Tim Sullivan

Junior Professor in Applied Mathematics:
Risk and Uncertainty Quantification

Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity

Preprint: Computing with dense kernel matrices at near-linear cost

Florian Schäfer, Houman Owhadi and I have just uploaded a preprint of our latest paper, “Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity” to the arXiv. This paper builds upon the probabilistic-numerical ideas of “gamblets” (elementary gables upon the solution of a PDE) introduced by Owhadi (2016) to provide near-linear cost \(\varepsilon\)-approximate compression, inversion and principal component analysis of dense kernel matrices, the entries of which come from Green's functions of suitable differential operators.

Abstract. Dense kernel matrices \(\Theta \in \mathbb{R}^{N \times N}\) obtained from point evaluations of a covariance function \(G\) at locations \(\{x_{i}\}_{1 \leq i \leq N}\) arise in statistics, machine learning, and numerical analysis. For covariance functions that are Green's functions elliptic boundary value problems and approximately equally spaced sampling points, we show how to identify a subset \(S \subset \{ 1,\dots, N \} \times \{ 1,\dots,N \}\), with \(\#S = O(N \log(N)\log^{d}(N/\varepsilon))\), such that the zero fill-in block-incomplete Cholesky decomposition of \(\Theta_{i,j} 1_{(i,j) \in S}\) is an \(\varepsilon\)-approximation of \(\Theta\). This block-factorisation can provably be obtained in \(O(N \log^{2}(N)(\log(1/\varepsilon)+\log^{2}(N))^{4d+1})\) complexity in time. Numerical evidence further suggests that element-wise Cholesky decomposition with the same ordering constitutes an \(O(N \log^{2}(N) \log^{2d}(N/\varepsilon))\) solver. The algorithm only needs to know the spatial configuration of the \(x_{i}\) and does not require an analytic representation of \(G\). Furthermore, an approximate PCA with optimal rate of convergence in the operator norm can be easily read off from this decomposition. Hence, by using only subsampling and the incomplete Cholesky decomposition, we obtain at nearly linear complexity the compression, inversion and approximate PCA of a large class of covariance matrices. By inverting the order of the Cholesky decomposition we also obtain a near-linear-time solver for elliptic PDEs.

Published on Tuesday 13 June 2017 at 07:00 UTC #publication #preprint #prob-num

Strong convergence rates of probabilistic integrators for ordinary differential equations

Preprint: Strong convergence rates of probabilistic integrators for ODEs

Han Cheng Lie, Andrew Stuart and I have just uploaded a preprint of our latest paper, “Strong convergence rates of probabilistic integrators for ordinary differential equations” to the arXiv. This paper is a successor to the convergence results presented by Conrad et al. (2016). We consider a generic probabilistic solver for an ODE, using a fixed time step of length \(\tau > 0\), where the mean of the solver has global error of order \(\tau^{q}\) and the variance of the truncation error model has order \(\tau^{1 + 2 p}\). Whereas Conrad et al. showed, for a Lipschitz driving vector field, that the mean-square error between the numerical solution \(U_{k}\) and the true solution \(u_{k}\) is bounded uniformly in time as

\( \displaystyle \max_{0 \leq k \leq T / \tau} \mathbb{E} [ \| U_{k} - u_{k} \|^{2} ] \leq C \tau^{2 \min \{ p, q \}} \)

(i.e. has the same order of convergence as the underlying deterministic method), we are able to relax the regularity assumptions on the vector field / flow an obtain a stronger mode of convergence (mean-square in the uniform norm) with the same convergence rate:

\( \displaystyle \mathbb{E} \left[ \max_{0 \leq k \leq T / \tau} \| U_{k} - u_{k} \|^{2} \right] \leq C \tau^{2 \min \{ p, q \}} \)

Abstract. Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al. (Stat. Comput., 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator.

Published on Monday 13 March 2017 at 13:00 UTC #publication #preprint #prob-num

Bayesian probabilistic numerical methods

Preprint: Bayesian probabilistic numerical methods

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Bayesian probabilistic numerical methods” to the arXiv. Following on from our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems”, our aim is to provide some rigorous theoretical underpinnings for the emerging field of probabilistic numerics, and in particular to define what it means for such a method to be “Bayesian”, by connecting with the established theories of Bayesian inversion and disintegration of measures.

Abstract. The emergent field of probabilistic numerics has thus far lacked rigorous statistical principals. This paper establishes Bayesian probabilistic numerical methods as those which can be cast as solutions to certain Bayesian inverse problems, albeit problems that are non-standard. This allows us to establish general conditions under which Bayesian probabilistic numerical methods are well-defined, encompassing both non-linear and non-Gaussian models. For general computation, a numerical approximation scheme is developed and its asymptotic convergence is established. The theoretical development is then extended to pipelines of computation, wherein probabilistic numerical methods are composed to solve more challenging numerical tasks. The contribution highlights an important research frontier at the interface of numerical analysis and uncertainty quantification, with some illustrative applications presented.

Published on Tuesday 14 February 2017 at 12:00 UTC #publication #preprint #prob-num

Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Preprint: Probabilistic numerical methods for PDE-constrained Bayesian inverse problems

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Probabilistic numerical methods for PDE-constrained Bayesian inverse problems” to the arXiv. This paper is intended to complement our earlier work “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” and to give a more concise presentation of the main ideas, aimed at a general audience.

Published on Wednesday 18 January 2017 at 12:00 UTC #publication #preprint #prob-num #inverse-problems

Well-posed Bayesian inverse problems and heavy-tailed stable Banach space priors

Preprint: Bayesian inversion with heavy-tailed stable priors

A revised version of “Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors” has been released on arXiv today. Among other improvements, the revised version incorporates additional remarks on the connection to the existing literature on stable distributions in Banach spaces, and generalises the results of the previous version of the paper to quasi-Banach spaces, which are like complete normed vector spaces in every respect except that the triangle inequality only holds in the weakened form

\( \| x + y \| \leq C ( \| x \| + \| y \| ) \)

for some constant \( C \geq 1 \).

Published on Monday 21 November 2016 at 11:30 UTC #publication #preprint #inverse-problems

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