The subject of inverse problems in differential equations is of enormous practical importance, and has also generated substantial mathematical and computational innovation. Typically some form of regularization is required to ameliorate ill-posed behaviour. In this article we review the Bayesian approach to regularization, developing a function space viewpoint on the subject. This approach allows for a full characterization of all possible solutions, and their relative probabilities, whilst simultaneously forcing significant modelling issues to be addressed in a clear and precise fashion. Although expensive to implement, this approach is starting to lie within the range of the available computational resources in many application areas. It also allows for the quantification of uncertainty and risk, something which is increasingly demanded by these applications. Furthermore, the approach is conceptually important for the understanding of simpler, computationally expedient approaches to inverse problems.

/pdf/inverse-problems-a-bayesian-perspective-1dknxttxjb.pdf

Inverse problems: A Bayesian perspective

Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive dependence measures quantify the degree of dependence of outputs on inputs in physical systems. The proposed dependence measures provide a natural framework for a limit theory for stationary processes. In particular, under conditions with quite simple forms, we present limit theorems for partial sums, empirical processes, and kernel density estimates. The conditions are mild and easily verifiable because they are directly related to the data-generating mechanisms.

https://www.pnas.org/content/pnas/102/40/14150.full.pdf

Nonlinear system theory: Another look at dependence

We commonly refer to state estimation theory in geosciences as data assimilation (DA). This term encompasses the entire sequence of operations that, starting from the observations of a system, and from additional statistical and dynamical information (such as a dynamical evolution model), provides an estimate of its state. DA is standard practice in numerical weather prediction, but its application is becoming widespread in many other areas of climate, atmosphere, ocean, and environment modeling; in all circumstances where one intends to estimate the state of a large dynamical system based on limited information. While the complexity of DA, and of the methods thereof, stands on its interdisciplinary nature across statistics, dynamical systems, and numerical optimization, when applied to geosciences, an additional difficulty arises by the continually increasing sophistication of the environmental models. Thus, in spite of DA being nowadays ubiquitous in geosciences, it has so far remained a topic mostly reserved to experts. We aim this overview article at geoscientists with a background in mathematical and physical modeling, who are interested in the rapid development of DA and its growing domains of application in environmental science, but so far have not delved into its conceptual and methodological complexities.

Data assimilation in the geosciences: An overview of methods, issues, and perspectives

Linear Inverse Problems.- Large-Scale Inverse Problems in Imaging.- Regularization Methods for Ill-Posed Problems.- Distance Measures and Applications to Multi-Modal Variational Imaging.- Energy Minimization Methods.- Compressive Sensing.- Duality and Convex Programming.- EM Algorithms.- Iterative Solution Methods.- Level Set Methods for Structural Inversion and Image Reconstructions.- Expansion Methods.- Sampling Methods.- Inverse Scattering.- Electrical Impedance Tomography.- Synthetic Aperture Radar Imaging.- Tomography.- Optical Imaging.- Photoacoustic and Thermoacoustic Tomography: Image Formation Principles.- Mathematics of Photoacoustic and Thermoacoustic Tomography.- Wave Phenomena.- Statistical Methods in Imaging.- Supervised Learning by Support Vector Machines.- Total Variation in Imaging.- Numerical Methods and Applications in Total Variation Image Restoration.- Mumford and Shah Model and its Applications in Total Variation Image Restoration.- Local Smoothing Neighbourhood Filters.- Neighbourhood Filters and the Recovery of 3D Information.- Splines and Multiresolution Analysis.- Gabor Analysis for Imaging.- Shaper Spaces.- Variational Methods in Shape Analysis.- Manifold Intrinsic Similarity.- Image Segmentation with Shape Priors: Explicit Versus Implicit Representations.- Starlet Transform in Astronomical Data Processing.- Differential Methods for Multi-Dimensional Visual Data Analysis.- Wave fronts in Imaging, Quinto.- Ultrasound Tomography, Natterer.- Optical Flow, Schnoerr.- Morphology, Petros.- Maragos.- PDEs, Weickert. - Registration, Modersitzki. - Discrete Geometry in Imaging, Bobenko, Pottmann.-Visualization, Hege.- Fast Marching and Level Sets, Osher.- Couple Physics Imaging, Arridge.- Imaging in Random Media, Borcea.- Conformal Methods, Gu.- Texture, Peyre.- Graph Cuts, Darbon.- Imaging in Physics with Fourier Transform (i.e. Phase Retrieval e.g Dark field imaging), J. R. Fienup.- Electron Microscopy, Oktem Ozan.- Mathematical Imaging OCT (this is also FFT based), Mark E. Brezinski.- Spect, PET, Faukas, Louis.

/pdf/handbook-of-mathematical-methods-in-imaging-mpoi7xkf0v.pdf

Handbook of mathematical methods in imaging

Part I. Deterministic Differential Equations: 1. Linear analysis 2. Galerkin approximation and finite elements 3. Time-dependent differential equations Part II. Stochastic Processes and Random Fields: 4. Probability theory 5. Stochastic processes 6. Stationary Gaussian processes 7. Random fields Part III. Stochastic Differential Equations: 8. Stochastic ordinary differential equations (SODEs) 9. Elliptic PDEs with random data 10. Semilinear stochastic PDEs.

/pdf/an-introduction-to-computational-stochastic-pdes-74xqtzrc5j.pdf

An Introduction to Computational Stochastic PDEs

Filtering Theory- The Stochastic Process ?- The Filtering Equations- Uniqueness of the Solution to the Zakai and the Kushner-Stratonovich Equations- The Robust Representation Formula- Finite-Dimensional Filters- The Density of the Conditional Distribution of the Signal- Numerical Algorithms- Numerical Methods for Solving the Filtering Problem- A Continuous Time Particle Filter- Particle Filters in Discrete Time

Fundamentals of Stochastic Filtering

Throughout this chapter, we take the signal X to be the solution of (3.9); that is, $ X = ({X^i})_{i = 1}^d $ is the solution of the stochastic differential equation 

$$ {d}{{X}_{t}}=f({{X}_{t}})\text{d}t+\sigma ({{X}_{t}})\text{d}{{V}_{t}}\text{, } $$

(9.1)

where f : ℝ d → ℝ d and σ : ℝ d → ℝ d×p are bounded and globally Lipschitz functions and $ V\, = \,({V^j})_{j = 1}^p $ is a p-dimensional Brownian motion.

A Continuous Time Particle Filter

The purpose of this chapter is to present a rigorous mathematical treatment of the convergence of particle filters in the (simpler) framework where both the signal X and the observation Y are discrete time processes. This restriction means that this chapter does not use stochastic calculus. The chapter is organized as follows. In the following section we describe the discrete time framework. In Section 10.2 we deduce the recurrence formula for the conditional distribution of the signal in discrete time. In Section 10.3 we deduce necessary and sufficient conditions for sequences of (random) measures to converge to the conditional distribution of the signal. In Section 10.4 we describe a generic class of particle filters which are shown to converge in the following section.

Particle Filters in Discrete Time

The principal aim of this chapter is to familiarize the reader with the fact that the conditional distribution of the signal can be viewed as a stochastic process with values in the space of probability measures. While it is true that this chapter sets the scene for the subsequent chapters, it can be skipped by those readers whose interests are biased towards the applied aspects of the subject. The gist of the chapter can be summarized by the following.

The Stochastic Process π

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Fundamentals of Stochastic Filtering

A Continuous Time Particle Filter

Particle Filters in Discrete Time

The Stochastic Process π