Applied Mathematics and Optimization最新文献

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.

This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions

in a bounded and smooth domain (Omega subset mathbb R^2), where the parameters (mu _i, alpha _i, a_i, b_i, gamma _i) ((i=1,2)) are positive constants, and the functions (D_1(v),S_1(v),D_2(z),S_2(z)) fulfill the following hypotheses: (Diamond ) (D_i(psi ),S_i(psi )in C^2([0,infty ))), (D_i(psi ),S_i(psi )>0) for all (psi ge 0), (D_i^{prime }(psi )<0) and (underset{psi rightarrow infty }{lim } D_i(psi )=0); (Diamond ) (underset{psi rightarrow infty }{lim } frac{S_i(psi )}{D_i(psi )}) and (underset{psi rightarrow infty }{lim } frac{D^{prime }_i(psi )}{D_i(psi )}) exist. We first confirm the global boundedness of the classical solution provided that the additional conditions (2gamma _1le 1+alpha _2) and (2gamma _2le 1+alpha _1) hold. Moreover, by constructing several suitable Lyapunov functionals, it is demonstrated that the global solution exponentially or algebraically converges to the constant stationary solutions and the corresponding convergence rates are determined under some specific stress conditions.

We propose an entropic approximation approach for optimal transportation problems with a supremal cost. We establish (Gamma )-convergence for suitably chosen parameters for the entropic penalization and that this procedure selects (infty )-cyclically monotone plans at the limit. We also present some numerical illustrations performed with Sinkhorn’s algorithm.

The main objective of this paper is to demonstrate the uniform large deviation principle (UDLP) for the solutions of two-dimensional stochastic Navier–Stokes equations (SNSE) in the vorticity form when perturbed by two distinct types of noises. We first consider an infinite-dimensional additive noise that is white in time and colored in space and then consider a finite-dimensional Wiener process with linear growth coefficient. In order to obtain the ULDP for 2D SNSE in the vorticity form, where the noise is white in time and colored in space, we utilize the existence and uniqueness result from B. Ferrario et. al., Stochastic Process. Appl., 129 (2019), 1568–1604, and the uniform contraction principle. For the finite-dimensional multiplicative Wiener noise, we first prove the existence of a unique local mild solution to the vorticity equation using a truncation and fixed point arguments. We then establish the global existence of the truncated system by deriving a uniform energy estimate for the local mild solution. By applying stopping time arguments and a version of Skorokhod’s representation theorem, we conclude the global existence and uniqueness of a solution to our model. We employ the weak convergence approach to establish the ULDP for the law of the solutions in two distinct topologies. We prove ULDP in the ({{textrm{C}}([0,T];{textrm{L}}^p({mathbb {T}}^2))}) topology, for (p>2), taking into account the uniformity of the initial conditions contained in bounded subsets of ({{textrm{L}}^p({mathbb {T}}^2)}). Finally, in ({{textrm{C}}([0,T]times {mathbb {T}}^2)}) topology, the uniformity of initial conditions lying in bounded subsets of ({{textrm{C}}({mathbb {T}}^2)}) is considered.

The convexification numerical method with the rigorously established global convergence property is constructed for a problem for the Mean Field Games System of the second order. This is the problem of the retrospective analysis of a game of infinitely many rational players. In addition to traditional initial and terminal conditions, one extra terminal condition is assumed to be known. Carleman estimates and a Carleman Weight Function play the key role. Numerical experiments demonstrate a good performance for complicated functions. Various versions of the convexification have been actively used by this research team for a number of years to numerically solve coefficient inverse problems.