Theory Of Viscosity Solutions Research Articles

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

AbstractWe study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

Stability and guaranteed error control of approximations to the Monge–Ampère equation

This paper analyzes a regularization scheme of the Monge–Ampère equation by uniformly elliptic Hamilton–Jacobi–Bellman equations. The main tools are stability estimates in the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document} norm from the theory of viscosity solutions which are independent of the regularization parameter ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}. They allow for the uniform convergence of the solution uε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_\\varepsilon $$\\end{document} to the regularized problem towards the Alexandrov solution u to the Monge–Ampère equation for any nonnegative Ln\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^n$$\\end{document} right-hand side and continuous Dirichlet data. The main application are guaranteed a posteriori error bounds in the L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^\\infty $$\\end{document} norm for continuously differentiable finite element approximations of u or uε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_\\varepsilon $$\\end{document}.

Regularity of viscosity solutions to fully nonlinear elliptic transmission problems

We develop the regularity theory of viscosity solutions to transmission problems for fully nonlinear second order uniformly elliptic equations. Our results give a complete theory of existence, uniqueness, comparison principle, and regularity of solutions to flat interface transmission problems; and the C0,α, C1,α and C2,α regularity of viscosity solutions up to the transmission surface for the case of curved interfaces.

An application of the theory of viscosity solutions to higher order differential equations

An application of the theory of viscosity solutions to higher order differential equations

Abstraction of Continuous-Time Systems Based on Feedback Controllers and Mixed Monotonicity

In this paper, we consider the problem of the computation of efficient symbolic abstractions for continuous-time control systems. The new abstraction algorithm builds symbolic models with the same number of states but fewer transitions in comparison to the one produced by the standard algorithm. At the same time, the new abstract system is at least as controllable as the standard one. The proposed algorithm is based on the solution of a region-to-region control synthesis problem. This solution is formally obtained using the theory of viscosity solutions of the dynamic programming equation and the theory of differential equations with discontinuous righthand side. In the new abstraction algorithm, the symbolic controls are essentially the feedback controllers that solve this control synthesis problem. The improvement in the number of transitions is achieved by reducing the number of successors for each symbolic control. For a certain class of control systems, with a suitable set of discretization parameters, the new algorithm may even produce deterministic abstract systems or systems with a singleton input alphabet. The approach is illustrated by examples that compare the two abstraction algorithms.

Singularity formation and regularization at multiple times in the viscous Hamilton–Jacobi equation

The Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation (VHJ) from stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after appearance of singularity, which occurs through gradient blow-up (GBU) on the boundary. Whereas viscosity solution theory has been extensively applied to many PDEs, there seem to be less results on refined singular behavior of solutions. Although occurrence of two types of GBU, with or without loss of boundary condition (LBC), are known, detailed behavior after GBU has remained open except for a strongly restricted special class of one dimensional solutions. In this paper, in general dimensions, we construct solutions which undergo GBU with LBC arbitrarily many times and then recover regularity, as well as solutions without LBC at first GBU time. In one space dimension, we obtain the complete classification of viscosity solutions at each time, which extends to radial case in higher dimensions. Furthermore we show the existence of solutions exhibiting an arbitrarily given combination of GBU types with/without LBC at multiple times, in which a new type of behavior called “bouncing” is discovered. Global weak solutions of VHJ with multiple times singularity turn out to display larger variety of behaviors than in Fujita equation. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires delicate arguments. Since we do not rely on any known special solution unlike in Fujita equation, our method is expected to apply to other equations. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. Our results imply that for certain spatial distributions of rewards, if a controlled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.

Limits and consistency of nonlocal and graph approximations to the Eikonal equation

Abstract In this paper, we study a nonlocal approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the nonlocal problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and nonlocal problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the nonlocal problem and that of the local one, both in continuous time and forward Euler time discretization. We then turn to studying continuum limits of nonlocal problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows then, almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.

New proofs of Khinchin's law of large numbers and Lindeberg's central limit theorem –PDE's approach

<p style='text-indent:20px;'>Both Khinchin's law of large numbers (Khinchin's LLN) and Lindeberg's central limit theorem (Lindeberg's CLT) are fundamental results in probability theory. In this paper, we give the new proofs of these two theorems. A law of large numbers and a central limit theorem are proved for independent and non-identical distributed random variables. Indeed, these results include the Khinchin's LLN and Lindeberg's CLT. Our main tool is the viscosity solution theory of partial differential equation (PDE).</p>

Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments

We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equations and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable x/t and in which the time and space derivatives are coupled together. We first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s=x/t. In terms of the standard Fisher-KPP equation of reaction-diffusion type, we give explicit formulas of the spreading speed when the environment has one or two shifting speeds. As a byproduct, we also introduce a novel class of “asymptotically homogeneous” environments which share the same spreading speed with the corresponding homogeneous environments.

Viscosity solutions for doubly nonlinear evolution equations

We extend the theory of viscosity solutions to treat scalar-valued doubly nonlinear evolution equations. Such equations arise naturally in many mechanical models including a dry friction. After providing a suitable definition for discontinuous viscosity solutions in this setting, we show that Perron's construction is still available, i.e. we prove an existence result. Moreover, we will prove comparison principles and stability results for these problems. The theoretical considerations are accompanied by several examples, e.g. we prove the existence of a solution to a rate-independent level-set mean curvature flow. Finally, we discuss in detail a rate-independent ordinary differential equation stemming from a problem with non-convex energy. We show that the solution obtained by maximal minimizing movements and the solution obtained by the vanishing viscosity method coincide with the upper and lower Perron solutions and show the emergence of a rate-independent hysteresis loop.

