Daniela Calvetti

• Case Western Reserve University

In this article, we exploit the similarities between Tikhonov regularization and Bayesian hierarchical models to propose a regularization scheme that acts like a distributed Tikhonov regularization where the amount of regularization varies from component to component, and a computationally efficient numerical scheme that is suitable for large-scale...

The inherent redundancy of the musculoskeletal systems is traditionally solved by optimizing a cost function. This approach may not be correct to model non-adult or pathological populations likely to adopt a “non-optimal” motor control strategy. Over the years, various methods have been developed to address this limitation, such as the stochastic a...

A recent method for determining accurate, dense velocity fields from a pair of particle images is to solve the optical flow problem, but the ill-posed inverse problem of optical flow velocimetry (OFV) generally entails minimizing a weighted sum of two terms--fidelity and regularization--and the weights in the sum are parameters that require manual...

We consider inverse problems estimating distributed parameters from indirect noisy observations through discretization of continuum models described by partial differential or integral equations. It is well understood that errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as co...

Dictionary learning methods continue to gain popularity for the solution of challenging inverse problems. In the dictionary learning approach, the computational forward model is replaced by a large dictionary of possible outcomes, and the problem is to identify the dictionary entries that best match the data, akin to traditional query matching in s...

Bayesian particle filters (PFs) are a viable alternative to sampling methods such as Markov chain Monte Carlo methods to estimate model parameters and related uncertainties when the forward model is a dynamical system, and the data are time series that depend on the state vector. PF techniques are particularly attractive when the dimensionality of...

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes an...

Stellar streams are sensitive probes of the Galactic potential. The likelihood of a stream model given stream data is often assessed using simulations. However, comparing to simulations is challenging when even the stream paths can be hard to quantify. Here we present a novel application of Self-Organizing Maps and first-order Kalman Filters to rec...

Bayesian hierarchical models have been demonstrated to provide efficient algorithms for finding sparse solutions to ill-posed inverse problems. The models comprise typically a conditionally Gaussian prior model for the unknown, augmented by a hyperprior model for the variances. A widely used choice for the hyperprior is a member of the family of ge...

After the lengthy technical introduction of the previous chapters, we are now ready to start estimating unknown quantities based on incomplete information and indirect observations. We adopt here the Bayesian point of view: Any quantity that is not known exactly, in the sense that a value can be attached to it with no uncertainty, is modeled as a r...

“The only relevant thing is uncertainty—the extent of our knowledge and ignorance. The actual fact of whether or not the events considered are in some sense determined, or known by other people, and so on, is of no consequence.”

Algorithms for finding sparse solutions are of importance in several application areas, and the design of penalty functionals that promote sparsity continues to be the topic of active research. In the Bayesian framework, searching for a sparse solution implicitly states that there is an a priori belief that only a few components of the unknown are...

Gaussian probability distributions are the workhorse in Bayesian scientific computing, providing a well understood subclass of distributions that often allow closed form solutions to inference problems. Not only do the Gaussian distributions play a role similar to that of linear operators in analysis and in inverse problems in particular, as pointe...

A large portion of statistical inference deals with statistical modeling and analysis, which often include the description and understanding of the data collection process. Therefore, statistical inference lies at the very core of scientific modeling and empirical testing of theories. Statistical modeling may lead to the conclusion that the underly...

Now that the basic principles guiding the design of likelihoods and priors have been introduced, we are ready to welcome on the stage the main character in the Bayesian play of inverse problems, the posterior distributionPosterior distribution, and in particular, the posterior density. Bayes’ formula is the way in which prior and likelihood combine...

In the previous chapter, we considered dynamic inverse problems where the posterior density is updated sequentially as new observations become available. The particle filter approach is fully general and does not assume anything particular about the probability densities, as they were approximated by particle-based point mass distributions. However...

In this chapter we return to the solution of linear systems of equations, a task that in scientific computing has a core role, either as a problem of interest per se or as part of a larger computational endeavor.

After the generic introduction to probability and linear algebra, in this chapter, we start to put together these concepts. In particular, we introduce the workhorse of computational statistics, the normal distribution, using elements from matrix factorizations. Normal distributions play a role in computational statistics similar to that of linear...

