A penalty scheme and policy iteration for nonlocal HJB variational inequalities with monotone nonlinearities

Reisinger, Christoph; Zhang, Yufei

doi:10.1016/j.camwa.2021.04.011

Cited by 6 publications

(2 citation statements)

References 37 publications

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“…In the setting of nonlocal Bellman-Isaacs equations, such approximations were introduced in [36,16,10] with further developments, including error estimates, in e.g. [8,21,43,27,9,10,36]. These approximations has fractional order accuracy, depending on the order of the fractional/nonlocal operator.…”

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confidence: 99%

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Chowdhury,

Jakobsen,

Lien

2024

Preprint

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We study discretizations by powers of discrete Laplacians of fully nonlinear equations. Our problems are parabolic and of order σ є (0,2) since they involve fractional Laplace operators (-Δ)σ/2. They arise e.g. in control and game theory as dynamic programming equations, and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations which are 2nd order accurate for all values of σ. The accuracy of previous approximations depend on σ and are worse when σ is close to 2. We show that the schemes are monotone, consistent, L∞-stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We present several numerical examples. 2020 Mathematics Subject Classification. 49L25, 35J60, 34K37, 35R11, 35J70, 45K05, 49L25, 49M25, 93E20, 65N06, 65R20, 65N12.

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Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Chowdhury,

Jakobsen,

Lien

2024

Preprint

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“…In the setting of HJB equations, it was introduced in [48,22,11] with further developments in e.g. [8,26,56,34]. We will give new results for this approximation here, and focus on a version based on semi-Lagrangian type approximations [21,30] of the nonlocal operators [22].…”

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confidence: 99%

On Numerical Approximations of Fractional and Nonlocal Mean Field Games

2022

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We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

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Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

Chowdhury,

Jakobsen,

Lien

2024

Dyn Games Appl

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We 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.

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A penalty scheme and policy iteration for nonlocal HJB variational inequalities with monotone nonlinearities

Cited by 6 publications

References 37 publications

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

Discretization of fractional fully nonlinear equations by powers of discrete Laplacians

On Numerical Approximations of Fractional and Nonlocal Mean Field Games

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

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