Infinite-dimensional Framework Research Articles

Global weak solutions to a time-periodic body-liquid interaction problem

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier–Stokes fluid interacting with a rigid body of finite size, subject to an undamped linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field {\boldsymbol V} far from the body. We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of {\boldsymbol V} and, rather remarkably, the frequency of {\boldsymbol V} is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical “invading domains” technique, our approach requires several new ideas. Indeed, due to the lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincaré map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.

Infinite dimensional Piecewise Deterministic Markov Processes

In this paper we aim to construct infinite dimensional versions of well established Piecewise Deterministic Monte Carlo methods, such as the Bouncy Particle Sampler, the Zig-Zag Sampler and the Boomerang Sampler. In order to do so we provide an abstract infinite dimensional framework for Piecewise Deterministic Markov Processes (PDMPs) with unbounded event intensities. We further develop exponential convergence to equilibrium of the infinite dimensional Boomerang Sampler, using hypocoercivity techniques. Furthermore we establish how the infinite dimensional Boomerang Sampler admits a finite dimensional approximation, rendering it suitable for computer simulation.

A convenient infinite dimensional framework for generative adversarial learning

In recent years, generative adversarial networks (GANs) have demonstrated impressive experimental results while there are only a few works that foster statistical learning theory for GANs. In this work, we propose an infinite dimensional theoretical framework for generative adversarial learning. We assume that the probability density functions of the underlying measure are uniformly bounded, k-times α-Hölder differentiable (Ck,α) and uniformly bounded away from zero. Under these assumptions, we show that the Rosenblatt transformation induces an optimal generator, which is realizable in the hypothesis space of Ck,α-generators. With a consistent definition of the hypothesis space of discriminators, we further show that the Jensen-Shannon divergence between the distribution induced by the generator from the adversarial learning procedure and the data generating distribution converges to zero. Under certain regularity assumptions on the density of the data generating process, we also provide rates of convergence based on chaining and concentration.

Consensus formation in first-order graphon models with time-varying topologies

In this paper, we investigate the asymptotic formation of consensus for several classes of time-dependent cooperative graphon dynamics. After motivating the use of this type of macroscopic models to describe multi-agent systems, we adapt the classical notion of scrambling coefficient to this setting, and leverage it to establish sufficient conditions ensuring the exponential convergence to consensus with respect to the [Formula: see text]-norm topology. We then shift our attention to consensus formation expressed in terms of the [Formula: see text]-norm, and prove three different results for symmetric, balanced and strongly connected topologies, which involve a suitable generalisation of the notion of algebraic connectivity to this infinite-dimensional framework. We then show that, just as in the finite-dimensional setting, the notion of algebraic connectivity that we propose encodes information about the connectivity properties of the underlying interaction topology. We finally use the corresponding results to shed some light on the relation between [Formula: see text]- and [Formula: see text]-consensus formation, and illustrate our contributions by a series of numerical simulations.

A topological proof of Sklar’s theorem in arbitrary dimensions

Abstract Copulas are appealing tools in multivariate probability theory and statistics. Nevertheless, the transfer of this concept to infinite dimensions entails some nontrivial topological and functional analytic issues, making a deeper theoretical understanding indispensable toward applications. In this short work, we transfer the well-known property of compactness of the set of copulas in finite dimensions to the infinite-dimensional framework. As an application, we prove Sklar’s theorem in infinite dimensions via a topological argument and the notion of inverse systems.

SPHARMA approximations for stationary functional time series on the sphere

In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in Caponera and Marinucci (Ann Stat 49(1):346–369, 2021) and Caponera et al. (Stoch Process Appl 137:167–199, 2021); more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.

Spatial Cox processes in an infinite-dimensional framework

We introduce a new class of spatial Cox processes driven by a Hilbert-valued random log-intensity. We adopt a parametric framework in the spectral domain, to estimate its spatial functional correlation structure. Specifically, we consider a spectral functional, approach based on the periodogram operator, inspired on Whittle estimation methodology. Strong consistency of the parametric estimator is proved in the linear case. We illustrate this property in a simulation study under a Gaussian first-order Spatial Autoregressive Hilbertian scenario for the log-intensity model. Our method is applied to the spatial functional prediction of respiratory disease mortality in the Spanish Iberian Peninsula, in the period 1980–2015.

Equilibrium statistical mechanics of barotropic quasi-geostrophic equations

We consider equations describing a barotropic inviscid flow in a channel with topography effects and beta-plane approximation of Coriolis force, in which a large-scale mean flow interacts with smaller scales. Gibbsian measures associated to the first integrals energy and enstrophy are Gaussian measures supported by distributional spaces. We define a suitable weak formulation for barotropic equations, and prove existence of a solution preserving Gibbsian measures, thus providing a rigorous infinite-dimensional framework for the equilibrium statistical mechanics of the model.