Richard Price, a Welsh philosopher and mathematician who has significantly contributed to the early development of Bayesian theory, published in 1764 an essay on Bayes’ work, in which he asked how to assign a subjective probability to the sunrise, given that the sun had been observed to rise a given number of times before. Price’s idea is that we l...

In Chapter 4, where we first introduced random sampling, we pointed out that sampling is used primarily to explore a given probability distribution and to calculate estimates of integrals via Monte Carlo integration. It was also indicated that sampling from a non-Gaussian probability density may be a challenging task. In this section we further dev...

There is no question that linear algebra is one of the cornerstones of scientific computing. Moreover, linear algebra provides an effective language for dealing with multidimensional phenomena, including multivariate statistics that without this language would become awkward and cumbersome. Instead of collecting all the linear algebra definitions a...

Bayesian scientific computing, as understood in this text, is a field of applied mathematics that combines numerical analysis and traditional scientific computingScientific computing to solve problems in science and engineering with the philosophy and language of Bayesian inferenceBayesian inference. An overarching goal is to apply these techniques...

Dictionary learning, aiming at representing a signal in terms of the atoms of a dictionary, has gained popularity in a wide range of applications, including, but not limited to, image denoising, face recognition, remote sensing, medical imaging and feature extraction. Dictionary learning can be seen as a possible data-driven alternative to solve in...

There are many factors in the current phase of the COVID-19 pandemic that signal the need for new modeling ideas. In fact, most traditional infectious disease models do not address adequately the waning immunity, in particular as new emerging variants have been able to brake the immune shield acquired either by previous infection by a different str...

We present a standalone Matlab software platform complete with visualization for the reconstruction of the neural activity in the brain from MEG or EEG data. The underlying inversion combines hierarchical Bayesian models and Krylov subspace iterative least squares solvers. The Bayesian framework of the underlying inversion algorithm allows to accou...

Stellar streams are sensitive probes of the Galactic potential. The likelihood of a stream model given stream data is often assessed using simulations. However, comparing to simulations is challenging when even the stream paths can be hard to quantify. Here we present a novel application of Self-Organizing Maps and first-order Kalman Filters to rec...

The different active roles of neurons and astrocytes during neuronal activation are associated with the metabolic processes necessary to supply the energy needed for their respective tasks at rest and during neuronal activation. Metabolism, in turn, relies on the delivery of metabolites and removal of toxic byproducts through diffusion processes an...

The transport of gases across cell membranes plays a key role in many different cell functions, from cell respiration to pH control. Mathematical models play a central role in understanding the factors affecting gas transport through membranes, and are the tool needed for testing the novel hypothesis of the preferential crossing through specific ga...

In this article, we consider a dynamic model for the spread of epidemics, in particular of COVID-19, and the inverse problem of estimating sequentially the time evolution of the unknown state and the model parameters based on noisy observations of the new daily infections. A characteristic of COVID-19 is the significant proportion of secondary infe...

Meditation practices have been claimed to have a positive effect on the regulation of mood and emotions for quite some time by practitioners, and in recent times there has been a sustained effort to provide a more precise description of the influence of meditation on the human brain. Longitudinal studies have reported morphological changes in corti...

Understanding the dynamics of the spread of COVID-19 between connected communities is fundamental in planning appropriate mitigation measures. To that end, we propose and analyze a novel metapopulation network model, particularly suitable for modeling commuter traffic patterns, that takes into account the connectivity between a heterogeneous set of...

Meditation practices have been claimed to have a positive effect on the regulation of mood and emotion for quite some time by practitioners, and in recent times there has been a sustained effort to provide a more precise description of the changes induced by meditation on human brain. Longitudinal studies have reported morphological changes in cort...

A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining \begin{document}$ \ell_1 $\end{document}-penalized least...

A common task in inverse problems and imaging is finding a solution that is sparse, in the sense that most of its components vanish. In the framework of compressed sensing, general results guaranteeing exact recovery have been proven. In practice, sparse solutions are often computed combining $\ell_1$-penalized least squares optimization with an ap...

Mathematical models of SARS-CoV-2 (the virus which causes COVID-19) spread are used for guiding the design of mitigation steps and helping identify impending breaches of health care system surge capacity. The challenges of having only lacunary information about daily new infections and mortality counts are compounded by geographic heterogeneity of...

