Analysis and computation for Bayesian inverse problems

Parents of children with genetically determined leukoencephalopathies play a major role in their children's health care. Because of the COVID-19 pandemic, many health care services were suspended, delayed or delivered remotely with telemedicine. We sought to explore the experience of parents of children with genetically determined leukoencephalopathies during the pandemic given the adapted health care services. We conducted semistructured interviews with 13 parents of 13 affected children. Three main themes were identified using thematic analysis: perceived impact of COVID-19 on health care services, benefits and challenges of telemedicine, and expectations of health care after the pandemic. Parents perceived a loss/delay in health care services while having a positive response to telemedicine. Parents wished telemedicine would remain in their care after the pandemic. This is the first study assessing the impact of COVID-19 on health care services in this population. Our results suggest that parents experience a higher level of stress owing to the shortage of services and the children's vulnerability.

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. This is especially the case as together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and quadratic Bayesian update on the small but taxing example of the chaotic Lorenz 84 model, where we experiment with the influence of dierent observation or measurement operators on the update.

This chapter is about metric spaces, an abstract generalization of the real line that allows discussion of open and closed sets, limits, convergence, continuity, and similar properties. The usual distance function for the real line becomes an example of a metric. The other notions are defined in terms of the metric. The advantage of the generalization is that proofs of certain properties of the real line immediately go over to all other examples. Section 1 gives the definition of metric space and open set, and it lists a number of important examples, including Euclidean spaces and certain spaces of functions. Sections 2 through 4 develop properties of open and closed sets, continuity, and convergence of sequences that are simple generalizations of known facts about $\mathbb{R}$. Section 5 shows how a subset of a metric space can be made into a metric space so that the restriction of a continuous function from the whole space to the subset remains continuous. It also shows that three natural metrics for the product of two metric spaces lead to the same open sets, continuous functions, and convergent sequences. Section 6 shows that any metric space is “Hausdorff,” “regular,” and “normal,” and it goes on to exhibit three different countability hypotheses about a metric space as equivalent. A metric space with these properties is called “separable.” Section 7 concerns compactness and completeness. A metric space is defined to be “compact” if every open cover has a finite subcover. This property is equivalent to the condition that every sequence has a convergent subsequence. The Heine–Borel Theorem says that the compact sets of $\mathbb{R}^n$ are exactly the closed bounded sets. A number of the results early in Chapter I that were proved by the Bolzano–Weierstrass Theorem in the context of the real line are seen to extend to any compact metric space. A metric space is “complete” if every Cauchy sequence is convergent. A metric space is compact if and only if it is complete and “totally bounded.” Section 8 concerns connectedness, which is an abstraction of the property of an interval of the line that accounts for the Intermediate Value Theorem. Section 9 proves a fundamental result known as the Baire Category Theorem. A sample consequence of the theorem is that the pointwise limit of a sequence of continuous complex-valued functions on a complete metric space must have points where it is continuous. Section 10 studies the spaces of real-valued and complex-valued continuous functions on a compact metric space. A generalization of Ascoli's Theorem from the setting of Chapter I provides a characterization of compact sets in either of these spaces of continuous functions. A generalization of the Weierstrass Approximation Theorem, known as the Stone–Weierstrass Theorem, gives sufficient conditions for a subalgebra of either of these spaces of continuous functions to be dense. One consequence is that these spaces of continuous functions are separable. Section 11 constructs the “completion” of a metric space out of Cauchy sequences in the given space. The result is a complete metric space and a distance-preserving map of the given metric space into the completion such that the image is dense.

This paper is mainly concerned with the study of the generators of those positive -semigroups on weighted continuous function spaces that leave invariant a given closed sublattice of bounded continuous functions and whose relevant restrictions are Feller semigroups. Additive and multiplicative perturbation results for this class of generators are also established. Finally, some applications concerning multiplicative perturbations of the Laplacian on , , and degenerate second-order differential operators on unbounded real intervals are showed.

