Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Agarwal, Ankush; Claisse, Julien

doi:10.48550/arxiv.1704.00328

Cited by 5 publications

(15 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In Section 2 we discussed stochastic representations for non-Dirichlet boundary conditions, unfortunately such boundary conditions have not been considered in this more general setting, but are future work for the authors. For the interested reader we mention recent analysis of the Monte Carlo Branching methodology to elliptic PDEs [9].…”

Section: Branching Diffusions and The Kpp Equationmentioning

confidence: 99%

Hybrid PDE solver for data-driven problems and modern branching

Bernal¹,

Reis²,

Smith³

2017

Eur. J. Appl. Math

View full text Add to dashboard Cite

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.

show abstract

Section: Branching Diffusions and The Kpp Equationmentioning

confidence: 99%

Hybrid PDE solver for data-driven problems and modern branching

Bernal¹,

Reis²,

Smith³

2017

Eur. J. Appl. Math

View full text Add to dashboard Cite

show abstract

“…Recently, several probabilistic approximation methods for high-dimensional nonlinear PDEs have been proposed in hopes of overcoming the curse of dimensionality in the numerical approximation of nonlinear PDEs. We refer, e.g., to [3,4,5,8,9,14,16,20,21,24,26,29,34,35,36,38,41,50,51,52,54,57] for deep learning based approximation methods for possibly nonlinear PDEs, e.g., to [1,10,12,13,15,37,39,40,53,55,58,59] for approximation methods for nonlinear secondorder parabolic PDEs based on branching diffusions, and, e.g., to [7,22,23,27,42,44,45,46] for full-history recursive multilevel Picard (MLP) approximation methods for nonlinear secondorder parabolic PDEs. Numerical experiments raise hopes that deep learning based approximation methods are able to approximate solutions of high-dimensional nonlinear PDE problems, but at the moment there are only partial explanations for the good performance of deep learning based approximation methods in numerical experiments for high-dimensional PDEs available (cf., e.g.,…”

Section: Introductionmentioning

confidence: 99%

“…In the following we add comments on some of the mathematical objects appearing in Theorem 1.1. The functions u d : R d Ñ R, d P N, in Theorem 1.1 above describe the solutions of the elliptic PDEs which we intend to solve approximately; see (1) above. The functions f d : R d ˆR Ñ R, d P N, represent the nonlinearities in the elliptic PDEs in (1).…”

Section: Introductionmentioning

confidence: 99%

“…The functions u d : R d Ñ R, d P N, in Theorem 1.1 above describe the solutions of the elliptic PDEs which we intend to solve approximately; see (1) above. The functions f d : R d ˆR Ñ R, d P N, represent the nonlinearities in the elliptic PDEs in (1). The nonlinearities are assumed to satisfy certain Lipschitz conditions which are formulated with the help of the real numbers L P r0, 8q and λ P pL, 8q in Theorem 1.1 above.…”

Section: Introductionmentioning

confidence: 99%

“…The nonlinearities are assumed to satisfy certain Lipschitz conditions which are formulated with the help of the real numbers L P r0, 8q and λ P pL, 8q in Theorem 1.1 above. The random fields U d,θ n : R d ˆΩ Ñ R, d P N, n P N 0 , θ P Θ, in (2) above describe the approximation algorithm which we employ in this work to approximately calculate the PDE solutions u d : R d Ñ R, d P N. The motivation for the specific form of the random fields U d,θ n : R d ˆΩ Ñ R, d P N, n P N 0 , θ P Θ, in (2) stems from multilevel Monte Carlo approximations of Picard approximations of certain stochastic fixed point equations (SFPEs) which are satisfied by the solutions u d : R d Ñ R, d P N, of the elliptic PDEs in (1). The different numbers of Monte Carlo samples in (2) are determined by the constant M P N which is restricted by the real numbers L P r0, 8q and λ P pL, 8q used to formulate the Lipschitz assumptions for the nonlinearities f d : R d ˆR Ñ R, d P N, in the PDEs in (1) above.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

Beck,

Gonon,

Jentzen

2020

Preprint

View full text Add to dashboard Cite

Recently, so-called full-history recursive multilevel Picard (MLP) approximation schemes have been introduced and shown to overcome the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations (PDEs) with Lipschitz nonlinearities. The key contribution of this article is to introduce and analyze a new variant of MLP approximation schemes for certain semilinear elliptic PDEs with Lipschitz nonlinearities and to prove that the proposed approximation schemes overcome the curse of dimensionality in the numerical approximation of such semilinear elliptic PDEs. Theorem 1.1. Let c, L P r0, 8q, λ P pL, 8q, M P NXpp ? λ`?Lq 2 p ? λ´?Lq ´2, 8q, Θ " Y nPN Z n , let u d P C 2 pR d , Rq, d P N, and f d P CpR d ˆR, Rq, d P N, satisfy for all d P N, x P R d that p∆u d qpxq " f d px, u d pxqq, (1) let pΩ, F , Pq be a probability space, let W d,θ : r0, 8q ˆΩ Ñ R d , θ P Θ, d P N, be i.i.d. standard Brownian motions, let R θ : Ω Ñ r0, 8q, θ P Θ, be i.i.d. random variables, assume that pR θ q θPΘ and pW d,θ q pd,θqPNˆΘ are independent, assume for all d P N, x " px 1 , . . . , x d q P R d , v, w P R, ε P p0, 8q that |f d px, vq ´fd px, wq ´λpv ´wq| ď L|v ´w|, |f d px, 0q| ď cd c r1 řd j"1 |x j |s c , sup y"py 1 ,...,y d qPR d r|u d pyq| expp´ε ř d j"1 |y j |qs ă 8, and PpR 0 ě εq " e ´λε , let U d,θ n " pU d,θ n pxqq xPR d : R d ˆΩ Ñ R, θ P Θ, d, n P N 0 , satisfy for all d, n P N, θ P Θ, x P R d that U d,θ 0 pxq " 0 and U d,θ n pxq " ´1 λM n « M n ÿ m"1 f d `x `?2 W d,pθ,0,mq R pθ,0,mq , 0 ˘ff `n´1 ÿ k"1 1 λM pn´kq « M pn´kq ÿ m"1 ˆλ" U d,pθ,k,mq k `x `?2 W d,pθ,k,mq R pθ,k,mq ˘´U d,pθ,k,´mq k´1 `x `?2 W d,pθ,k,mq R pθ,k,mq ˘ı ´"f d ´x `?2 W d,pθ,k,mq R pθ,k,mq , U d,pθ,k,mq k `x `?2 W d,pθ,k,mq R pθ,k,mq ˘f d ´x `?2 W d,pθ,k,mq R pθ,k,mq , U d,pθ,k,´mq k´1 `x `?2 W d,pθ,k,mq R pθ,k,mq

show abstract

An Unbiased Itô Type Stochastic Representation for Transport PDEs: A Toy Example

Reis

Smith

2019

Springer Proceedings in Mathematics &Amp; Statistics

View full text Add to dashboard Cite

We propose a stochastic representation for a simple class of transport PDEs based on Itô representations. We detail an algorithm using an estimator stemming for the representation that, unlike regularization by noise estimators, is unbiased. We rely on recent developments on branching diffusions, regime switching processes and their representations of PDEs.There is a loose relation between our technique and regularization by noise, but contrary to the latter, we add a perturbation and immediately its correction. The method is only possible through a judicious choice of the diffusion coefficient σ. A key feature is that our approach does not rely on the smallness of σ, in fact, our σ is strictly bounded from below which is in stark contrast with standard perturbation techniques. This is critical for extending this method to non-toy PDEs which have nonlinear terms in the first derivative where the usual perturbation technique breaks down.The examples presented show the algorithm outperforming alternative approaches. Moreover, the examples point toward a potential algorithm for the fully nonlinear case where the method of characteristics break down.

show abstract

Branching diffusion representation of semi-linear elliptic PDEs and estimation using Monte Carlo method

Cited by 5 publications

References 0 publications

Hybrid PDE solver for data-driven problems and modern branching

Hybrid PDE solver for data-driven problems and modern branching

Overcoming the curse of dimensionality in the numerical approximation of high-dimensional semilinear elliptic partial differential equations

An Unbiased Itô Type Stochastic Representation for Transport PDEs: A Toy Example

Contact Info

Product

Resources

About