Game theoretical methods in PDEs

Lewicka, Marta; Manfredi, Juan J.

doi:10.1007/s40574-014-0011-z

Cited by 17 publications

(13 citation statements)

References 13 publications

(14 reference statements)

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“…Theorem 1 proves that graph-based semi-supervised learning with the game theoretic p-Laplacian is well-posed in the limit of finite labeled data and infinite unlabeled data when p > d. We prove in Section 5 that weak distributional solutions of ( 22) are equivalent to viscosity solutions, so we can also interpret u as the unique weak solution of (22). By regularity theory for weak solutions of the weighted p-Laplace equation [33,34], we have u ∈ C 1,α loc (T d \O) and if f is smooth then u ∈ C ∞ (B(x, r)) near any point where ∇u(x) = 0.…”

Section: Resultsmentioning

confidence: 92%

The game theoreticp-Laplacian and semi-supervised learning with few labels

Calder

2018

Nonlinearity

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We study the game theoretic p-Laplacian for semi-supervised learning on graphs, and show that it is well-posed in the limit of finite labeled data and infinite unlabeled data. In particular, we show that the continuum limit of graph-based semi-supervised learning with the game theoretic p-Laplacian is a weighted version of the continuous p-Laplace equation. We also prove that solutions to the graph p-Laplace equation are approximately Hölder continuous with high probability. Our proof uses the viscosity solution machinery and the maximum principle on a graph.

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Section: Resultsmentioning

confidence: 92%

The game theoreticp-Laplacian and semi-supervised learning with few labels

Calder

2018

Nonlinearity

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“…See [11] for a discussion on this topic. Such approximations were first presented in [29] (see also [20,30,31] for a probabilistic game theoretical approach). The basic idea of these approximations is to combine the classical mean value property (MVP) for the Laplacian with a MVP for the normalized infinity Laplacian motivated by Tug-of-War games [35].…”

Section: Related Resultsmentioning

confidence: 99%

A Finite Difference Method for the Variational p-Laplacian

Teso

Lindgren

2021

J Sci Comput

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We propose a new monotone finite difference discretization for the variational p-Laplace operator, $$\Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u),$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.

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“…Juutinen [37] investigated the asymptotic behavior for (1.2). For more results on the stochastic tug-of-war game and the p-Laplacian operators, see for instance [40,42,46,54].…”

Section: Introductionmentioning

confidence: 99%

Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity

Fang¹,

Zhang²

2021

Preprint

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We introduce a new class of quasi-linear parabolic equations involving nonhomogeneous degeneracy or/and singularitywhere 1 < p < ∞, −1 < q ≤ s < ∞ and a(x, t) ≥ 0. The motivation to investigate this model stems not only from the connections to tug-of-war like stochastic games with noise, but also from the non-standard growth problems of double phase type. According to different values of q, s, such equations include nonhomogeneous degeneracy or singularity, and may involve these two features simultaneously. In particular, when q = p − 2 and q < s, it will encompass the parabolic p-Laplacian both in divergence form and in non-divergence form. We aim to explore the from L ∞ to C 1,α regularity theory for the aforementioned problem. To be precise, under some proper assumptions, we use geometrical methods to establish the local Hölder regularity of spatial gradients of viscosity solutions.

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Game theoretical methods in PDEs

Cited by 17 publications

References 13 publications

The game theoreticp-Laplacian and semi-supervised learning with few labels

The game theoreticp-Laplacian and semi-supervised learning with few labels

A Finite Difference Method for the Variational p-Laplacian

Regularity for quasi-linear parabolic equations with nonhomogeneous degeneracy or singularity

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