Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Multilevel Monte Carlo Method for Ergodic SDEs without Contractivity

This paper proposes a new multilevel Monte Carlo (MLMC) method for the ergodic SDEs which do not satisfy the contractivity condition. By introducing the change of measure technique, we simulate the path with contractivity and add the Radon-Nykodim derivative to the estimator. We can show the strong error of the path is uniformly bounded with respect to $T.$ Moreover, the variance of the new level estimators increase linearly in $T,$ which is a great reduction compared with the exponential increase in standard MLMC. Then the total computational cost is reduced to $O(\varepsilon^{-2}|\log \varepsilon|^{2})$ from $O(\varepsilon^{-3}|\log \varepsilon|)$ of the standard Monte Carlo method. Numerical experiments support our analysis.Comment: 39 pages, 7 figure

Multilevel Estimation of Expected Exit Times and Other Functionals of Stopped Diffusions

This paper proposes and analyses a new multilevel Monte Carlo method for the estimation of mean exit times for multi-dimensional Brownian diffusions, and associated functionals which correspond to solutions to high dimensional parabolic PDEs through the Feynman-Kac formula. In particular, it is proved that the complexity to achieve an ε root-mean-square error is O (ε−2 |log ε|3)

Multilevel estimation of expected exit times and other functionals of stopped diffusions

See README file within datase

Multilevel estimation of expected exit times and other functionals of stopped diffusions

This paper proposes and analyzes a new multilevel Monte Carlo method for the estimation of mean exit times for multidimensional Brownian diffusions and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman–Kac formula. In particular, it is proved that the complexity to achieve an ε root-mean-square error is O(ε −2 |log ε| 3 )

Multilevel estimation of expected exit times and other functionals of stopped diffusions

This paper proposes and analyzes a new multilevel Monte Carlo method for the estimation of mean exit times for multidimensional Brownian diffusions and associated functionals which correspond to solutions to high-dimensional parabolic PDEs through the Feynman–Kac formula. In particular, it is proved that the complexity to achieve an ε root-mean-square error is O(ε −2 |log ε| 3 )

Multilevel estimation of expected exit times and other functionals of stopped diffusions

See README file within datase