problem with fully explicit solution and present numerical results to illustrate the accuracy and robustness of the approach.

Didactic situations in project-based learning: The case of numerical patterns and sequences

The article presents a multiple-case study examining the project-based learning model where mathematical projects of middle school student teams originate from different tasks on the topic of numerical sequences. Learning situations in PBL are identified and characterized using the analytical apparatus of the Theory of Didactic Situations. A long-term project can be conceptualized as a complex system of alternating didactic situations shaped by the interaction of students and teachers with the global milieu and the local milieu of each student team. Students' transition from open to a structured inquiry by self-constraining the milieu was a successful strategy that promoted acculturation to ways of mathematical exploration. The student project's milieu can be modified by introducing auxiliary problems containing elements necessary for future mathematical situations. The situations of institutionalization in PBL are dedicated to the processes of mathematical exploration as well as mathematical content. The practical principles for implementing PBL into mathematics instruction are presented.

An inverse boundary value problem for a two-dimensional pseudo-parabolic equation of third order

In the present work, we consider an inverse boundary value problem for a two-dimensional pseudo-parabolic equation of the third-order. Using analytical and operator-theoretic methods, as well as the Fourier method, the existence and uniqueness of the classical solution of this problem is proved. This inverse problem is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis. In addition, the two-dimensional pseudo-parabolic problem is discretized using the FDM and reshaped as nonlinear least-squares optimization of the Tikhonov regularization function. This is numerically solved by means of the MATLAB subroutine lsqnonlin tool. Both analytical and perturbed data are inverted. Numerical outcomes for benchmark test example is reported and discussed.

Ambika approach for solving matrix games with payoffs of single-valued trapezoidal neutrosophic numbers

The principal objective of this article is to develop an effective approach to solve matrix games with payoffs of single-valued trapezoidal neutrosophic numbers (SVTNNs). In this approach, the concepts and suitable ranking function of SVTNNs are defined. Hereby, the optimal strategies and game values for both players can be determined by solving the parameterized mathematical programming problems, which are obtained from two novel auxiliary SVTNNs programming problems based on the proposed Ambika approach. In this approach, it is verified that any matrix game with SVTNN payoffs always has a SVTNN game value. Moreover, an application example is examined to verify the effectiveness and superiority of the developed algorithm. Finally, a comparison analysis between the proposed and the existing approaches is conducted to expose the advantages of our work.

The policy iteration algorithm for a compound Poisson process applied to optimal dividend strategies under a Cramér–Lundberg risk model

In this paper, we focus on the policy iteration algorithm (PIA) for the optimal dividend problem under the Cramér–Lundberg risk model. We conclude that the optimal value function is the minimum nonnegative solution of an optimization equation. Under any conditions, it can be approximated by iteration starting with the initial zero-valued policy, i.e., the policy of no dividend payment at all. An auxiliary optimization problem of maximizing dividend payment with terminal reward up to the first claim arrival time is introduced. This auxiliary optimization problem is solved completely in the sense that, no matter what the claim-size distribution is, the explicit formulae for both optimal strategy and value function are presented in general situation. These ensure that, at each iteration, we can get the explicit formulae of the improved strategy and its evaluation function. Finally, the PIA is illustrated with some numerical examples.

Approximation of random diffusion equations by nonlocal diffusion equations in free boundary problems of one space dimension

We show how the Stefan type free boundary problem with random diffusion in one space dimension can be approximated by the corresponding free boundary problem with nonlocal diffusion. The approximation problem is a slightly modified version of the nonlocal diffusion problem with free boundaries considered in [J. Cao, Y. Du, F. Li and W.-T. Li, The dynamics of a Fisher–KPP nonlocal diffusion model with free boundaries, J. Functional Anal. 277 (2019) 2772–2814; C. Cortazar, F. Quiros and N. Wolanski, A nonlocal diffusion problem with a sharp free boundary, Interfaces Free Bound. 21 (2019) 441–462]. The proof relies on the introduction of several auxiliary free boundary problems and constructions of delicate upper and lower solutions for these problems. As usual, the approximation is achieved by choosing the kernel function in the nonlocal diffusion term of the form [Formula: see text] for small [Formula: see text], where [Formula: see text] has compact support. We also give an estimate of the error term of the approximation by some positive power of [Formula: see text].

