Sequential Monte Carlo methods for Bayesian elliptic inverse problems

Beskos, Alexandros; Jasra, Ajay; Muzaffer, Ege; Stuart, Andrew M.

doi:10.1007/s11222-015-9556-7

Cited by 55 publications

(87 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The application of those ideas to the solution of PDE inverse problems was first demonstrated in [50], where the inverse problem is to determine the initial condition of the Navier-Stokes equations from observations. The method is applied to the elliptic inverse problem, with uniform priors, in [10]. The proof of Theorem 23 follows the very clear exposition given in [84] in the context of filtering for hidden Markov models.…”

Section: Bibliographic Notesmentioning

confidence: 95%

The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

Self Cite

141

222

View full text Add to dashboard Cite

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

show abstract

Section: Bibliographic Notesmentioning

confidence: 95%

The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

Self Cite

141

222

View full text Add to dashboard Cite

show abstract

“…In more general cases, such as when the forward map is non-linear or the prior is only conditionally Gaussian, sampling typically cannot be performed directly, and methods such as MCMC or SMC must be used instead to target the posterior. We note here that when the prior is Gaussian, MCMC and SMC methods are available for targeting the posterior that are well-defined on function space and possess dimension-independent convergence properties [9,13,8] -the existence of such methods is important when considering the choice of hierarchical parameterization in the next subsection.…”

Section: 12mentioning

confidence: 99%

Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

Dunlop¹,

Helin²,

Stuart³

2020

The SMAI Journal of Computational Mathematics

Self Cite

View full text Add to dashboard Cite

The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; means for uncertainty quantification; and it allows for principled learning of hyperparameters. The posterior distribution may be explored by sampling methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization. This is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck process.

show abstract

“…For this reason we approximate the posterior measure by sequential Monte Carlo (SMC) with tempering, see e.g. [7] for the application of SMC in the context of elliptic equations in three dimensions, and [34] for the Navier-Stokes equations, respectively. We point out that sequential Monte Carlo with tempering has not been employed for the calibration of tumour models; note that Markov chain Monte Carlo (MCMC) methods were employed in previous works [28,39,40,46,48].…”

Section: Bayesian Inversion and Main Contributionsmentioning

confidence: 99%

Bayesian Parameter Identification in Cahn--Hilliard Models for Biological Growth

Kahle¹,

Lam²,

Latz³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

View full text Add to dashboard Cite

We consider the inverse problem of parameter estimation in a diffuse interface model for tumour growth. The model consists of a fourth-order Cahn-Hilliard system and contains three phenomenological parameters: the tumour proliferation rate, the nutrient consumption rate, and the chemotactic sensitivity. We study the inverse problem within the Bayesian framework and construct the likelihood and noise for two typical observation settings. One setting involves an infinite-dimensional data space where we observe the full tumour. In the second setting we observe only the tumour volume, hence the data space is finite-dimensional. We show the well-posedness of the posterior measure for both settings, building upon and improving the analytical results in [C. Kahle and K.F. Lam, Appl. Math. Optim. (2018)]. A numerical example involving synthetic data is presented in which the posterior measure is numerically approximated by the sequential Monte Carlo approach with tempering. are able to ignore apoptosis (programmed cell death) signals, remain elusive to attacks from the immune system, and, most dangerously, have the ability to induce the growth of new blood vessels towards itself (angiogenesis). This leads to the spreading of cancer to other parts of the body, and the formation of secondary tumours (metastasis).The study of tumour growth can be roughly divided according to the physical and chemical phenomena occuring at three scales [47]: the tissue scale which is commonly observed in experiments involving movement of cells (such as metastasis and growth into the extracellular matrix) and nutrient diffusion; the cellular scale consisting of activities and interactions between individual cells such as mitosis and the activation of receptors; and sub-cellular scale where genetic mutations and DNA degradation occur. We focus on the tissue-scaled phenomena, as they are the first to be detected in a routine diagnosis, and can be described fairly well with help of continuum models consisting of differential equations.Since the seminal work in [11] and [27] where simple mathematical models for tumour growth are employed, there has been an explosion in the number of models proposed for modelling the multiscale nature of cancer, see for instance [19,21,47] and the references cited therein. The diversity of model variants reflects the difficulties when we try to identify key biological phenomena that are responsible for experimental observations.As metastasis is an important hallmark of cancer, we restrict our attention to continuum models that can capture such events. Continuum models often rely on a mathematical description to distinguish tumour tissue from healthy host tissues. To be able to capture metastasis the models have to allow for some form of topological change of the separation layers between the tumour and the host tissues. The classical description represents the separation layers as idealised hypersurfaces, known also as the sharp interface approach. In this case complicated boundary conditions have to be imposed t...

show abstract

Sequential Monte Carlo methods for Bayesian elliptic inverse problems

Cited by 55 publications

References 26 publications

The Bayesian Approach to Inverse Problems

The Bayesian Approach to Inverse Problems

Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

Bayesian Parameter Identification in Cahn--Hilliard Models for Biological Growth

Contact Info

Product

Resources

About