Discrete gradients for computational Bayesian inference

Pathiraja, Sahani; Reich, Sebastian

doi:10.3934/jcd.2019019

Cited by 11 publications

(21 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the linear setting, ensemble Kalman based methods may be viewed as Monte Carlo approximations of the Kalman filter; in the nonlinear case ensemble Kalman based methods do not converge to the filtering or posterior distribution in the large particle limit (Ernst, Sprungk and Starkloff, 2015). Related interacting particle based methodologies of current interest include Stein variational gradient descent (Lu, Lu and Nolen, 2018;Liu and Wang, 2016;Detommaso et al, 2018) and the Fokker-Planck particle dynamics of Reich (Reich, 2018;Pathiraja and Reich, 2019), both of which map an arbitrary initial measure into the desired posterior measure over an infinite time horizon s ∈ [0, ∞). A related approach is to introduce an artificial time s ∈ [0, 1] and a homotopy between the prior at time s = 0 and the posterior measure at time s = 1 and write an evolution equation for the measures (Daum and Huang, 2011;Reich, 2011;El Moselhy and Marzouk, 2012;Laugesen et al, 2015); this evolution equation can be approximated by particle methods.…”

Section: Literature Reviewmentioning

confidence: 99%

“…Solutions to the Fokker-Planck equation are gradient flows of the relative entropy in the density manifold (Otto, 2001;Jordan, Kinderlehrer and Otto, 1998). Designing time-stepping methods which preserve gradient structure is also of current interest: see (Pathiraja and Reich, 2019) and, within the context of Wasserstein gradient flows, (Li and Montufar, 2018;Tong Lin et al, 2018;Li, Lin and Montúfar, 2019). The subject of discrete gradients for time-integration of gradient and Hamiltonian systems is developed in (Humphries and Stuart, 1994;Gonzalez, 1996;McLachlan, Quispel and Robidoux, 1999;Hairer and Lubich, 2013).…”

Section: Literature Reviewmentioning

confidence: 99%

“…We emphasize that the practical derivative-free algorithm that we propose rests on a particle-based approximation of a specific preconditioned gradient flow, as described in section 4.3 of the paper (Kovachki and Stuart, 2018); we add a judiciously chosen noise to this setting and it is this additional noise which enables approximate posterior sampling. Related approximations are also studied in the paper (Pathiraja and Reich, 2019) in which the effect of both time-discretization and particle approximation are discussed when applied to various deterministic interacting particle systems with gradient structure. In order to frame the analysis of our methods, we introduce a new metric, named the Kalman-Wasserstein metric, defined through both the covariance matrix of the mean field limit and the Wasserstein metric.…”

Section: Literature Reviewmentioning

confidence: 99%

See 2 more Smart Citations

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Iñigo¹,

Hoffmann²,

Li³

et al. 2020

SIAM J. Appl. Dyn. Syst.

141

253

View full text Add to dashboard Cite

Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of over-damped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker-Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker-Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

show abstract

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

See 1 more Smart Citation

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Garbuno-Iñigo¹,

Hoffmann²,

Li³

et al. 2020

SIAM J. Appl. Dyn. Syst.

141

253

View full text Add to dashboard Cite

show abstract

“…which constitutes the desired particle approximation to the Fokker-Planck equation (29) with drift term (145). Time-stepping methods for such gradient flow systems have been discussed by Pathiraja and Reich (2019). We also remark that an alternative interacting particle system, approximating the same asymptotic PDF π * in the limit s → ∞, has been proposed recently by Liu and Wang (2016) under the notion of Stein variational descent.…”

Section: Discussionmentioning

confidence: 90%

Data assimilation: The Schrödinger perspective

Reich

2019

Acta Numerica

Self Cite

View full text Add to dashboard Cite

Data assimilation addresses the general problem of how to combine model-based predictions with partial and noisy observations of the process in an optimal manner. This survey focuses on sequential data assimilation techniques using probabilistic particle-based algorithms. In addition to surveying recent developments for discrete-and continuous-time data assimilation, both in terms of mathematical foundations and algorithmic implementations, we also provide a unifying framework from the perspective of coupling of measures, and Schrödinger's boundary value problem for stochastic processes in particular.

show abstract

“…Benning, Celledoni, Ehrhardt, Owren, and Schönlieb [2] apply partitioned symplectic Runge-Kutta methods to an ODE formulation of deep learning; the time steps are regarded as parameters to be learned. Likewise, Pathiraja and Reich [20] apply discrete gradient methods to a gradient ODE formulation of Bayesian inference. Here we are moving into the realm of data analysis via geometric numerical techniques.…”

mentioning

confidence: 99%

Preface Special issue in honor of Reinout Quispel

Celledoni¹,

McLachlan²

2019

Journal of Computational Dynamics

View full text Add to dashboard Cite

Reinout Quispel was born on 8 October 1953 in Bilthoven, a small town near Utrecht in the Netherlands. He studied both chemistry and physics, gaining bachelor's degrees at the University of Utrecht in 1973 and 1976 respectively, and then specialized in theoretical physics, with a Master's degree in 1979 (on solitons in the Heisenberg spin chain, supervised by Theodorus Ruijgrok) and a PhD, Linear Integral Equations and Soliton Systems [22], in 1983, supervised by Hans Capel.This thesis, which begins with a study of integrable PDEs, arrives in Chapter 4 (later published in [24]) with the discovery of a method for obtaining fully discrete integrable systems on square lattices, that have as continuum limits the Korteweg-de Vries, nonlinear Schrödinger, and complex sine-Gordon equations, and the Heisenberg spin chain. Thus several of Reinout's lifelong research interests -continuous and discrete integrability, and the relationship between the continuous and the discrete -were present right from the start.The next stop was a postdoc at Twente University, working with Robert Helleman, the founder of the 'Dynamics Days' conference series, before a long-distance move to the Australian National University, working with Rodney Baxter. Reinout and Nel expected this southern sojourn to last for three years; thirty-three years later they are still happily resident in Australia. In 1990 Reinout moved to La Trobe University, Melbourne, where he became a Professor in 2004.Reinout's three main research areas are discrete integrable systems, dynamical systems, and geometric numerical integration, along with interactions between these topics.In discrete integrable systems, having introduced a major new direction in his PhD thesis -his novel reductions to Painlevé equations led to the Clarkson-Kruskal non-classical reduction method -he continued by codiscovering the QRT map [25,26], an 18-parameter family of completely integrable maps of the plane. These turned out to have far-reaching implications in dynamical systems theory, geometry, and integrability. For example, the modern construction of nonautonomous dynamical systems known as discrete Painlevé equations rely on them. Their geometry is explored at length in the 2010 book QRT and Elliptic Surfaces by Hans Duistermaat and is still being investigated today.In dynamical systems, his work has centred on systems with discrete and/or continuous symmetries. His review [28] marked the emergence of reversible dynamical i ii ELENA CELLEDONI AND ROBERT I. MCLACHLAN

show abstract

Discrete gradients for computational Bayesian inference

Cited by 11 publications

References 30 publications

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Interacting Langevin Diffusions: Gradient Structure and Ensemble Kalman Sampler

Data assimilation: The Schrödinger perspective

Preface Special issue in honor of Reinout Quispel

Contact Info

Product

Resources

About