On the Stokes-Type Resolvent Problem Associated with Time-Periodic Flow Around a Rotating Obstacle

Consider the resolvent problem associated with the linearized viscous flow around a rotating body. Within a setting of classical Sobolev spaces, this problem is not well posed on the whole imaginary axis. Therefore, a framework of homogeneous Sobolev spaces is introduced where existence of a unique solution can be guaranteed for every purely imaginary resolvent parameter. For this purpose, the problem is reduced to an auxiliary problem, which is studied by means of Fourier analytic tools in a group setting. In the end, uniform resolvent estimates can be derived, which lead to the existence of solutions to the associated time-periodic linear problem.

Linearly stable and unstable complex soliton solutions with real energies in the Bullough-Dodd model

We investigate different types of complex soliton solutions with regard to their stability against linear perturbations. In the Bullough-Dodd scalar field theory we find linearly stable complex PT-symmetric solutions and linearly unstable solutions for which the PT-symmetry is broken. Both types of solutions have real energies. The auxiliary Sturm-Liouville eigenvalue equation in the stability analysis for the PT-symmetric solutions can be solved exactly by supersymmetrically mapping it to an isospectral partner system involving a shifted and scaled inverse cosh-squared potential. We identify exactly one shape mode in form of a bound state solution and scattering states which when used as linear perturbations leave the solutions stable. The auxiliary problem for the solutions with broken PT-symmetry involves a complex shifted and scaled inverse sin-squared potential. The corresponding bound and scattering state solutions have complex eigenvalues, such that when used as linear perturbations for the corresponding soliton solutions lead to their decay or blow up as time evolves.

Social optima in mean field linear–quadratic–Gaussian models with control input constraint

This paper studies linear quadratic social optimal problems with control input constraints, where both dynamics and costs involve the coupled term of states, and the diffusion terms contain control variables. In virtue of the variational method, we obtain the corresponding auxiliary stochastic control problem by introducing the limiting state-average process. Furthermore, we design a set of decentralized control strategies by using the stochastic maximum principle, which is given by a fully coupled non-linear mean-field forward–backward stochastic differential equation, and they are further shown to be asymptotically social optimality by perturbation analysis. As applications, we apply the theoretical results to deal with the optimal consumption problem and obtain the related optimal consumption strategy.

The Fundamental Formulation for Inhomogeneous Inclusion Problems with the Equivalent Eigenstrain Principle

In this paper, and on the basis of the equivalent eigenstrain principle, a fundamental formulation for inhomogeneous inclusion problems is proposed, which is to transform the inhomogeneous inclusion problems into auxiliary equivalent homogenous inclusion problems. Then, the analysis, which is based on the equivalent homogenous inclusions, would significantly reduce the workload and would enable the analytical solutions that are possible for a series of inhomogeneous inclusion problems. It also provides a feasible way to evaluate the effective properties of composite materials in terms of their equivalent homogenous materials. This formulation allows for solving the problems: (i) With an arbitrarily connected and shaped inhomogeneous inclusion; (ii) Under an arbitrary internal load by means of the nonuniform eigenstrain distribution; and (iii) With any kind of external load, such as singularity, uniform far field, and so on. To demonstrate the implementation of the formulation, an oblate inclusion that interacts with a dilatational eigenstrain nucleus is analyzed, and an explicit solution is obtained. The fundamental formulation that is introduced here will find application in the mechanics of composites, inclusions, phase transformation, plasticity, fractures, etc.

