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Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Indranil Chowdhury, Espen Robstad Jakobsen, Robin Østern Lien

Abstract

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $σ\in(0,2)$ since they involve fractional Laplace operators $(-Δ)^{σ/2}$. 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 $σ$. The accuracy of previous approximations of fractional fully nonlinear equations depend on $σ$ and are worse when $σ$ is close to $2$. We show that the schemes are monotone, consistent, $L^\infty$-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.

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Abstract

We study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order since they involve fractional Laplace operators . 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 . The accuracy of previous approximations of fractional fully nonlinear equations depend on and are worse when is close to . We show that the schemes are monotone, consistent, -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.
Paper Structure (18 sections, 14 theorems, 83 equations, 5 figures, 2 tables)

This paper contains 18 sections, 14 theorems, 83 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. On nonlocal PDEs
  3. Wellposedness of nonlocal PDEs
  4. A differential game related to nonlocal PDEs
  5. Discretization by powers of discrete Laplacian
  6. Powers of the discrete Laplacian
  7. Numerical approximation of the nonlocal PDE
  8. Convergence of the scheme
  9. Numerical experiments
  10. Example 1: A degenerate diffusion equation in 1d
  11. Example 2: A degenerate diffusion equation in 2d
  12. Example 3: Limits of solutions as $\sigma\to0$ and $\sigma\to 2$
  13. Example 4: Convergence rate in $h$
  14. Extensions
  15. On time discretizations
  16. ...and 3 more sections

Key Result

Proposition 2.1

Assume F1 and F3. (i) (Comparison) If $u\in \text{USC}_b$ and $v\in\text{LSC}_b$ are bounded viscosity subsolution and supersolution of eq::simple_parabolic_eq respectively, then (ii) (Existence and uniqueness) There exists a unique bounded continuous viscosity solution $u$ of eq::simple_parabolic_eq. (iii) ($L^\infty$-stability) The solution $u$ in (ii) satisfies:

Figures (5)

  • Figure 1: Initial conditions $g_1$ (left), $g_2$ (middle), and $g_3$ (right).
  • Figure 2: Experiment 1a.
  • Figure 3: Experiment 1b.
  • Figure 4: Experiment 2. The figures to the right show the solution for $x \leq 0$ (top) and $y \leq 0$ (bottom).
  • Figure 5: Experiment 3.

Theorems & Definitions (32)

  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: EJT18b
  • proof
  • Theorem 3.3: Comparison
  • proof
  • Theorem 3.4: Existence, uniqueness, and $L^\infty$-stability
  • proof
  • ...and 22 more