The Bayesian Approach to Inverse Problems

摘要: These lecture notes highlight the mathematical and computational structure relating to formulation of, development of algorithms for, Bayesian approach inverse problems in differential equations. This approach is fundamental in quantification uncertainty within applications in volving blending models with data. The finite dimensional situation described first, along with some motivational examples. Then probability measures on separable Banach space undertaken, using a random series over an infinite set functions to construct draws; these are used as priors in Bayesian approach to problems. Regularity draws from the priors is studied natural Sobolev or Besov spaces implied by choice construction, Kolmogorov continuity theorem extend regularity considerations the space Holder continuous functions. Bayes’ theorem de rived this prior setting, here interpreted as finding conditions under which posterior absolutely respect prior, determining formula for Radon-Nikodym derivative terms the likelihood Having established form the posterior, we then describe various properties common it infinite setting. These properties include well-posedness, approximation theory, existence maximum posteriori estimators. We measure-preserving dynamics, again the infinite space, including Markov chain-Monte C arlo sequential Monte Carlo methods, reversible stochastic By formulating theory underlying obtain framework suitable rigorous analysis accuracy reconstructions, of computational complexity, well naturally constructing perform mesh refinement, since they inherently well-defined dimensions.