In this chapter, we consider Bayesian inverse problems in an infinite-dimensional Hilbert space setting. We focus on key aspects of formulating such inverse problems and provide a brief discussion of the underlying theory. The study of the associated computational methods is deferred to Chapters 16–18. As seen shortly, the formulation of a Bayesian inverse problem in a function space is rather abstract. One may pose the following question: why not first “discretize the problem” and then consider a finite-dimensional Bayesian inverse problem? We can then perform Bayesian inference as done usually in finite dimensions. Would that not be more straightforward compared to the foray into function space formulations? We need to discretize the problem to facilitate numerical computations anyway. As seen shortly, things are not so simple.

It is well known that microbials play a major role controlling many redox-sensitive \ngeochemical processes. In the context of radioactive waste disposal it has been \nrecognized recently the need to evaluate the relevance of microbial processes in \nthe performance of a HLW repository. A large amount of data on the structure, \ndiversity and activity of native microbial populations has been collected in \nunderground laboratories such as those in deep Swedish granites. Laboratory and \nfield experiments such as the REX experiment have confirmed the relevance of \nmicrobial processes and have been useful to obtain relevant hydrochemical and \nmicrobial parameters and test conceptual biogeochemical models. Here we present \ndirect and inverse models for coupled flow, reactive solute transport and \nbiogeochemical processes in porous and fractured media. The inverse problem is \nformulated as the minimization of a generalized least-squares criterion by means of \na Gauss-Newton- Levenberg-Marquardt method. Different types of data can be taken \ninto account including hydraulic heads, aqueous and total concentrations, water \nfluxes and water contents as well as prior information on model parameters. The \nmethodology can cope with the simultaneous estimation of flow, transport, microbial \nand geochemical parameters as well as initial and boundary waters, such as pH, pE, \nand abiotic and biotic concentrations. The mathematical formulation of inverse THBG \nmodels has been implemented in a finite element code, INVERSE-BIOCORE2D, which has \nbeen verified with synthetic examples and tested for the estimation of microbial \nparameters from the in situ REX experiment performed at a packed section of a \nborehole drilled from the access tunnel in Äspö Hard Rock laboratory in Sweden The \nhydro-biogeochemical model of the REX experiment considers microbially-mediated \nmethane oxidation and organic matter respiration. Numerical results indicate that \naerobic respiration of organic matter is much more efficient in consuming dissolved \noxygen than pyrite dissolution. Initial concentration of heterotrophic bacteria and \ntheir growth rate constant are the two most sensitive parameters. Calibrated THBG \nnumerical models have been later used for testing quantitatively several scenarios \nfor a HLW repository in Sweden including: 1) Aerobic respiration associated to the \noxygen supplied during the pre-closure stage; 2) Hydrochemical perturbations caused \nby the construction of the repository during its operational stage; and (3) \nScenario of a glaciation in which melting waters rich in oxygen and no organic \nmatter might reach the repository. Results of these models should be relevant for \nthe performance assessment of a HLW repository in granites for the long-evolution \nof groundwater chemistry and redox conditions during pre- and post- repository \nclosure.

