Additional Regularity Properties Research Articles

Finite-Time Nonconvex Optimization Using Time-Varying Dynamical Systems

In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka–Łojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence.

The Stochastic Klausmeier System and A Stochastic Schauder-Tychonoff Type Theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.

Multiobjective optimal control of a non-smooth semilinear elliptic partial differential equation

This paper is concerned with the derivation and analysis of first-order necessary optimality conditions for a class of multiobjective optimal control problems governed by an elliptic non-smooth semilinear partial differential equation. Using an adjoint calculus for the inverse of the non-linear and non-differentiable directional derivative of the solution map of the considered PDE, we extend the concept of strong stationarity to the multiobjective setting and demonstrate that the properties of weak and proper Pareto stationarity can also be characterized by suitable multiplier systems that involve both primal and dual quantities. The established optimality conditions imply in particular that Pareto stationary points possess additional regularity properties and that mollification approaches are – in a certain sense – exact for the studied problem class. We further show that the obtained results are closely related to rather peculiar hidden regularization effects that only reveal themselves when the control is eliminated and the problem is reduced to the state. This observation is also new for the case of a single objective function. The paper concludes with numerical experiments that illustrate that the derived optimality systems are amenable to numerical solution procedures.

Algebraic Calderón-Zygmund theory

Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of ‘Markov metric’ governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderón-Zygmund theory for arbitrary von Neumann algebras, but is also valid in classical settings like Riemannian manifolds with nonnegative Ricci curvature or doubling/nondoubling spaces. Other less standard commutative scenarios like fractals or abstract probability spaces are also included. Among our applications in the noncommutative setting, we improve recent results for quantum Euclidean spaces and group von Neumann algebras, respectively linked to noncommutative geometry and geometric group theory.

Semigroup properties of solutions of SDEs driven by Lévy processes with independent coordinates

We study the stochastic differential equation dXt=A(Xt−)dZt, X0=x, where Zt=(Zt(1),…,Zt(d))T and Zt(1),…,Zt(d) are independent one-dimensional Lévy processes with characteristic exponents ψ1,…,ψd. We assume that each ψi satisfies a weak lower scaling condition WLSC(α,0,C̲), a weak upper scaling condition WUSC(β,1,C¯) (where 0<α≤β<2) and some additional regularity properties. We consider two mutually exclusive assumptions: either (i) all ψ1,…,ψd are the same and α,β are arbitrary, or (ii) not all ψ1,…,ψd are the same and α>(2∕3)β. We also assume that the determinant of A(x)=(aij(x)) is bounded away from zero, and aij(x) are bounded and Lipschitz continuous. In both cases (i) and (ii) we prove that for any fixed γ∈(0,α)∩(0,1] the semigroup Pt of the process X satisfies |Ptf(x)−Ptf(y)|≤ct−γ∕α|x−y|γ||f||∞ for arbitrary bounded Borel function f. We also show the existence of a transition density of the process X.

Poincaré and logarithmic Sobolev constants for metastable Markov chains via capacitary inequalities

We investigate the metastable behavior of reversible Markov chains on possibly countable infinite state spaces. Based on a new definition of metastable Markov processes, we compute precisely the mean transition time between metastable sets. Under additional size and regularity properties of metastable sets, we establish asymptotic sharp estimates on the Poincaré and logarithmic Sobolev constant. The main ingredient in the proof is a capacitary inequality along the lines of V. Maz’ya that relates regularity properties of harmonic functions and capacities. We exemplify the usefulness of this new definition in the context of the random field Curie–Weiss model, where metastability and the additional regularity assumptions are verifiable.

Regularity and long-time behavior for a thermodynamically consistent model for complex fluids in two space dimensions

We consider a thermodynamically consistent model for the evolution of thermally conducting two-phase incompressible fluids. Complementing previous results, we prove additional regularity properties of solutions in the case when the evolution takes place in the two-dimensional flat torus with periodic boundary conditions. Thanks to improved regularity, we can also prove uniqueness and characterize the long-time behavior of trajectories showing existence of the global attractor in a suitable phase-space.

Lyapunov Functions for Weak Solutions of Reaction-Diffusion Equations with Discontinuous Interaction Functions and its Applications

AbstractIn this paper we investigate additional regularity properties for global and trajectory attractors of all globally defined weak solutions of semi-linear parabolic differential reaction-diffusion equations with discontinuous nonlinearities, when initial data uτ ∈ L2(Ω). The main contributions in this paper are: (i) sufficient conditions for the existence of a Lyapunov function for all weak solutions of autonomous differential reaction-diffusion equations with discontinuous and multivalued interaction functions; (ii) convergence results for all weak solutions in the strongest topologies; (iii) new structure and regularity properties for global and trajectory attractors. The obtained results allow investigating the long-time behavior of state functions for the following problems: (a) a model of combustion in porous media; (b) a model of conduction of electrical impulses in nerve axons; (c) a climate energy balance model; (d) a parabolic feedback control problem.

Quantitative spectral gap for thin groups of hyperbolic isometries

Let \Lambda be a subgroup of an arithmetic lattice in \mathrm{SO}(n+1 , 1) . The quotient \mathbb{H}^{n+1} / \Lambda has a natural family of congruence covers corresponding to ideals in a ring of integers. We establish a super-strong approximation result for Zariski-dense \Lambda with some additional regularity and thickness properties. Concretely, this asserts a quantitative spectral gap for the Laplacian operators on the congruence covers. This generalizes results of Sarnak and Xue (1991) and Gamburd (2002).

