Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems

Cockayne, Jon; Oates, Chris J.; Sullivan, Tim

doi:10.48550/arxiv.1605.07811

Cited by 25 publications

(53 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Some numerical examples for the one-dimensional Poisson equation are given in Section 5. The proofs of these results are based on reproducing kernel Hilbert space (RKHS) techniques which are commonly used to analyse approximation properties of GPs (van der Vaart and van Zanten, 2011;Cialenco et al, 2012;Cockayne et al, 2017;Karvonen et al, 2020;Teckentrup, 2020;Wang et al, 2020;Wynne et al, 2021). Our central tool is Theorem 3.7, which describes the RKHS associated to the prior u GP under the assumptions that the RKHS for f GP is a Sobolev space and L is a second-order elliptic differential operator.…”

Section: Contributionsmentioning

confidence: 99%

See 1 more Smart Citation

Error analysis for a statistical finite element method

Karvonen¹,

Cirak²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The recently proposed statistical finite element (statFEM) approach synthesises measurement data with finite element models and allows for making predictions about the true system response. We provide a probabilistic error analysis for a prototypical statFEM setup based on a Gaussian process prior under the assumption that the noisy measurement data are generated by a deterministic true system response function that satisfies a second-order elliptic partial differential equation for an unknown true source term. In certain cases, properties such as the smoothness of the source term may be misspecified by the Gaussian process model. The error estimates we derive are for the expectation with respect to the measurement noise of the L 2 -norm of the difference between the true system response and the mean of the statFEM posterior. The estimates imply polynomial rates of convergence in the numbers of measurement points and finite element basis functions and depend on the Sobolev smoothness of the true source term and the Gaussian process model. A numerical example for Poisson's equation is used to illustrate these theoretical results.

show abstract

Section: Contributionsmentioning

confidence: 99%

“…For convergence results in a well-specified setting, see for example Theorem 16.15 in Wendland (2005). In a GP setting similar methods have been proposed and analysed in Graepel (2003); Cialenco et al (2012); Cockayne et al (2017);and Raissi et al (2017). For some error estimates, see Lemma 3.4 and Proposition 3.5 in Cialenco et al (2012).…”

Section: Related Workmentioning

confidence: 99%

Error analysis for a statistical finite element method

Karvonen¹,

Cirak²

2022

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…Our first example concerns the probabilistic numerical solution of Poisson's equation with Dirichlet boundary conditions; the intention is to validate our methodology on a problem that is well-understood. SED for such problems was investigated with bespoke code in Cockayne et al (2016). The PDE we consider is defined on X = [−1, 1] 2 and takes the form…”

Section: Probabilistic Solution Of Pdesmentioning

confidence: 99%

GaussED: A Probabilistic Programming Language for Sequential Experimental Design

Fisher,

Teymur,

Oates

2021

Preprint

View full text Add to dashboard Cite

Sequential algorithms are popular for experimental design, enabling emulation, optimisation and inference to be efficiently performed. For most of these applications bespoke software has been developed, but the approach is general and many of the actual computations performed in such software are identical. Motivated by the diverse problems that can in principle be solved with common code, this paper presents GaussED, a simple probabilistic programming language coupled to a powerful experimental design engine, which together automate sequential experimental design for approximating a (possibly nonlinear) quantity of interest in Gaussian processes models. Using a handful of commands, GaussED can be used to: solve linear partial differential equations, perform tomographic reconstruction from integral data and implement Bayesian optimisation with gradient data.

show abstract

“…Probabilistic numerical methods have been developed for a series of diverse numerical tasks, including the numerical solution of linear systems, or the computation of integrals. Most notably, a series of works focused on the probabilistic quantification of approximation errors in the solution of ordinary [10,15,24,25,28,29,31,[46][47][48][51][52][53] and partial differential equations [10,12,13,15,20,35,[37][38][39]44,45]. One of the advantages of probabilistic numerical methods is that they allow "propagating uncertainty in computational pipelines" [36], where uncertainty is due to numerical discretization.…”

Section: Introductionmentioning

confidence: 99%

“…One of the advantages of probabilistic numerical methods is that they allow "propagating uncertainty in computational pipelines" [36], where uncertainty is due to numerical discretization. A notable example of such computational pipelines is given by inverse problems, especially in their Bayesian interpretation, for which the beneficial effects of adopting a probabilistic approach has been demonstrated in a series of works [3,4,10,12,13,15,35].…”

Section: Introductionmentioning

confidence: 99%

Sampling Methods for Bayesian Inference Involving Convergent Noisy Approximations of Forward Maps

Garegnani¹

2021

Preprint

View full text Add to dashboard Cite

We present Bayesian techniques for solving inverse problems which involve mean-square convergent random approximations of the forward map. Noisy approximations of the forward map arise in several fields, such as multiscale problems and probabilistic numerical methods. In these fields, a random approximation can enhance the quality or the efficiency of the inference procedure, but entails additional theoretical and computational difficulties due to the randomness of the forward map. A standard technique to address this issue is to combine Monte Carlo averaging with Markov chain Monte Carlo samplers, as for example in the pseudo-marginal Metropolis-Hastings methods. In this paper, we consider mean-square convergent random approximations, and quantify how Monte Carlo errors propagate from the forward map to the solution of the inverse problems. Moreover, we review and describe simple techniques to solve such inverse problems, and compare performances with a series of numerical experiments.

show abstract

Probabilistic Numerical Methods for Partial Differential Equations and Bayesian Inverse Problems

Cited by 25 publications

References 0 publications

Error analysis for a statistical finite element method

Error analysis for a statistical finite element method

GaussED: A Probabilistic Programming Language for Sequential Experimental Design

Sampling Methods for Bayesian Inference Involving Convergent Noisy Approximations of Forward Maps

Contact Info

Product

Resources

About