关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

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.

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Inverse problems: A Bayesian perspective

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezic, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

This paper gives a systematic introduction to HMM, the heterogeneous multiscale methods, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be ...

Heterogeneous multiscale methods: A review

Optimal prediction methods compensate for a lack of resolution in the numerical solution of complex problems through the use of prior statistical information. We point out a relation between optimal prediction and the statistical mechanics of irreversible processes, and use a version of the Mori–Zwanzig formalism to produce a higher-order optimal prediction method.

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Optimal prediction and the Mori-Zwanzig representation of irreversible processes.

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Optimal prediction with memory

On considere des problemes de Sturm-Liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur. On montre que si le potentiel est connu sur la moitie de l'intervalle et si une condition limite est donnee, alors le potentiel et l'autre condition limite sont determines de facon unique par les valeurs propres

Discontinuous inverse eigenvalue problems

Optimal prediction methods compensate for a lack of resolutionin the numerical solution of complex problems through the use ofprior statistical information. We point out a relation between op-timal prediction and the statistical mechanics of irreversible pro-cesses, and use a version of the MoriŒZwanzig formalism to pro-duce a higher-order optimal prediction method.

Optimal prediction and the MoriñZwanzig representation of irreversible processes

Preliminaries.- Introduction to Probability.- Brownian Motion and Its Applications.- Stationary Stochastic.-Processes.- Statistical Mechanics.- Time-Dependent Statistical Mechanics.

Ole H. Hald

Papers

Optimal prediction and the Mori-Zwanzig representation of irreversible processes.

Optimal prediction with memory

Discontinuous inverse eigenvalue problems

Optimal prediction and the MoriñZwanzig representation of irreversible processes

Stochastic Tools in Mathematics and Science