Discrete gradients for computational Bayesian inference

Sahani Pathiraja; Sebastian Reich

doi:10.3934/jcd.2019019

Article Contents

2019, Volume 6, Issue 2: 385-400. Doi: 10.3934/jcd.2019019

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Discrete gradients for computational Bayesian inference

Sahani Pathiraja and
Sebastian Reich

University of Potsdam, Institute of Mathematics, Karl-Liebknecht-Str. 24/25, Potsdam, D-14476, Germany

Received: February 28, 2019

Revised: June 30, 2019

Published: November 2019

This research has been partially funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1294 'Data Assimilation' (projects A02 and B04)

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Abstract

In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman–Bucy filter and a particle discretisation of the Fokker–Planck equation associated to Brownian dynamics. Both formulations can lead to stiff differential equations which require special numerical methods for their efficient numerical implementation. We compare discrete gradient methods to alternative semi-implicit and other iterative implementations of the underlying Bayesian inference problems.

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Mathematics Subject Classification: Primary: 62F15, 65C05, 82C22, 65P99; Secondary: 65C35.

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Figure 1. Exact solution and numerical approximations to the mean $ m_\tau $ and variance $ \sigma_\tau $ estimated by the EnKBF using the discrete gradient (DG) method with $ \theta = 1 $ and the semi-implicit (SI) Euler method with step-sizes (a) $ \Delta \tau = 0.1 $, (b) $ \Delta \tau = 0.2 $, (c) $ \Delta \tau = 0.5 $, (d) $ \Delta \tau = 1.0 $

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Figure 2. Exact final values and numerical approximations to the mean $ m_\tau $ and variance $ \sigma_\tau $ estimated by the EnKBF using the discrete gradient (DG) with $ \theta = 1 $, the semi-implicit (SI) Euler, and the gradient-free, explicit (IEnKF) method with step-sizes (a) $ \Delta \tau = 0.01 $, (b) $ \Delta \tau = 0.1 $, (c) $ \Delta \tau = 0.2 $, (d) $ \Delta \tau = 0.5 $. The reference time evolution of the mean and variance is given by the explicit Euler method with step-size $ \Delta \tau = 0.00025 $

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Figure 3. Exact final values and numerical approximations to the mean $ m_t $ and variance $ \sigma_t $ for the linear problem. Posterior is estimated by the Fokker-Planck dynamics with $ M = 10 $ particles using the discrete gradient (DG) method with $ \theta = 1 $ and the semi-implicit (SI) Euler method with step-sizes (a) $ \Delta \tau = 0.004 $, (b) $ \Delta \tau = 0.01 $, (c) $ \Delta \tau = 0.04 $, (d) $ \Delta \tau = 0.1 $. The reference time evolution of the mean and variance is given by the Explicit Euler (EE) method with step-size $ \Delta \tau = 2 \times 10^{-4} $

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Figure 4. Exact final values and numerical approximations to the mean $ m_t $ and variance $ \sigma_t $ for the non-linear problem. Posterior is estimated by the Fokker-Planck dynamics using the discrete gradient (DG) method with $ \theta = 1 $ and the semi-implicit (SI) Euler method with step-sizes (a) $ \Delta \tau = 0.002 $, (b) $ \Delta \tau = 0.005 $, (c) $ \Delta \tau = 0.02 $, (d) $ \Delta \tau = 0.05 $. The reference time evolution of the mean and variance is given by the Explicit Euler (EE) method with step-size $ \Delta \tau = 2.5 \times 10^{-6} $

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Table 1. RMSE for sequential data assimilation applied to the Lorenz-63 model. The parameter $ \alpha = 1 $ corresponds to a standard square root ensemble Kalman filter, while $ \alpha<1 $ results in a Gaussian mixture approximation to the prior and posterior distributions. Small improvements over the ensemble Kaman filter can be found for ensemble sizes $ M\ge 35 $

$ \alpha\backslash M $	$ 15 $	$ 20 $	$ 25 $	$ 30 $	$ 35 $	$ 40 $	$ 45 $	$ 50 $
$ 0.8 $	$ 2.5145 $	$ 2.4726 $	$ 2.4471 $	$ 2.4406 $	$ 2.4221 $	$ 2.4173 $	$ 2.4162 $	$ 2.4112 $
$ 0.85 $	$ 2.5048 $	$ 2.4585 $	$ 2.4357 $	$ 2.4330 $	$ 2.4186 $	$ 2.4161 $	$ 2.4020 $	$ 2.4026 $
$ 0.9 $	$ 2.4989 $	$ 2.4493 $	$ 2.4396 $	$ 2.4259 $	$ 2.4177 $	$ 2.4103 $	$ 2.4106 $	$ 2.4075 $
$ 0.95 $	$ 2.4998 $	$ 2.4605 $	$ 2.4399 $	$ 2.4327 $	$ 2.4242 $	$ 2.4132 $	$ 2.4162 $	$ 2.4131 $
$ 1.0 $	$ 2.4804 $	$ 2.4597 $	$ 2.4518 $	$ 2.4354 $	$ 2.4334 $	$ 2.4247 $	$ 2.4244 $	$ 2.4203 $

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