Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A Case Study for the Navier--Stokes Equations

Kantas, Nikolas; Beskos, Alexandros; Jasra, Ajay

doi:10.1137/130930364

Cited by 85 publications

(164 citation statements)

References 41 publications

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“…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].…”

Section: Bibliographic Notesmentioning

confidence: 99%

The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

141

222

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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.

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Section: Bibliographic Notesmentioning

confidence: 99%

The Bayesian Approach to Inverse Problems

Dashti

Stuart

2015

Handbook of Uncertainty Quantification

141

222

View full text Add to dashboard Cite

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“…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%

“…, n, are positive weights that sum to one, and {u (i) } n i=1 ∈ X n is an ensemble of particles. In the following, we briefly review two methods that are popular in Bayesian statistics and Bayesian inversion, namely importance sampling and sequential Monte Carlo (SMC), see [2,7,22,34] for more details.…”

Section: Sequential Monte Carlo With Temperingmentioning

confidence: 99%

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

Kahle¹,

Lam²,

Latz³

et al. 2019

SIAM/ASA J. Uncertainty Quantification

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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...

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“…Due to the nonlinearity of the parameter-to-output maps G m which appears in (19), these normalisation constants, in general, cannot be computed analytically, and so the resulting posterior distribution P(u|q 1:m ) cannot be expressed in closed form. Sampling/particle methods then need to be applied for the sequential approximation of the Bayesian posterior [25,29]. A generic particle-based approach, applied to the present problem, is displayed in Algorithm 1.…”

Section: The Computational Approach To the Bayesian Inference Frameworkmentioning

confidence: 99%

Quantifying uncertainty in thermophysical properties of walls by means of Bayesian inversion

Simon

Iglesias

Jones

et al. 2018

Energy and Buildings

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We introduce a computational framework to statistically infer thermophysical properties of any given wall from in-situ measurements of air temperature and surface heat fluxes. The proposed framework uses these measurements, within a Bayesian calibration approach, to sequentially infer input parameters of a one-dimensional heat diffusion model that describes the thermal performance of the wall. These inputs include spatially-variable functions that characterise the thermal conductivity and the volumetric heat capacity of the wall. We encode our computational framework in an algorithm that sequentially updates our probabilistic knowledge of the thermophysical properties as new measurements become available, and thus enables an on-the-fly uncertainty quantification of these properties. In addition, the proposed algorithm enables us to investigate the effect of the discretisation of the underlying heat diffusion model on the accuracy of estimates of thermophysical properties and the corresponding predictive distributions of heat flux. By means of virtual/synthetic and real experiments we show the capabilities of the proposed approach to (i) characterise heterogenous thermophysical properties associated with, for example, unknown cavities and insulators; (ii) obtain rapid and accurate uncertainty estimates of effective thermal properties (e.g. thermal transmittance); and (iii) accurately compute an statistical description of the thermal performance of the wall which is, in turn, crucial in evaluating possible retrofit measures.

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Sequential Monte Carlo Methods for High-Dimensional Inverse Problems: A Case Study for the Navier--Stokes Equations

Cited by 85 publications

References 41 publications

The Bayesian Approach to Inverse Problems

The Bayesian Approach to Inverse Problems

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

Quantifying uncertainty in thermophysical properties of walls by means of Bayesian inversion

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