A crucial decision in designing a spatial sample for soil survey is the number of sampling locations required to answer, with sufficient accuracy and precision, the questions posed by decision makers at different levels of geographic aggregation. In the Indian Soil Health Card (SHC) scheme, many thousands of locations are sampled per district. In this paper the SHC data are used to estimate the mean of a soil property within a defined study area, e.g., a district, or the areal fraction of the study area where some condition is satisfied, e.g., exceedence of a critical level. The central question is whether this large sample size is needed for this aim. The sample size required for a given maximum length of a confidence interval can be computed with formulas from classical sampling theory, using a prior estimate of the variance of the property of interest within the study area. Similarly, for the areal fraction a prior estimate of this fraction is required. In practice we are uncertain about these prior estimates, and our uncertainty is not accounted for in classical sample size determination (SSD). This deficiency can be overcome with a Bayesian approach, in which the prior estimate of the variance or areal fraction is replaced by a prior distribution. Once new data from the sample are available, this prior distribution is updated to a posterior distribution using Bayes’ rule. The apparent problem with a Bayesian approach prior to a sampling campaign is that the data are not yet available. This dilemma can be solved by computing, for a given sample size, the predictive distribution of the data, given a prior distribution on the population and design parameter. Thus we do not have a single vector with data values, but a finite or infinite set of possible data vectors. As a consequence, we have as many posterior distribution functions as we have data vectors. This leads to a probability distribution of lengths or coverages of Bayesian credible intervals, from which various criteria for SSD can be derived. Besides the fully Bayesian approach, a mixed Bayesian-likelihood approach for SSD is available. This is of interest when, after the data have been collected, we prefer to estimate the mean from these data only, using the frequentist approach, ignoring the prior distribution. The fully Bayesian and mixed Bayesian-likelihood approach are illustrated for estimating the mean of log-transformed Zn and the areal fraction with Zn-deficiency, defined as Zn concentration <0.9 mg kg −1, in the thirteen districts of Andhra Pradesh state. The SHC data from 2015–2017 are used to derive prior distributions. For all districts the Bayesian and mixed Bayesian-likelihood sample sizes are much smaller than the current sample sizes. The hyperparameters of the prior distributions have a strong effect on the sample sizes. We discuss methods to deal with this. Even at the mandal (sub-district) level the sample size can almost always be reduced substantially. Clearly SHC over-sampled, and here we show how to reduce the effort while still providing information required for decision-making. R scripts for SSD are provided as supplementary material.

For any compact Hausdorff space K and any Banach space K , let C(K, X) be the space of all continuous X-valued functions on K with the norm $$ \left\| f \right\| = \mathop {\sup }\limits_{t \in K} \left\| {f(t)} \right\|_X . $$

At the start of new operation, a production process is commonly unstable. Then, a process condition is into being stable gradually over time. For the purpose of shifting to mass production, the stability in a process should be evaluated. This paper proposes a method of evaluating the process stability based on Bayesian statistics and information theory. In Bayesian statistics, the knowledge for a process is renewed by deriving posterior distribution based on current prior distribution and observations. Here, equivalency between prior and posterior distributions could be considered to be a criterion for the stability in a process. It is needed to define difference between prior and posterior distributions in order to evaluate the equivalency between prior and posterior distributions. We first formulate the relation between the prior distribution and the posterior distribution for process mean using Bayesian statistics. Secondly, we evaluate the difference between the both distributions based on Kullback‐Leibler (K‐L) divergence in information theory. Finally, some numerical examples in the method of evaluating the stability in process mean are illustrated. Also, we discuss about a decision rule for the stability in process mean through the numerical investigation.

The similarity between maximum entropy (MaxEnt) and minimum relative entropy (MRE) allows recent advances in probabilistic inversion to obviate some of the shortcomings in the former method. The purpose of this paper is to review and extend the theory and practice of minimum relative entropy. In this regard, we illustrate important philosophies on inversion and the similarly and differences between maximum entropy, minimum relative entropy, classical smallest model (SVD) and Bayesian solutions for inverse problems. MaxEnt is applicable when we are determining a function that can be regarded as a probability distribution. The approach can be extended to the case of the general linear problem and is interpreted as the model which fits all the constraints and is the one model which has the greatest multiplicity or “spreadout” that can be realized in the greatest number of ways. The MRE solution to the inverse problem differs from the maximum entropy viewpoint as noted above. The relative entropy formulation provides the advantage of allowing for non-positive models, a prior bias in the estimated pdf and `hard' bounds if desired. We outline how MRE can be used as a measure of resolution in linear inversion and show that MRE provides us with a method to explore the limits of model space. The Bayesian methodology readily lends itself to the problem of updating prior probabilities based on uncertain field measurements, and whose truth follows from the theorems of total and compound probabilities. In the Bayesian approach information is complete and Bayes' theorem gives a unique posterior pdf. In comparing the results of the classical, MaxEnt, MRE and Bayesian approaches we notice that the approaches produce different results. In␣comparing MaxEnt with MRE for Jayne's die problem we see excellent comparisons between the results. We compare MaxEnt, smallest model and MRE approaches for the density distribution of an equivalent spherically-symmetric earth and for the contaminant plume-source problem. Theoretical comparisons between MRE and Bayesian solutions for the case of the linear model and Gaussian priors may show different results. The Bayesian expected-value solution approaches that of MRE and that of the smallest model as the prior distribution becomes uniform, but the Bayesian maximum aposteriori (MAP) solution may not exist for an underdetermined case with a uniform prior.