A variational inequality based stochastic approximation for estimating the flexural rigidity in random fourth-order models

This paper develops a stochastic approximation approach for estimating the flexural rigidity in the framework of variational inequalities. The nonlinear inverse problem is analyzed as a stochastic optimization problem using an energy least-squares formulation. A necessary and sufficient optimality condition for the optimization problem is a stochastic variational inequality solved by a stochastic auxiliary problem principle-based iterative scheme. Exhaustive convergence analysis for the proposed iterative scheme is given under quite general conditions on the random noise. Detailed computational results demonstrate the feasibility and the efficacy of the proposed methodology.

Adaptive event‐triggered fault detection for nonlinear networked systems in finite‐frequency domain under stochastic cyber‐attacks

SummaryThis article investigates the fault detection problem in finite‐frequency domain for a class of nonlinear networked systems under stochastic cyber‐attacks. A novel adaptive event‐triggered scheme is introduced to mitigate the transmission burden of the network. A unified measurement model is proposed to take the randomly occurring cyber‐attacks and the transmission delays into account simultaneously. Under the consideration of fault sensitivity and disturbance robustness, the addressed fault detection problem is converted into an auxiliary filtering problem by properly augmenting the states of the original system and the fault detection filter. Intensive stochastic analysis is carried out to obtain sufficient conditions for the existence of the desired fault detection filter, and the corresponding optimal filter parameters can be easily derived by solving a convex optimization problem. Finally, an illustrative example is presented to show the effectiveness and applicability of the proposed method.

M-stationarity for a class of MPCCs in Lebesgue spaces

We show that an optimality condition of M-stationarity type holds for minimizers of a class of mathematical programs with complementarity constraints (MPCCs) in Lebesgue spaces. We apply these results also to local minimizers of an inverse optimal control problem (which is an instance of an infinite-dimensional bilevel optimization problem). The multipliers for the M-stationarity system can be constructed via convex combinations of various multipliers to auxiliary, linear problems. However, proving the existence of the multipliers to these auxiliary problems is difficult and only possible in some situations.

Solution of one optimum control problem regarding the depletion of gas reservoir

Using the methods of the optimal control theory, the problem of determining the optimal technological mode of gas deposits’ exploitation under the condition of their depletion by a given point in time is solved. This task is of particular interest for the exploitation of offshore fields, the activity of which is limited by the service life of the field equipment. The considered problem is also of certain mathematical interest as an objective of optimal control of nonlinear systems with distributed parameters. The usefulness and importance of solving such problems are determined by the richness of the class of major tasks that have a practical result. As an optimality criterion, a quadratic functional characterizing the conditions of reservoir depletion is considered. By introducing an auxiliary boundary value problem, and taking into account the stationarity conditions for the Lagrange functions at the optimal point, a formula for the gradient of the minimized functional is obtained. To obtain a solution to this specific optimization problem, which control function is sought in the class of a piecewise continuous and bounded function with discontinuities of the first kind, the Pontryagin’s maximum principle is subjected. The calculation of the gradient of the functional for the original and adjoint problems with partial differential equations is carried out by the method of straight lines. The numerical solution of the problem was carried out by two methods – the method of gradient projection with a special choice of step and the method of successive approximations. Despite the incorrectness of optimal control problems with a quadratic functional, the gradient projection method did not show a tendency to «dispersion» and gave a convergent sequence of controls.

Darboux transformations of the Camassa-Holm type systems

We propose a unified approach to establish Darboux-Bäcklund transformations of the Camassa-Holm (CH) type equations. The method simplifies the approach presented by Xia, Zhou and Qiao [J. Math. Phys. 57, 103502 (2016)] to obtain soliton solutions of the CH equation by the Darboux transformation, because it does not need to solve spectral problem of the CH equation at λ=0 which is a singular point in the auxiliary problem, and avoids computing asymptotic properties of the wave functions. We also study many other CH type equations such as the 2-CH, the modified CH, the Degasperis-Procesi and the Novikov via the method. Especially, we present three Bäcklund transformations of the 2-CH system.

