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Abstract: Abstract In a Hilbert space setting, this paper is devoted to the study of a class of first-order algorithms which aim to solve structured monotone equations involving the sum of potential and nonpotential operators. Precisely, we are looking for the zeros of an operator \(A= \nabla f +B \) , where \(\nabla f\) is the gradient of a differentiable convex function f, and B is a nonpotential monotone and cocoercive operator. This study is based on the inertial autonomous dynamic previously studied by the authors, which involves dampings controlled respectively by the Hessian of f, and by a Newton-type correction term attached to B. These geometric dampings attenuate the oscillations which occur with the inertial methods with viscous damping. Temporal discretization of this dynamic provides fully splitted proximal-gradient algorithms. Their convergence properties are proven using Lyapunov analysis. These results open the door to the design of first-order accelerated algorithms in numerical optimization taking into account the specific properties of potential and nonpotential terms. PubDate: 2022-05-10

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Abstract: Abstract We study the asymptotic organization among many optimizing individuals interacting in a suitable “moderate" way. We justify this limiting game by proving that its solution provides approximate Nash equilibria for large but finite player games. This proof depends upon the derivation of a law of large numbers for the empirical processes in the limit as the number of players tends to infinity. Because it is of independent interest, we prove this result in full detail. We characterize the solutions of the limiting game via a verification argument. PubDate: 2022-05-10

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Abstract: Abstract Several engineering problems result in a PDE-constrained optimization problem that aims at finding the shape of a solid inside a fluid which minimizes a given cost function. These problems are categorized as Topology Optimization (TO) problems. In order to tackle these problems, the solid may be located with a penalization term added in the constraints equations that vanishes in fluid regions and becomes large in solid regions. This paper addresses a TO problem for anisothermal flows modelled by the steady-state incompressible Navier–Stokes system coupled to an energy equation, with mixed boundary conditions, under the Boussinesq approximation. We first prove the existence and uniqueness of a solution to these equations as well as the convergence of its finite element discretization. Next, we show that our TO problem has at least one optimal solution for cost functions that satisfy general assumptions. The convergence of discrete optimum toward the continuous one is then proved as well as necessary first order optimality conditions. Eventually, all these results let us design a numerical algorithm to solve a TO problem approximating solids with piecewise constant thermal diffusivities also refered as multi-materials. A physical problem solved numerically for varying parameters concludes this paper. PubDate: 2022-05-10

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Abstract: Abstract In this paper, we deal with the following double phase problem $$\begin{aligned} \left\{ \begin{array}{lll} -\text{ div }\left( \nabla u ^{p-2}\nabla u+a(x) \nabla u ^{q-2}\nabla u\right) =&{} \gamma \left( \displaystyle \frac{ u ^{p-2}u}{ x ^p}+a(x)\displaystyle \frac{ u ^{q-2}u}{ x ^q}\right) \\ &{}+f(x,u) &{} \text{ in } \Omega ,\\ u=0&{} &{} \text{ in } \partial \Omega , \end{array} \right. \end{aligned}$$ where \(\Omega \subset {\mathbb {R}}^N\) is an open, bounded set with Lipschitz boundary, \(0\in \Omega \) , \(N\ge 2\) , \(1<p<q<N\) , weight \(a(\cdot )\ge 0\) , \(\gamma \) is a real parameter and f is a subcritical function. By variational method, we provide the existence of a non-trivial weak solution on the Musielak-Orlicz-Sobolev space \(W^{1,{\mathcal {H}}}_0(\Omega )\) , with modular function \({\mathcal {H}}(t,x)=t^p+a(x)t^q\) . For this, we first introduce the Hardy inequalities for space \(W^{1,{\mathcal {H}}}_0(\Omega )\) , under suitable assumptions on \(a(\cdot )\) . PubDate: 2022-05-10

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Abstract: Abstract In this paper, we study a non-autonomous damped wave equation with a nonlinear memory term. By using the theory of evolution process and sectorial operators, we ensure sufficient conditions for well-posedness and spatial regularity to the problem. PubDate: 2022-05-10