Regularity of Weak Solutions and Their Attractors for a Parabolic Feedback Control Problem

We investigate additional regularity properties of all globally defined weak solutions, their global and trajectory attractors for a class of autonomous differential inclusion with upper semi-continuous interaction function, when initial data \(u_{\tau}\in L^2(\Omega)\). The main contributions of this paper are: (i) additional regularity and new topological properties of all weak solutions of parabolic feedback control problem with upper semi-continuous interaction function, (ii) a sufficient condition for regularity of global and trajectory attractors, and (iii) new a priory estimates for all weak solutions.

Well‐posedness and Conservativity for Linear Control Systems (Part 2)

AbstractUsing the results obtained in Part 1, we study conservativity of a certain class of linear control problems. For this purpose we require additional regularity properties of our solution operator in order to allow point‐wise evaluations of our solution. We apply the results to the linear control system with unbounded observation and control operators mentioned in Part 1. (© 2012 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

State Constrained Optimal Control Problems with States of Low Regularity

We consider first order optimality conditions for state constrained optimal control problems. In particular, we study the case where the state equation does not have enough regularity to admit existence of a Slater point in function space. We overcome this difficulty by a special transformation. Under a density condition we show existence of Lagrange multipliers, which have a representation via measures and additional regularity properties.

Optimal Error Estimates for a Semi-Implicit Euler Scheme for Incompressible Fluids with Shear Dependent Viscosities

Certain rheological behaviors of fluids in engineering sciences are modeled by power law ansatz with $p\in(1,2]$. In the present paper a semi-implicit time discretization scheme for such fluids is proposed. The main result is the optimal $\mathcal{O}(k)$ error estimate, where k is the time step size. Our results hold in the range $p\in(3/2,2]$ (in the three-dimensional setting) for strong solutions of the continuous problem, whose existence is guaranteed under appropriate assumptions on the data. The estimates are uniform with respect to the degeneracy parameter $\delta\in[0,\delta_0]$ of the extra stress tensor. Additional regularity properties of the solution of the discrete problem are proved.

Degenerate semiconductor device equations with temperature effect

In this paper, we establish an existence assertion for the following elliptic system: - div ( a ( v ) ∇ n - a ( v ) n ∇ ψ ) = 0 in Ω , - div ( a ( v ) ∇ p + a ( v ) p ∇ ψ ) = 0 in Ω , - div ( a ( v ) ∇ ψ ) = p - n + f in Ω , - div ( a ( v ) ∇ v ) = a ( v ) | ∇ ψ | 2 in Ω coupled with suitable boundary conditions. This problem arises from the study of semiconductor devices with temperature effect and without recombination. We only assume that a is continuous and positive. This gives rise to the possibility that the system may be degenerate and/or singular. We show that, if a ( s ) does not go to zero too fast as s → ∞ , there exists a bounded weak solution, and therefore neither degeneracy nor singularity really occur. This also immediately implies some additional regularity properties for the weak solution.

Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation

We consider a semilinear wave equation, defined on a two-dimensional bounded domain Ω, with a nonlinear dissipation. Our main result is that the flow generated by the model is attracted by a finite dimensional global attractor. In addition, this attractor has additional regularity properties that depend on regularity properties of nonlinear functions in the equation. To our knowledge this is a first result of this type in the context of higher dimensional wave equations.

Variational Analysis for the Black and Scholes Equation with Stochastic Volatility

We propose a variational analysis for a Black and Scholes equation with stochastic volatility. This equation gives the price of a European option as a function of the time, of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The variational analysis involves weighted Sobolev spaces. It enables to prove qualitative properties of the solution, namely a maximum principle and additional regularity properties. Finally, we make numerical simulations of the solution, by finite element and finite difference methods.

On the Cauchy Problem for Stochastic Stokes Equations

We extend Krylov's Lp-solvability theory of the second order quasi-linear parabolic stochastic differential equations to the stochastic Stokes equation. Some additional integrability and regularity properties are also presented.

A Note on Krylov's $L_p$-Theory for Systems of SPDEs

We extend Krylov's $L_p$-solvability theory to the Cauchy problem for systems of parabolic stochastic partial differential equations. Some additional integrability and regularity properties are also presented.

Global Solutions to Maxwell Equations in a Ferromagnetic Medium

We study the Cauchy problem for the Landau-Lifschitz model in ferromagnetism without exchange energy. Once existence of global finite energy solutions is obtained, we study additional uniqueness and regularity properties of these solutions.

The Cauchy Problem in Local Spaces for the Complex Ginzburg--Landau Equation¶II. Contraction Methods

We continue the study of the initial value problem for the complex Ginzburg—Landau equation (with a > 0, b > 0, g≥ 0) in initiated in a previous paper [I]. We treat the case where the initial data and the solutions belong to local uniform spaces, more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms, but no decay conditions (or arbitrarily weak decay conditions) at infinity in . In [I] we used compactness methods and an extended version of recent local estimates [3] and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces L r for r≥ 2 or H 1. Here we treat the same problem by contraction methods. This allows us in particular to prove that the solutions obtained in [I] are unique under suitable subcriticality conditions, and to obtain for them additional regularity properties and uniform bounds. The method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces.