Multi-Index Sequential Monte Carlo Methods for Partially Observed Stochastic Partial Differential Equations

Jasra, Ajay; Law, Kody J. H.; Xu, Yaxian

doi:10.1615/int.j.uncertaintyquantification.2020033219

Cited by 9 publications

(16 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Another potential direction is consider the extension and development of a a multi-index Monte Carlo method, cf. [16,22], based on the correctly coupled exponential Euler method. This has the potential of further improving tractability in higher-dimensional physical space and low-regularity settings.…”

Section: Discussionmentioning

confidence: 99%

Improved efficiency of multilevel Monte Carlo for stochastic PDE through strong pairwise coupling

Chada¹,

Hoel²,

Jasra³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Multilevel Monte Carlo (MLMC) has become an important methodology in applied mathematics for reducing the computational cost of weak approximations. For many problems, it is well-known that strong pairwise coupling of numerical solutions in the multilevel hierarchy is needed to obtain efficiency gains. In this work, we show that strong pairwise coupling indeed is also important when (MLMC) is applied to stochastic partial differential equations (SPDE) of reaction-diffusion type, as it can improve the rate of convergence and thus improve tractability. For the (MLMC) method with strong pairwise coupling that was developed and studied numerically on filtering problems in [Chernov et al., Numer. Math., 147 (2021), 71-125], we prove that the rate of computational efficiency is higher than for existing methods. We also provide numerical comparisons with alternative coupling ideas illustrate the importance of this feature. The comparisons are conducted on a range of SPDE, which include a linear SPDE, a stochastic reaction-diffusion equation, and stochastic Allen-Cahn equation.

show abstract

Section: Discussionmentioning

confidence: 99%

Improved efficiency of multilevel Monte Carlo for stochastic PDE through strong pairwise coupling

Chada¹,

Hoel²,

Jasra³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Over the past decade, there has been an explosion of interest in applying these methods to inference, e.g. see [25,15,3,26,33] for examples of MLMC and [31,35] for MIMC. A notable benefit of MC methods is easy parallelizability, however typically MLMC and MIMC methods for inference are much more synchronous, or even serial in the case of MCMC.…”

Section: The Sweet and The Bitter Of Bayesmentioning

confidence: 99%

“…Achieving such estimates with efficient inverse MC methods has been the focus of a large body of work. These methods can be classified according to 3 primary strategies: importance sampling [25,3,2,42,37], coupled algorithms [15,26,29,21,34], and approximate couplings [30,31,35]. See e.g.…”

Section: Now One Leverages the Telescopic Summentioning

confidence: 99%

See 1 more Smart Citation

Randomized multilevel Monte Carlo for embarrassingly parallel inference

Jasra¹,

Law²,

Tarakanov³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

This position paper summarizes a recently developed research program focused on inference in the context of data centric science and engineering applications, and forecasts its trajectory forward over the next decade. Often one endeavours in this context to learn complex systems in order to make more informed predictions and high stakes decisions under uncertainty. Some key challenges which must be met in this context are robustness, generalizability, and interpretability. The Bayesian framework addresses these three challenges, while bringing with it a fourth, undesirable feature: it is typically far more expensive than its deterministic counterparts. In the 21st century, and increasingly over the past decade, a growing number of methods have emerged which allow one to leverage cheap low-fidelity models in order to precondition algorithms for performing inference with more expensive models and make Bayesian inference tractable in the context of high-dimensional and expensive models. Notable examples are multilevel Monte Carlo (MLMC), multi-index Monte Carlo (MIMC), and their randomized counterparts (rMLMC), which are able to provably achieve a dimension-independent (including ∞−dimension) canonical complexity rate with respect to mean squared error (MSE) of 1/MSE. Some parallelizability is typically lost in an inference context, but recently this has been largely recovered via novel double randomization approaches. Such an approach delivers i.i.d. samples of quantities of interest which are unbiased with respect to the infinite resolution target distribution. Over the coming decade, this family of algorithms has the potential to transform data centric science and engineering, as well as classical machine learning applications such as deep learning, by scaling up and scaling out fully Bayesian inference.

show abstract

“…The MIMC method has very recently been applied to the inference context [44,19,39], however the estimates required for increments of increments has proven challenging from a theoretical perspective, and this has severely limited progress thus far. In particular, an MIMC method for inference with provable convergence guarantees does not currently exist, except a ratio estimator using simple importance sampling, as considered for MLMC and QMC in this work [57].…”

Section: Introductionmentioning

confidence: 99%

Multi-index Sequential Monte Carlo ratio estimators for Bayesian Inverse problems

Law¹,

Walton²,

Yang³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

We consider the problem of estimating expectations with respect to a target distribution with an unknown normalizing constant, and where even the unnormalized target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index Sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of Sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of MSE −1 , while single level methods require MSE −ξ for ξ > 1. This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where ξ = 5/4 and ξ = 3/2, respectively. It is also illustrated on a more challenging log Gaussian process models, where single level complexity is approximately ξ = 9/4 and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives ξ = 5/4 + ω, for any ω > 0, whereas our method is again canonical.

show abstract

Multi-Index Sequential Monte Carlo Methods for Partially Observed Stochastic Partial Differential Equations

Cited by 9 publications

References 26 publications

Improved efficiency of multilevel Monte Carlo for stochastic PDE through strong pairwise coupling

Improved efficiency of multilevel Monte Carlo for stochastic PDE through strong pairwise coupling

Randomized multilevel Monte Carlo for embarrassingly parallel inference

Multi-index Sequential Monte Carlo ratio estimators for Bayesian Inverse problems

Contact Info

Product

Resources

About