On a vector-valued generalisation of viscosity solutions for general PDE systems

We propose a theory of non-differentiable solutions which applies to fully nonlinear PDE systems and extends the theory of viscosity solutions of Crandall–Ishii–Lions to the vectorial case. Our key ingredient is the discovery of a notion of extremum for maps which extends min-max and allows “nonlinear passage of derivatives” to test maps. This new PDE approach supports certain stability and convergence results, preserving some basic features of the scalar viscosity counterpart. In this introductory work, we focus on studying the analytical foundations of this new theory.

Crandall–Lions viscosity solutions for path-dependent PDEs: The case of heat equation

We address our interest to the development of a theory of viscosity solutions à la Crandall–Lions for path-dependent partial differential equations (PDEs), namely PDEs in the space of continuous paths C([0,T];Rd). Path-dependent PDEs can play a central role in the study of certain classes of optimal control problems, as for instance optimal control problems with delay. Typically, they do not admit a smooth solution satisfying the corresponding HJB equation in a classical sense, it is therefore natural to search for a weaker notion of solution. While other notions of generalized solution have been proposed in the literature, the extension of the Crandall–Lions framework to the path-dependent setting is still an open problem. The question of uniqueness of the solutions, which is the most delicate issue, will be based on early ideas from the theory of viscosity solutions and a suitable variant of Ekeland’s variational principle. This latter is based on the construction of a smooth gauge-type function, where smooth is meant in the horizontal/vertical (rather than Fréchet) sense. In order to make the presentation more readable, we address the path-dependent heat equation, which in particular simplifies the smoothing of its natural “candidate” solution. Finally, concerning the existence part, we provide a functional Itô formula under general assumptions, extending earlier results in the literature.

Pricing Bermudan Option with Variable Transaction Costs under the Information-Based Model

The Bermudan option pricing problem with variable transaction costs is considered for a risky asset whose price process is derived under the information-based model. The price is formulated as the value function of an optimal stopping problem, which is the value function of a stochastic control problem given by a non-linear second order partial differential equation. The theory of viscosity solutions is applied to solve the stochastic control problem such that the value function is also the solution of the corresponding Bellman equation. Under some regularity assumptions, the existence and uniqueness of the solution of the pricing equation are derived by the application of the Perron method and Banach Fixed Point theorem.

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

A viscosity solution approach to regularity properties of the optimal value function

We analyze the optimal value function v associated with a general parametric optimization problem via the theory of viscosity solutions. The novelty is that we obtain regularity properties of v by showing that it is a viscosity solution to a set of first-order equations. As a consequence, in Banach spaces, we provide sufficient conditions for local and global Lipschitz properties of v. We also derive, in finite dimensions, conditions for optimality through a comparison principle. Finally, we study the relationship between viscosity and Clarke generalized solutions to get further differentiability properties of v in Euclidean spaces.

Finite time extinction for the 1D stochastic porous medium equation with transport noise

We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate smash {O(t^frac{1}{2})} whereas the support of solutions to the deterministic PME grows only with rate smash {O(t^{frac{1}{m{+}1}})}. The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.

Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions

We study a stochastic optimal control problem for a two scale system driven by an infinite dimensional stochastic differential equation which consists of “slow” and “fast” components. We use the theory of viscosity solutions in Hilbert spaces to show that as the speed of the fast component goes to infinity, the value function of the optimal control problem converges to the viscosity solution of a reduced effective equation. We consider a rather general case where the evolution is given by an abstract semilinear stochastic differential equation with nonlinear dependence on the controls. The results of this paper generalize to the infinite dimensional case the finite dimensional results of Alvarez and Bardi [SIAM J. Control Optim.40(2001/02) 1159–1188] and complement the results in Hilbert spaces obtained recently in Guatteri and Tessitore [To appear in:Appl. Math. Optim.(2019)https://doi.org/10.1007/s00245-019-09577-y].

Singular Neumann Boundary Problems for a Class of Fully Nonlinear Parabolic Equations in One Dimension

In this paper, we discuss a singular Neumann boundary problem for a class of nonlinear parabolic equations in one space dimension. Our boundary problem describes the motion of a planar curve sliding along the boundary with a zero contact angle, which can be viewed as a limiting model for the capillary phenomenon. We study the uniqueness and existence of solutions by using the viscosity solution theory. We also show the convergence of the solution to a traveling wave as time proceeds to infinity when the initial value is assumed to be convex.

Viscosity solutions to an initial value problem for a Hamilton–Jacobi equation with a degenerate Hamiltonian occurring in the dynamics of peakons

We consider an initial value problem for a Hamilton–Jacobi equation with a quadratic and degenerate Hamiltonian. Our Hamiltonian comes from the dynamics of N-peakon in the Camassa–Holm equation. It is given by a quadratic form with a singular positive semi-definite matrix. Such a problem does not fall into the standard theory of viscosity solutions. Also viability related results, sometimes used to deal with degenerate Hamiltonians, do not seem applicable in our case. We prove the global existence of a viscosity solution by looking at the associated optimal control problem and showing that the value function is a viscosity solution. The most complicated part is the continuity of a viscosity solution which is obtained in the two-peakon case only. The source of the difficulties is the non-uniqueness of solutions to the state equation in the optimal control problem. We prove that the viscosity solution is Lipschitz continuous and unique on some short time interval if the initial condition is Lipschitz continuous. We end the paper with an example showing the loss of Lipschitz continuity of a viscosity solution in the one-dimensional case.

The limit shape of convex hull peeling

We prove that the convex peeling of a random point set in dimension d approximates motion by the 1/(d + 1) power of Gaussian curvature. We use viscosity solution theory to interpret the limiting partial differential equation. We use the Martingale method to solve the cell problem associated to convex peeling. Our proof follows the program of Armstrong-Cardaliaguet for homogenization of geometric motions, but with completely different ingredients.