Stochastic system for generalized polytropic filtration

In this paper, we study a certain class of stochastic quasilinear parabolic equations describing a generalized polytropic elastic filtration in the framework of variable exponents Lebesgue and Sobolev spaces. We establish an existence result in the infinite dimensional framework of weak probabilistic solutions when the forcing terms do not satisfy Lipschitz conditions, and the governing equations are subjected to cylindrical Wiener processes. We use a Galerkin method, derive crucial a priori estimates for the approximate solutions, and combine profound analytic and probabilistic compactness results in order to pass to the limit. Several difficulties arise in obtaining these uniform bounds and passing to the limit since the nonlinear elliptic part of the leading operator admits nonstandard growth. Apart from adapting the above essential tools, we extend classical methods of monotonicity to the present situation.

Stability in shape optimization with second variation

We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the Hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.

Bayesian inversion in resin transfer molding

We study a Bayesian inverse problem arising in the context of resin transfer molding (RTM), which is a process commonly used for the manufacturing of fiber-reinforced composite materials. The forward model is described by a moving boundary problem in a porous medium. During the injection of resin in RTM, our aim is to update, on the fly, our probabilistic knowledge of the permeability of the material as soon as pressure measurements and observations of the resin moving domain become available. A probabilistic on-the-fly characterisation of the material permeability via the inversion of those measurements/observations is crucial for optimal real-time control aimed at minimising both process duration and the risk of defects formation within RTM. We consider both one-dimensional (1D) and two-dimensional (2D) forward models for RTM. Based on the analytical solution for the 1D case, we prove existence of the sequence of posteriors that arise from a sequential Bayesian formulation within the infinite-dimensional framework. For the numerical characterisation of the Bayesian posteriors in the 1D case, we investigate the application of a fully-Bayesian sequential Monte Carlo method (SMC) for high-dimensional inverse problems. By means of SMC we construct a benchmark against which we compare performance of a novel regularizing ensemble Kalman algorithm (REnKA) that we propose to approximate the posteriors in a computationally efficient manner under practical scenarios. We investigate the robustness of the proposed REnKA with respect to tuneable parameters and computational cost. We demonstrate advantages of REnKA compared with SMC with a small number of particles. We further investigate, in both the 1D and 2D settings, practical aspects of REnKA relevant to RTM, which include the effect of pressure sensors configuration and the observational noise level in the uncertainty in the log-permeability quantified via the sequence of Bayesian posteriors. The results of this work are also useful for other applications than RTM, which can be modelled by a random moving boundary problem.

Distance estimates for state constrained trajectories of infinite dimensional differential inclusions

This paper concerns estimates on the distance between a trajectory of a differential inclusion and the set of feasible trajectories of the same inclusion, feasible meaning confined to a given set of constraints. We apply these estimates to investigate Lipschitz continuity of the value functions arising in optimal control, and to variational inclusions, useful for proving non degenerate necessary optimality conditions. The main feature of our analysis is the infinite dimensional framework, which can be applied to models involving PDEs.

Software demonstration of quasi on line parametric identification related to thermal state in tokamak plasmas

In this paper, estimation of space and time varying thermal diffusivity and heating source in a nonlinear parabolic partial differential equation (PDE) describing heat transport in tokamak plasma is considered. An iterative regularization method based on a Conjugate Gradient Method (CGM) is implemented to deal with model and measurement errors. The estimation strategy is investigated in the infinite dimensional framework. Real data from the Tore Supra (TS) tokamak are provided to evaluate the performance of the proposed strategy.

Retraction statement: Adaptive distributed parameter and input estimation in linear parabolic PDEs

Summary In this paper, we discuss the on-line estimation of distributed source term, diffusion, and reaction coefficients of a linear parabolic partial differential equation using both distributed and interior-point measurements. First, new sufficient identifiability conditions of the input and the parameter simultaneous estimation are stated. Then, by means of Lyapunov-based design, an adaptive estimator is derived in the infinite-dimensional framework. It consists of a state observer and gradient-based parameter and input adaptation laws. The parameter convergence depends on the plant signal richness assumption, whereas the state convergence is established using a Lyapunov approach. The results of the paper are illustrated by simulation on tokamak plasma heat transport model using simulated data. Copyright © 2016 John Wiley & Sons, Ltd.

