Numerical analysis of degenerate Kolmogorov equations of constrained stochastic Hamiltonian systems

Laurent Mertz; Olivier Pironneau

doi:10.1016/j.camwa.2019.04.013

Numerical analysis of degenerate Kolmogorov equations of constrained stochastic Hamiltonian systems

Laurent Mertz
, Olivier Pironneau^*

^*Corresponding author for this work

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

2 Citations (Scopus)

Abstract

In this work, we propose a method to compute numerical approximations of the invariant measures and Rice's formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in Mertz et al. (2018). A functional analysis framework is presented; regularization and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.

Original language	English
Pages (from-to)	2719-2733
Journal	Computers and Mathematics with Applications
Volume	78
Issue number	8
Online published	13 May 2019
DOIs	https://doi.org/10.1016/j.camwa.2019.04.013
Publication status	Published - 15 Oct 2019
Externally published	Yes

Funding

L. Mertz is supported by the National Natural Science Foundation of China, Research Fund for International Young Scientists under the project #1161101053 titled “Computational methods for non-smooth dynamical systems excited by random forces” and the Young Scientist Program, China under the project #11601335 titled “Stochastic Control Method in Probabilistic Engineering Mechanics”. L.M. also thanks the LJLL for its hospitality.

Research Keywords

Constrained stochastic Hamiltonian system
Nonlocal boundary conditions
Partial differential equations
Rice's formula

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https://www.sciencedirect.com/science/article/am/pii/S0898122119302111Licence: Elsevier user license

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abstract = "In this work, we propose a method to compute numerical approximations of the invariant measures and Rice's formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in Mertz et al. (2018). A functional analysis framework is presented; regularization and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.",

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T1 - Numerical analysis of degenerate Kolmogorov equations of constrained stochastic Hamiltonian systems

AU - Mertz, Laurent

AU - Pironneau, Olivier

PY - 2019/10/15

Y1 - 2019/10/15

N2 - In this work, we propose a method to compute numerical approximations of the invariant measures and Rice's formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in Mertz et al. (2018). A functional analysis framework is presented; regularization and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.

AB - In this work, we propose a method to compute numerical approximations of the invariant measures and Rice's formula (frequency of threshold crossings) for a certain type of stochastic Hamiltonian system constrained by an obstacle and subjected to white or colored noise. As an alternative to probabilistic Monte-Carlo simulations, our approach relies on solving a class of degenerate partial differential equations with non-local Dirichlet boundary conditions, as derived in Mertz et al. (2018). A functional analysis framework is presented; regularization and approximation by the finite element method is applied; numerical experiments on these are performed and show good agreement with probabilistic simulations.

KW - Constrained stochastic Hamiltonian system

KW - Nonlocal boundary conditions

KW - Partial differential equations

KW - Rice's formula

UR - https://www.scopus.com/pages/publications/85065334940

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85065334940&origin=recordpage

U2 - 10.1016/j.camwa.2019.04.013

DO - 10.1016/j.camwa.2019.04.013

M3 - RGC 21 - Publication in refereed journal

SN - 0898-1221

VL - 78

SP - 2719

EP - 2733

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

IS - 8

ER -

Numerical analysis of degenerate Kolmogorov equations of constrained stochastic Hamiltonian systems

Abstract

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