Applied Mathematics and Optimization最新文献

This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn–Hilliard models.

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-(*)) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

This work focuses on moderate deviations for two-time scale systems with mixed fractional Brownian motion. Our proof uses the weak convergence method which is based on the variational representation formula for mixed fractional Brownian motion. Throughout this paper, the Hurst parameter of fractional Brownian motion is larger than 1/2 and the integral along the fractional Brownian motion is understood as the generalized Riemann-Stieltjes integral. First, we consider single-time scale systems with fractional Brownian motion. The key of our proof is showing the weak convergence of the controlled system. Next, we extend our method to show moderate deviations for two-time scale systems. To this goal, we combine the Khasminskii-type averaging principle and the weak convergence approach.

We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.

We investigate the strong convergence properties of a Nesterov type algorithm with two Tikhonov regularization terms in connection to the minimization problem of a smooth convex function f. We show that the generated sequences converge strongly to the minimal norm element from (text {argmin}f). We also show fast convergence for the potential energies (f(x_n)-text {min}f) and (f(y_n)-text {min}f), where ((x_n),,(y_n)) are the sequences generated by our algorithm. Further we obtain fast convergence to zero of the discrete velocity and some estimates concerning the value of the gradient of the objective function in the generated sequences. Via some numerical experiments we show that we need both Tikhonov regularization terms in our algorithm in order to obtain the strong convergence of the generated sequences to the minimum norm minimizer of our objective function.

The upper semicontinuity of random attractors for stochastic/random (partial) differential equations with nonlinear diffusion term is an unsolved problem. In this paper, we first show the existence of random attractor for the random differential equation with nonlinear diffusion term driven by the approximation of the fractional noise, and then prove the upper semicontinuity of the random attractors when the intensity of the approximations tends to zero. The obtained result partly gives an answer to this problem.

In this work we further develop a nonlocal calculus theory (initially introduced in Bellido et al. (Adv Nonlinear Anal 12:20220316, 2023)) associated with singular fractional-type operators which exhibit kernels with finite support of interactions. The applicability of the framework to nonlocal elasticity and the theory of peridynamics has attracted increased interest and motivation to study it and find connections with its classical counterpart. In particular, a critical contribution of this paper is producing vector identities, integration by part type theorems (such as the Divergence Theorem, Green identities), as well as a Helmholtz–Hodge decomposition. The estimates, together with the analysis performed along the way provide stepping stones for proving additional results in the framework, as well as pathways for numerical implementations.

Motivated by many applications, optimal control problems with integer controls have recently received a significant attention. Some state-of-the-art work uses perimeter-regularization to derive stationarity conditions and trust-region algorithms. However, the discretization is difficult in this case because the perimeter is concentrated on a set of dimension (d - 1) for a domain of dimension d. This article proposes a potential way to overcome this challenge by using the fractional nonlocal perimeter with fractional exponent (0<alpha <1). In this way, the boundary integrals in the perimeter regularization are replaced by volume integrals. Besides establishing some non-trivial properties associated with this perimeter, a (Gamma )-convergence result is derived. This result establishes convergence of minimizers of fractional perimeter-regularized problem, to the standard one, as the exponent (alpha ) tends to 1. In addition, the stationarity results are derived and algorithmic convergence analysis is carried out for (alpha in (0.5,1)) under an additional assumption on the gradient of the reduced objective. The theoretical results are supplemented by a preliminary computational experiment. We observe that the isotropy of the total variation may be approximated by means of the fractional perimeter functional.

In this paper we prove the existence of multiple positive solutions for a quasilinear elliptic problem with unbalanced growth in expanding domains by using variational methods and the Lusternik–Schnirelmann category theory. Based on the properties of the category, we introduce suitable maps between the expanding domains and the critical levels of the energy functional related to the problem, which allow us to estimate the number of positive solutions by the shape of the domain.

We consider a finite number of N statistically equal agents, each moving on a finite set of states according to a continuous-time Markov Decision Process (MDP). Transition intensities of the agents and generated rewards depend not only on the state and action of the agent itself, but also on the states of the other agents as well as the chosen action. Interactions like this are typical for a wide range of models in e.g. biology, epidemics, finance, social science and queueing systems among others. The aim is to maximize the expected discounted reward of the system, i.e. the agents have to cooperate as a team. Computationally this is a difficult task when N is large. Thus, we consider the limit for (Nrightarrow infty .) In contrast to other papers we treat this problem from an MDP perspective. This has the advantage that we need less regularity assumptions in order to construct asymptotically optimal strategies than using viscosity solutions of HJB equations. The convergence rate is (1/sqrt{N}). We show how to apply our results using two examples: a machine replacement problem and a problem from epidemics. We also show that optimal feedback policies from the limiting problem are not necessarily asymptotically optimal.