Exponential stability of positive semigroups in Banach spaces

The paper establishes a link between the stability of the semigroup e(−Γ+M)t and the spectral radius of Γ−1M in ordered Banach spaces. On the one hand our result allows utilizing simple estimates for the eigenvalues of −Γ+M in order to provide general conditions for the convergence of the successive approximation scheme for semilinear operator equations. On the other hand, this paper helps examining the stability of the semigroup e(−Γ+M)t for those classes of matrices −Γ and M, which lead to observable expressions for Γ−1M, e.g. when M is a coupling applied to disjoint systems representing Γ. The novelty of the paper is in the development of an infinite-dimensional framework, where an absolute value function induced by a cone is introduced and a way to deal with the lack of global continuity of eigenvalues is presented.

Infinite Dimensional Infinite-horizon Pontryagin Principles for Discrete-time Problems

The aim of this paper is the extension to setting of the infinite-dimensional state and control spaces of the method of the reduction at finite horizon. This method is a way to establish Pontryagin principles for infinite-horizon discrete-time optimal control problems. We establish several new Pontryagin principles using this approach in the infinite-dimensional framework.

Adaptive Distributed Parameter and Input Estimation in Plasma Tokamak Heat Transport

: In this paper, the adaptive estimation of spatially varying diffusion and source term coefficients for a linear parabolic partial differential equation describing tokamak plasma heat transport is considered. An estimator is defined in the infinite-dimensional framework having the system state and the parameters’ estimate as its states. Our scheme allows to estimate constant, spatially distributed and spatio-temporally distributed parameters as well as input with known upper bounds in time. While the parameters convergence depends on the plant signal richness assumption, the state convergence is established using the Lyapunov approach. Since the estimator is infinite-dimensional, the Galerkin finite-dimensional technique is used to implement it. In silico simulations are provided to illustrate the performance of the proposed approach.

Explicit approximate controllability of the Schrödinger equation with a polarizability term

We consider a controlled Schrödinger equation with a dipolar and a polarizability term, used when the dipolar approximation is not valid. The control is the amplitude of the external electric field, it acts nonlinearly on the state. We extend in this infinite dimensional framework previous techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in finite dimension. We consider a highly oscillating control and prove the semi-global weak $$H^2$$ stabilization of the averaged system using a Lyapunov function introduced by Nersesyan. Then it is proved that the solutions of the Schrödinger equation and of the averaged equation stay close on every finite time horizon provided that the control is oscillating enough. Combining these two results, we get approximate controllability to the ground state for the polarizability system with explicit controls. Numerical simulations are presented to illustrate those theoretical results.

A second-order differential system with Hessian-driven damping; Application to non-elastic shock laws

We consider the second-order differential system with Hessian-driven damping ¨ u + α ˙ u +β∇ 2 Φ(u)˙ u +∇Φ(u )+ ∇Ψ(u )= 0, where H is a real Hilbert space, Φ,Ψ : H → R are scalar potentials, and α,β are positive parameters. An interesting property of this system is that, after introduction of an auxiliary variable y, it can be equivalently written as a first- order system involving only the time derivatives ˙ u, ˙ y and the gradient operators ∇Φ, ∇Ψ. This allows to extend our analysis to the case of a convex lower semicontinuous function Φ : H → R ∪{ +∞}, and so to introduce constraints in our model. When Φ = δK is the indicator function of a closed convex set K ⊆ H , the subdifferential operator ∂Φ takes account of the contact forces, while ∇Ψ takes account of the driving forces. In this setting, by playing with the geometrical damping parameter β , we can describe nonelastic shock laws with restitution coefficient. Taking advantage of the infinite dimensional framework, we introduce a nonlinear hyperbolic PDE describing a damped oscillating system with obstacle. The first-order system is dissipative; each trajectory weakly converges to a minimizer of Φ + Ψ, provided that Φ and Φ +Ψ are convex functions. Exponential stabilization is obtained under strong convexity assumptions.

PODa-posteriorierror based inexact SQP method for bilinear elliptic optimal control problems

An optimal control problem governed by a bilinear elliptic equation is considered. This problem is solved by the sequential quadratic programming (SQP) method in an infinite-dimensional framework. In each level of this iterative method the solution of linear-quadratic subproblem is computed by a Galerkin projection using proper orthogonal decomposition (POD). Thus, an approximate (inexact) solution of the subproblem is determined. Based on a POD a-posteriori error estimator developed by Troltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115] the difference of the suboptimal to the (unknown) optimal solution of the linear-quadratic subproblem is estimated. Hence, the inexactness of the discrete solution is controlled in such a way that locally superlinear or even quadratic rate of convergence of the SQP is ensured. Numerical examples illustrate the efficiency for the proposed approach.