### #preprint

### Bayesian numerical methods for nonlinear PDEs

Junyang Wang, Jon Cockayne, Oksana Chkrebtii, Chris Oates, and I have just uploaded a preprint of our recent work “Bayesian numerical methods for nonlinear partial differential equations” to the arXiv. This paper continues our study of (approximate) Bayesian probabilistic numerical methods (Cockayne et al., 2019), in this case for the challenging setting of nonlinear PDEs, with the goal of realising a posterior distribution over the solution of the PDE that carries a meaningful expression of uncertainty about the solution's true value given the discretisation error that has been incurred.

**Abstract.**
The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied.
However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula.
This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost.
The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator.
Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated.
A suitable prior model for the solution of the PDE is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.

Published on Tuesday 27 April 2021 at 10:00 UTC #preprint #prob-num #wang #cockayne #chkrebtii #oates

### Randomised integration for deterministic operator differential equations

Han Cheng Lie, Martin Stahn, and I have just uploaded a preprint of our recent work “Randomised one-step time integration methods for deterministic operator differential equations” to the arXiv. In this paper, we extend the analysis of Conrad et al. (2016) and Lie et al. (2019) to the case of evolutionary systems in Banach spaces or even Gel′fand triples, this being the right setting for many evolutionary partial differential equations.

**Abstract.**
Uncertainty quantification plays an important role in applications that involve simulating ensembles of trajectories of dynamical systems.
Conrad et al. (*Stat. Comput.*, 2017) proposed randomisation of deterministic time integration methods as a strategy for quantifying uncertainty due to time discretisation.
We consider this strategy for systems that are described by deterministic, possibly non-autonomous operator differential equations defined on a Banach space or a Gel′fand triple.
We prove pathwise and expected error bounds on the random trajectories, given an assumption on the local truncation error of the underlying deterministic time integration and an assumption that the absolute moments of the random variables decay with the time step. Our analysis shows that the error analysis for differential equations in finite-dimensional Euclidean space carries over to infinite-dimensional settings.

Published on Wednesday 31 March 2021 at 09:00 UTC #preprint #prob-num #lie #stahn

### Linear conditional expectation in Hilbert space

Ilja Klebanov, Björn Sprungk, and I have just uploaded a preprint of our recent work “The linear conditional expectation in Hilbert space” to the arXiv. In this paper, we study the best approximation \(\mathbb{E}^{\mathrm{A}}[U|V]\) of the conditional expectation \(\mathbb{E}[U|V]\) of an \(\mathcal{G}\)-valued random variable \(U\) conditional upon a \(\mathcal{H}\)-valued random variable \(V\), where “best” means \(L^{2}\)-optimality within the class \(\mathrm{A}(\mathcal{H}; \mathcal{G})\) of affine functions of the conditioning variable \(V\). This approximation is a powerful one and lies at the heart of the Bayes linear approach to statistical inference, but its analytical properties, especially for \(U\) and \(V\) taking values in infinite-dimensional spaces \(\mathcal{G}\) and \(\mathcal{H}\), are only partially understood — which this article aims to rectify.

**Abstract.**
The *linear conditional expectation* (LCE) provides a best linear (or rather, affine) estimate of the conditional expectation and hence plays an important rôle in approximate Bayesian inference, especially the *Bayes linear* approach. This article establishes the analytical properties of the LCE in an infinite-dimensional Hilbert space context. In addition, working in the space of affine Hilbert–Schmidt operators, we establish a regularisation procedure for this LCE. As an important application, we obtain a simple alternative derivation and intuitive justification of the *conditional mean embedding* formula, a concept widely used in machine learning to perform the conditioning of random variables by embedding them into reproducing kernel Hilbert spaces.

Published on Friday 28 August 2020 at 09:00 UTC #preprint #tru2 #bayesian #rkhs #mean-embedding #klebanov #sprungk

### Adaptive reconstruction of monotone functions

Luc Bonnet, Jean-Luc Akian, Éric Savin, and I have just uploaded a preprint of our recent work “Adaptive reconstruction of imperfectly-observed monotone functions, with applications to uncertainty quantification” to the arXiv. In this work, motivated by the computational needs of the optimal uncertainty quantification (OUQ) framework, we present and develop an algorithm for reconstructing a monotone function \(F\) given the ability to interrogate \(F\) pointwise but subject to partially controllable one-sided observational errors of the type that one would typically encounter if the observations would arise from a numerical optimisation routine.

**Abstract.**
Motivated by the desire to numerically calculate rigorous upper and lower bounds on deviation probabilities over large classes of probability distributions, we present an adaptive algorithm for the reconstruction of increasing real-valued functions.
While this problem is similar to the classical statistical problem of isotonic regression, we assume that the observational data arise from optimisation problems with partially controllable one-sided errors, and this setting alters several characteristics of the problem and opens natural algorithmic possibilities.
Our algorithm uses imperfect evaluations of the target function to direct further evaluations of the target function either at new sites in the function's domain or to improve the quality of evaluations at already-evaluated sites.
We establish sufficient conditions for convergence of the reconstruction to the ground truth, and apply the method both to synthetic test cases and to a real-world example of uncertainty quantification for aerodynamic design.

Published on Monday 13 July 2020 at 10:00 UTC #preprint #daad #ouq #isotonic #bonnet #akian #savin

### A rigorous theory of conditional mean embeddings

Ilja Klebanov, Ingmar Schuster, and I have just uploaded a preprint of our recent work “A rigorous theory of conditional mean embeddings” to the arXiv.
In this work we take a close mathematical look at the method of conditional mean embedding.
In this approach to non-parametric inference, a random variable \(Y \sim \mathbb{P}_{Y}\) in a set \(\mathcal{Y}\) is represented by its *kernel mean embedding*, the reproducing kernel Hilbert space element

\( \displaystyle \mu_{Y} = \int_{\mathcal{Y}} \psi(y) \, \mathrm{d} \mathbb{P}_{Y} (y) \in \mathcal{G}, \)

and conditioning with respect to an observation \(x\) of a related random variable \(X \sim \mathbb{P}_{X}\) in a set \(\mathcal{X}\) with RKHS \(\mathcal{H}\) is performed using the Woodbury formula\( \displaystyle \mu_{Y|X = x} = \mu_Y + (C_{XX}^{\dagger} C_{XY})^\ast \, (\varphi(x) - \mu_X) . \)

Here \(\psi \colon \mathcal{Y} \to \mathcal{G}\) and \(\varphi \colon \mathcal{X} \to \mathcal{H}\) are the canonical feature maps and the \(C\)'s denote the appropriate centred (cross-)covariance operators of the embedded random variables \(\psi(Y)\) in \(\mathcal{G}\) and \(\varphi(X)\) in \(\mathcal{H}\).

Our article aims to provide rigorous mathematical foundations for this attractive but apparently naïve approach to conditional probability, and hence to Bayesian inference.

**Abstract.**
Conditional mean embeddings (CME) have proven themselves to be a powerful tool in many machine learning applications. They allow the efficient conditioning of probability distributions within the corresponding reproducing kernel Hilbert spaces (RKHSs) by providing a linear-algebraic relation for the kernel mean embeddings of the respective probability distributions. Both centered and uncentered covariance operators have been used to define CMEs in the existing literature. In this paper, we develop a mathematically rigorous theory for both variants, discuss the merits and problems of either, and significantly weaken the conditions for applicability of CMEs. In the course of this, we demonstrate a beautiful connection to Gaussian conditioning in Hilbert spaces.

Published on Tuesday 3 December 2019 at 07:00 UTC #preprint #mathplus #tru2 #rkhs #mean-embedding #klebanov #schuster