scholarly journals A general convergence result for viscosity solutions of Hamilton-Jacobi equations and non-linear semigroups

2021 ◽  
pp. 109346
Author(s):  
Richard C. Kraaij
Keyword(s):  
Viscosity Solutions ◽  
Convergence Result ◽  
Non Linear ◽  
Jacobi Equations ◽  
General Convergence

scholarly journals VISCOSITY SOLUTIONS OF HAMILTON–JACOBI EQUATIONS WITH DISCONTINUOUS COEFFICIENTS

2007 ◽  
Vol 04 (04) ◽  
pp. 771-795 ◽  
Author(s):  
GIUSEPPE MARIA COCLITE ◽  
NILS HENRIK RISEBRO
Keyword(s):  
Cauchy Problem ◽  
Viscosity Solution ◽  
Viscosity Solutions ◽  
Front Tracking ◽  
Discontinuous Coefficients ◽  
Comparison Result ◽  
The Cauchy Problem ◽  
Temporal Location ◽  
Additional Regularity ◽  
Jacobi Equations

We consider Hamilton–Jacobi equations, where the Hamiltonian depends discontinuously on both the spatial and temporal location. Our main result is the existence of viscosity solution to the Cauchy problem, and that the front tracking algorithm yields an L∞ contractive semigroup. We define a viscosity solution by treating the discontinuities in the coefficients analogously to "internal boundaries". The existence of viscosity solutions is established constructively via a front tracking approximation, whose limits are viscosity solutions, where by "viscosity solution" we mean a viscosity solution that posses some additional regularity at the discontinuities in the coefficients. We then show a comparison result that is valid for these viscosity solutions.

scholarly journals Convergence rates for estimators of geodesic distances and Frèchet expectations

2018 ◽  
Vol 55 (4) ◽  
pp. 1001-1013
Author(s):  
Catherine Aaron ◽  
Olivier Bodart
Keyword(s):  
Probability Distribution ◽  
Convergence Rate ◽  
Compact Manifold ◽  
Convergence Rates ◽  
Geodesic Distance ◽  
Convergence Result ◽  
Regularity Conditions ◽  
Compact Set ◽  
Geodesic Distances ◽  
General Convergence

Abstract Consider a sample 𝒳n={X1,…,Xn} of independent and identically distributed variables drawn with a probability distribution ℙX supported on a compact set M⊂ℝd. In this paper we mainly deal with the study of a natural estimator for the geodesic distance on M. Under rather general geometric assumptions on M, we prove a general convergence result. Assuming M to be a compact manifold of known dimension d′≤d, and under regularity assumptions on ℙX, we give an explicit convergence rate. In the case when M has no boundary, knowledge of the dimension d′ is not needed to obtain this convergence rate. The second part of the work consists in building an estimator for the Fréchet expectations on M, and proving its convergence under regularity conditions, applying the previous results.

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