Bayesian inverse problems for functions and applications to fluid mechanics

Abstract

In this paper we establish a mathematical framework for a range of inverse problems for functions, given a finite set of noisy observations. The problems are hence underdetermined and are often ill-posed. We study these problems from the viewpoint of Bayesian statistics, with the resulting posterior probability measure being defined on a space of functions. We develop an abstract framework for such problems which facilitates application of an infinite-dimensional version of Bayes theorem, leads to a well-posedness result for the posterior measure (continuity in a suitable probability metric with respect to changes in data), and also leads to a theory for the existence of maximizing the posterior probability (MAP) estimators for such Bayesian inverse problems on function space. A central idea underlying these results is that continuity properties and bounds on the forward model guide the choice of the prior measure for the inverse problem, leading to the desired results on well-posedness and MAP estimators; the PDE analysis and probability theory required are thus clearly dileneated, allowing a straightforward derivation of results. We show that the abstract theory applies to some concrete applications of interest by studying problems arising from data assimilation in fluid mechanics. The objective is to make inference about the underlying velocity field, on the basis of either Eulerian or Lagrangian observations. We study problems without model error, in which case the inference is on the initial condition, and problems with model error in which case the inference is on the initial condition and on the driving noise process or, equivalently, on the entire time-dependent velocity field. In order to undertake a relatively uncluttered mathematical analysis we consider the two-dimensional Navier-Stokes equation on a torus. The case of Eulerian observationsdirect observations of the velocity field itself-is then a model for weather forecasting. The case of Lagrangian observations-observations of passive tracers advected by the flow-is then a model for data arising in oceanography. The methodology which we describe herein may be applied to many other inverse problems in which it is of interest to find, given observations, an infinitedimensional object, such as the initial condition for a PDE. A similar approach might be adopted, for example, to determine an appropriate mathematical

Variational data assimilation using targetted random walks

“…Although only indicative, being a markedly simpler model than any model that is currently used in practical meteorological and oceanographical scenarios, it still gives us a better idea of how to attack these problems, and of how much information is contained within different types of noisy observation in them. Furthermore, in the current computational practise various simplifications are used-primarily filtering or variational methods [33]-and the posterior measure that we compute in this paper could be viewed as an 'ideal solution' against which these simpler and more practical methods can be compared. To that end it would be of interest to apply the ideas in this paper to a number of simple model problems from data assimilation in meteorology or oceanography and to compare the results from filtering and variational methods with the ideal solution found from MCMC methods.…”

“…Implementation of gradient-based methods such as the (Metropolis-Adjusted Langevin) MALA algorithm, as shown in [33], is also of great interest, and may be useful in tackling these more complex posterior densities.…”

“…In this paper, we stick to the simpler Random Walk method as a proof of concept, but the reader should be aware that gradient methods, such as the Langevin Algorithm, or Hybrid Monte-Carlo, could be implemented to increase the efficiency. The basic mathematical framework within which we work is that described in [33] in which we formulate Bayesian inverse problems on the function space, and then study data assimilation problems arising in fluid mechanics from this point of view. The abstract framework of [33] also provides a natural setting for the design of random walk-type MCMC methods, appropriate to the function space setting.…”

“…The basic mathematical framework within which we work is that described in [33] in which we formulate Bayesian inverse problems on the function space, and then study data assimilation problems arising in fluid mechanics from this point of view. The abstract framework of [33] also provides a natural setting for the design of random walk-type MCMC methods, appropriate to the function space setting. The probabilistic approximation theory associated to such algorithms is studied in [34].…”

Space–time adaptive solution of inverse problems with the discrete adjoint method

“…IPs arise in various applications of engineering and mathematics, e.g., seismography, meteorology, oceanography, medical imaging, systems biology, and fluid dynamics (see, e.g., [1,2,3,4,5,6]). IPs are usually described as constrained optimization problems, where the constraints are ordinary (ODE) or partial differential equations (PDEs).…”