Various computational methods have been proposed for inference of protein-protein interactions since protein-protein interaction plays an essential role in many cellular processes. One of well-studied approaches is to infer protein-protein interactions based on domain-domain interactions. To extend this approach, we proposed a method called LPNM to infer ratios of interactions, which outperformed other existing methods in terms of error of predicted ratios. However, since the LPNM method is based on the linear programming approach, it may require a large amount of time to infer interactions for a large data set. In this paper, we propose a simple method to infer the ratios of protein-protein interactions based on the association method by Sprinzak et al. In an experiment with a data set of protein-protein interactions in yeast, it runs more than 150 times as fast as the LPNM method, and achieves almost the same accuracy. On implementing algorithms for the inference problem, it is essential to understand how difficult the problem is. Even though various methods for the problem have been already proposed, it has not been analyzed rigorously from a computational point of view. We hence define a problem to maximize correctly classified examples, and prove the problem is MAX SNP-hard, which also means the problem is NP-hard.

We prove that the existence of a solution operator for a convolution operator from the space of ultradifferntiable functions to the corresponding space of ultradistributions is equivalent to the existence of a continuous solution operator in the space of functions. Our results are in the spirit of a classical characterization of the surjectivity of convolution operators due to Hormander. The behaviour of a fixed convolution operator in different classes of ultradifferentiable functions of Beurling type concerning the existence of a continuous linear right inverse is also considered.

The Bayesian inference is widely used in many scientific and engineering problems, especially in the linear inverse problems in infinite-dimensional setting where the unknowns are functions. In such problems, choosing an appropriate prior distribution is an important task. Especially when the function to infer has much detail information, such as many sharp jumps, corners, and the discontinuous and nonsmooth oscillation, the so-called total variation-Gaussian (TG) prior is proposed in function space to address it. However, the TG prior is easy to lead the blocky (staircase) effect in numerical results. In this work, we present a fractional order-TG (FTG) hybrid prior to deal with such problems, where the fractional order total variation (FTV) term is used to capture the detail information of the unknowns and simultaneously uses the Gaussian measure to ensure that it results in a well-defined posterior measure. For the numerical implementations of linear inverse problems in function spaces, we also propose an efficient independence sampler based on a transport map, which uses a proposal distribution derived from a diagonal map, and the acceptance probability associated to the proposal is independent of discretization dimensionality. And in order to take full advantage of the transport map, the hierarchical Bayesian framework is applied to flexibly determine the regularization parameter. Finally we provide some numerical examples to demonstrate the performance of the FTG prior and the efficiency and robustness of the proposed independence sampler method.

Inverse problems arise from ill-posed mappings in many areas of science and engineering. For every inverse problem, there exist some fundamental statistical limits for the accuracy of solutions. These theoretical limits cannot be superseded in any way without incorporating some additional prior information. A major "secret" of inversion methodologies consists in identifying and applying all forms of prior information available, both hidden and explicit. Meta-Evolutionary Optimization and Bayesian Statistics are central concepts behind a very general inversion methodology recently developed. In this article, some underlying ideas of this robust inversion methodology are reviewed. As illustrated, uncertainty of the inversions may be reduced by incorporating very general forms of prior information: non-linear parametric models, probabilistic noise distributions, deterministic constraint conditions, etc. This inversion methodology is applied to an important problem from Magnetic Resonance Spectroscopy (MRS): quantitative in vivo MRS beyond inhomogeneity and noise limits. After analyzing the principal role of prior information for solving inverse problems, some limitations of two popular approaches (Sampled Pattern Matching, Artificial Neural Networks) are elucidated as well.

On the universal completion of pointfree function spaces