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Abstract: Abstract It is well known that the optimal transportation plan between two probability measures \(\mu \) and \(\nu \) is induced by a transportation map whenever \(\mu \) is an absolutely continuous measure supported over a compact set in the Euclidean space and the cost function is a strictly convex function of the Euclidean distance. However, when \(\mu \) and \(\nu \) are both discrete, this result is generally false. In this paper, we prove that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as the sum of two deterministic plans, i.e., plans induced by transportation maps. As an application, we estimate the infinity-Wasserstein distance between two discrete probability measures \(\mu \) and \(\nu \) with the p-Wasserstein distance, times a constant depending on \(\mu \) , on \(\nu \) , and on the fixed cost function. PubDate: 2022-05-10

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Abstract: Abstract In this article, relying on Foster–Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the \(\text {L}^p\) -Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov Lévy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the \(\text {L}^p\) -Wasserstein distance for a class of Itô processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein–Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds. PubDate: 2022-05-10

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Abstract: Abstract In this paper we study a general optimal liquidation problem with a control-dependent stopping time which is the first time the stock holding becomes zero or a fixed terminal time, whichever comes first. We prove a stochastic maximum principle (SMP) which is markedly different in its Hamiltonian condition from that of the standard SMP with fixed terminal time. We present a simple example in which the optimal solution satisfies the SMP in this paper but fails the standard SMP in the literature. PubDate: 2022-05-10

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Abstract: Abstract In this paper, we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface, we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually, we apply the Hilbert Uniqueness Method, introduced by Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally, we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives. PubDate: 2022-05-10

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Abstract: Abstract In this paper, we analyze linear-quadratic stochastic differential games with a continuum of players interacting through graphon aggregates, each state being subject to idiosyncratic Brownian shocks. The major technical issue is the joint measurability of the player state trajectories with respect to samples and player labels, which is required to compute for example costs involving the graphon aggregate. To resolve this issue we set the game in a Fubini extension of a product probability space. We provide conditions under which the graphon aggregates are deterministic and the linear state equation is uniquely solvable for all players in the continuum. The Pontryagin maximum principle yields equilibrium conditions for the graphon game in the form of a forward-backward stochastic differential equation, for which we establish existence and uniqueness. We then study how graphon games approximate games with finitely many players over graphs with random weights. We illustrate some of the results with a numerical example. PubDate: 2022-05-10

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Abstract: Abstract This paper explores a novel connection between two areas: shape analysis of surfaces and unbalanced optimal transport. Specifically, we characterize the square root normal field (SRNF) shape distance as the pullback of the Wasserstein–Fisher–Rao (WFR) unbalanced optimal transport distance. In addition we propose a new algorithm for computing the WFR distance and present numerical results that highlight the effectiveness of this algorithm. As a consequence of our results we obtain a precise method for computing the SRNF shape distance directly on piecewise linear surfaces and gain new insights about the degeneracy of this distance. PubDate: 2022-05-10

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Abstract: Abstract An initial-boundary value problem for the modified 2D Zakharov–Kuznetsov equation posed on a right-hand half-strip is considered. Studied here is the critical power in nonlinearity which is a novelty for unbounded domains even with the homogeneous boundary condition. The results on existence, uniqueness and asymptotic behavior of solutions are presented. Partially, exponential-in-time decay of strong solutions in appropriated norms is proved. PubDate: 2022-05-05

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Abstract: Abstract In this work we study from the mathematical and numerical point of view a problem arising in vector-borne plant diseases. The model is written as a nonlinear system composed of a parabolic partial differential equation for the vector abundance function and a first-order ordinary differential equation for the plant health function. An existence and uniqueness result is proved using backward finite differences, uniform estimates and passing to the limit. The regularity of the solution is also obtained. Then, using the finite element method and the implicit Euler scheme, fully discrete approximations are introduced. A discrete stability property and a main a priori error estimates result are proved using a discrete version of Gronwall’s lemma and some estimates on the different approaches. Finally, some numerical results, in one and two dimensions, are presented to demonstrate the accuracy of the approximation and the behaviour of the solution. PubDate: 2022-04-19