Difference-based Equivalent Static Load Method with adaptive time selection and local stiffness adaption

Structural optimization of crash-related problems usually involves nonlinearities in geometry, material, and contact. The Equivalent Static Load (ESL) method provides a method to solve such problems. It has previously been extended to employ an individual Finite Element model describing the deformed geometry at each considered time step under the name Difference-based Equivalent Static Load (DiESL) method. This paper demonstrates how an appropriate selection of the time steps in each cycle can further improve the convergence behavior of the DiESL method. It is shown that the adaptive selection of time steps leads to better objective values and more reliable convergence to the presumed global optimum. Furthermore, the DiESL extension enables the adaption of path-dependent structural properties of the original nonlinear problem like material stiffness in each linear auxiliary load case. In this paper, an adaption of the Young’s modulus on element level in the linear auxiliary problem corresponding to the local plasticization in the nonlinear dynamic problem is successfully implemented. Here, the test examples indicate that an observable improvement can only be obtained if neither the elements in the elastic nor in the plastic range are dominating the structure’s behavior.

An improved Multiscale FEM for the free vibrations of heterogeneous solids

We present an improvement of the Multiscale Finite Element Method (MsFEM). MsFEM incorporates fine-scale features of a domain through special oscillatory coarse-scale trial functions and its essential step is an online computation of these special functions defined using local auxiliary problems. Our enhancement makes use of the dynamic condensation that transmits the construction of the oscillatory basis functions exclusively to the coarse mesh skeleton (element interfaces) where we defined special 1D eigenproblems. Consequently, both the approximation error and computational cost are reduced. Moreover, arbitrary coarse mesh element geometries, like polytopal ones, may be used without any additional modifications. We illustrate the accuracy and convergence of the improved method with higher order approximation through selected plane strain examples.

Localization of eigenfunctions in a thin cylinder with a locally periodic oscillating boundary

We study a Dirichlet spectral problem for a second-order elliptic operator with locally periodic coefficients in a thin cylinder. The lateral boundary of the cylinder is assumed to be locally periodic. When the thickness of the cylinder ε tends to zero, the eigenvalues are of order ε−2 and described in terms of the first eigenvalue μ(x1) of an auxiliary spectral cell problem parametrized by x1, while the eigenfunctions localize with rate ε.

Inventory Balancing with Online Learning

We study a general problem of allocating limited resources to heterogeneous customers over time under model uncertainty. Each type of customer can be serviced using different actions, each of which stochastically consumes some combination of resources and returns different rewards for the resources consumed. We consider a general model in which the resource consumption distribution associated with each customer type–action combination is not known but is consistent and can be learned over time. In addition, the sequence of customer types to arrive over time is arbitrary and completely unknown. We overcome both the challenges of model uncertainty and customer heterogeneity by judiciously synthesizing two algorithmic frameworks from the literature: inventory balancing, which “reserves” a portion of each resource for high-reward customer types that could later arrive based on competitive ratio analysis, and online learning, which “explores” the resource consumption distributions for each customer type under different actions based on regret analysis. We define an auxiliary problem, which allows for existing competitive ratio and regret bounds to be seamlessly integrated. Furthermore, we propose a new variant of upper confidence bound (UCB), dubbed lazyUCB, which conducts less exploration in a bid to focus on “exploitation” in view of the resource scarcity. Finally, we construct an information-theoretic family of counterexamples to show that our integrated framework achieves the best possible performance guarantee. We demonstrate the efficacy of our algorithms on both synthetic instances generated for the online matching with stochastic rewards problem under unknown probabilities and a publicly available hotel data set. Our framework is highly practical in that it requires no historical data (no fitted customer choice models or forecasting of customer arrival patterns) and can be used to initialize allocation strategies in fast-changing environments. This paper was accepted by J. George Shanthikumar, Management Science Special Section on Data-Driven Prescriptive Analytics.