Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- Part I Inverse Problems
- 1 Bayesian Inverse Problems andWell-Posedness
- 2 The Linear-Gaussian Setting
- 3 Optimization Perspective
- 4 Gaussian Approximation
- 5 Monte Carlo Sampling and Importance Sampling
- 6 Markov Chain Monte Carlo
- Exercises for Part I
- Part II Data Assimilation
- 7 Filtering and Smoothing Problems and Well-Posedness
- 8 The Kalman Filter and Smoother
- 9 Optimization for Filtering and Smoothing: 3DVAR and 4DVAR
- 10 The Extended and Ensemble Kalman Filters
- 11 Particle Filter
- 12 Optimal Particle Filter
- Exercises for Part II
- Part III Kalman Inversion
- 13 Blending Inverse Problems and Data Assimilation
- References
- Index
3 - Optimization Perspective
Published online by Cambridge University Press: 27 July 2023
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- Part I Inverse Problems
- 1 Bayesian Inverse Problems andWell-Posedness
- 2 The Linear-Gaussian Setting
- 3 Optimization Perspective
- 4 Gaussian Approximation
- 5 Monte Carlo Sampling and Importance Sampling
- 6 Markov Chain Monte Carlo
- Exercises for Part I
- Part II Data Assimilation
- 7 Filtering and Smoothing Problems and Well-Posedness
- 8 The Kalman Filter and Smoother
- 9 Optimization for Filtering and Smoothing: 3DVAR and 4DVAR
- 10 The Extended and Ensemble Kalman Filters
- 11 Particle Filter
- 12 Optimal Particle Filter
- Exercises for Part II
- Part III Kalman Inversion
- 13 Blending Inverse Problems and Data Assimilation
- References
- Index
Summary
In this chapter we explore the properties of Bayesian inversion from the perspective of an optimization problem which corresponds to maximizing the posterior probability; that is, to finding a maximum a posteriori (MAP) estimator, or mode of the posterior distribution. We demonstrate the properties of the point estimator resulting from this optimization problem, showing its positive and negative attributes, the latter motivating our work in the following three chapters. We also introduce, and study, basic gradient-based optimization algorithms.
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- Information
- Inverse Problems and Data Assimilation , pp. 32 - 48Publisher: Cambridge University PressPrint publication year: 2023