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Abstract: Abstract We study in this paper the well-posedness and stability for two linear Schrödinger equations in d-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the sense of semigroup theory. Then, a decay estimate depending on the smoothness of initial data and the arbitrarily growth at infinity of the relaxation function is established for each equation with the help of multipliers method and some arguments devised in (Guesmia in J Math Anal Appl 382:748–760, 2011) and (Guesmia in Applicable Anal 94:184–217, 2015). PubDate: 2022-04-19

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Abstract: Abstract This paper deals with a nonlinear filtering problem in which a multi-dimensional signal process is additively affected by a process \(\nu \) whose components have paths of bounded variation. The presence of the process \(\nu \) prevents from directly applying classical results and novel estimates need to be derived. By making use of the so-called reference probability measure approach, we derive the Zakai equation satisfied by the unnormalized filtering process, and then we deduce the corresponding Kushner–Stratonovich equation. Under the condition that the jump times of the process \(\nu \) do not accumulate over the considered time horizon, we show that the unnormalized filtering process is the unique solution to the Zakai equation, in the class of measure-valued processes having a square-integrable density. Our analysis paves the way to the study of stochastic control problems where a decision maker can exert singular controls in order to adjust the dynamics of an unobservable Itô-process. PubDate: 2022-04-13

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Abstract: Abstract In this paper, we consider a large class of constrained non-cooperative stochastic Markov games with countable state spaces and discounted cost criteria. In one-player case, i.e., constrained discounted Markov decision models, it is possible to formulate a static optimisation problem whose solution determines a stationary optimal strategy (alias control or policy) in the dynamical infinite horizon model. This solution lies in the compact convex set of all occupation measures induced by strategies, defined on the set of state–action pairs. In case of n-person discounted games the occupation measures are induced by strategies of all players. Therefore, it is difficult to generalise the approach for constrained discounted Markov decision processes directly. It is not clear how to define the domain for the best-response correspondence whose fixed point induces a stationary equilibrium in the Markov game. This domain should be the Cartesian product of compact convex sets in a locally convex topological vector space. One of our main results shows how to overcome this difficulty and define a constrained non-cooperative static game whose Nash equilibrium induces a stationary Nash equilibrium in the Markov game. This is done for games with bounded cost functions and positive initial state distribution. An extension to a class of Markov games with unbounded costs and arbitrary initial state distribution relies on an approximation of the unbounded game by bounded ones with positive initial state distributions. In the unbounded case, we assume the uniform integrability of the discounted costs with respect to all probability measures induced by strategies of the players, defined on the space of plays (histories) of the game. Our assumptions are weaker than those applied in earlier works on discounted dynamic programming or stochastic games using the so-called weighted norm approach. PubDate: 2022-04-13

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Abstract: Abstract We study monotone P1 finite element methods on unstructured meshes for fully non-linear, degenerately parabolic Isaacs equations with isotropic diffusions arising from stochastic game theory and optimal control and show uniform convergence to the viscosity solution. Elliptic projections are used to manage singular behaviour at the boundary and to treat a violation of the consistency conditions from the framework by Barles and Souganidis by the numerical operators. Boundary conditions may be imposed in the viscosity or in the strong sense, or in a combination thereof. The presented monotone numerical method has well-posed finite dimensional systems, which can be solved efficiently with Howard’s method. PubDate: 2022-04-13

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Abstract: Abstract A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. In this way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via \(\varGamma \) -convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some numerical experiments. PubDate: 2022-04-13

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Abstract: Abstract Based on the fact that dissipative dynamical systems possess finite degrees of freedom, a new continuous data assimilation algorithm for the two dimensional Cahn–Hilliard–Navier–Stokes system is introduced. In this paper, we provide some suitable conditions on the nudging parameters and the size of the spatial coarse mesh observables, which are sufficient to show that the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system under the assumption that the observed data are free of error. Thus, we can make the future predictions of the exact solution by the approximation solution of the continuous data assimilation algorithm if the initial data is missing, which usually appears in the fields of geophysical and biological sciences. PubDate: 2022-04-13

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Abstract: Abstract We show that the asynchronous block-iterative primal-dual projective splitting framework introduced by P. L. Combettes and J. Eckstein in their 2018 Math. Program. paper can be viewed as an instantiation of the recently proposed warped proximal algorithm. PubDate: 2022-04-13