The mechanism of gas transport across cell membranes remains a topic of considerable interest, particularly regarding the extent to which lipids vs. specific membrane proteins provide conduction pathways. Studies of transmembrane (CO2) transport often rely on data collected under controlled conditions, using pH-sensitive microelectrodes at the extr...

Mathematical models of SARS-CoV-2 spread are used for guiding the design of mitigation steps aimed at containing and decelerating the contagion, and at identifying impending breaches of health care system surge capacity. The challenges of having only lacunary information about daily new infections are compounded by the geographic heterogeneity of t...

In this article, we consider a dynamic epidemiology model for the spread of the COVID-19 infection. Starting from the classical SEIR model, the model is modified so as to better describe characteristic features of the underlying pathogen and its infectious modes. In line with the large number of secondary infections not related to contact with docu...

Layer stripping is a method for solving inverse boundary value problems for elliptic PDEs, originally proposed in the literature for solving the Calderón problem of electrical impedance tomography (EIT), where the data consist of the Neumann-to-Dirichlet operator on the boundary. Defining a tangent–normal coordinate system near the boundary, the da...

The recovery of sparse generative models from few noisy measurements is an important and challenging problem. Many deterministic algorithms rely on some form of $\ell_1$-$\ell_2$ minimization to combine the computational convenience of the $\ell_2$ penalty and the sparsity promotion of the $\ell_1$. It was recently shown within the Bayesian framewo...

Solving inverse problems with sparsity promoting regularizing penalties can be recast in the Bayesian framework as finding a maximum a posteriori (MAP) estimate with sparsity promoting priors. In the latter context, a computationally convenient choice of prior is the family of conditionally Gaussian hierarchical models for which the prior variances...

The slow propagating waves of strong depolarization of neural cells characterizing cortical spreading depression, or depolarization, (SD) are known to break cerebral homeostasis and induce significant hemodynamic and electro-metabolic alterations. Mathematical models of cortical spreading depression found in the literature tend to focus on the chan...

The energetic needs of brain cells at rest and during elevated neuronal activation has been the topic of many investigations where mathematical models have played a significant role providing a context for the interpretation of experimental findings. A recently proposed mathematical model, comprising a double feedback between cellular metabolism an...

A recently proposed IAS MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its sensitivity and specificity as a function of the activity location i...

The electrical impedance tomography (EIT) in its classical formulation seeks to estimate the electric conductivity distribution inside the body from the knowledge of the Dirichlet-to-Neumann (DtN) map of the conductivity equation at the boundary. Numerical methods for the solution of the EIT problem have been developed based on this formulation, mo...

Sparse recovery seeks to estimate the support and the non-zero entries of a sparse signal from possibly incomplete noisy observations , with , . It has been shown that under various restrictive conditions on the matrix , the problem can be reduced to the regularized problem where is the size of the error , and the approximation error is well contro...

Several key brain imaging modalities that are intended for retrieving information about neuronal activity in brain, the BOLD fMRI as a foremost example, rely on the assumption that elevated neuronal activity elicits spatiotemporally well localized increase of the oxygenated blood volume, which in turn can be monitored non-invasively. The details of...

The human brain is a small organ which uses a disproportional amount of the total metabolic energy production in the body. While it is well understood that the most significant energy sink is the maintenance of the neuronal membrane potential during the brain signaling activity, the role of astrocytes in the energy balance continues to be the topic...

Inverse problems deal with the quest for unknown causes of observed consequences, based on predictive models, known as the forward models, that associate the former quantities to the latter in the causal order. Forward models are usually well‐posed, as causes determine consequences in a unique and stable way. Inverse problems, on the other hand, ar...

In computational inverse problems, it is common that a detailed and accurate forward model is approximated by a computationally less challenging substitute. The model reduction may be necessary to meet constraints in computing time when optimization algorithms are used to find a single estimate, or to speed up Markov chain Monte Carlo (MCMC) calcul...

The solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the soluti...

In multiscale studies of emergent phenomena, a common approach is to define a microscopic scale generative model and subsequently, by passing to a diffusion limit, to derive a mesoscopic or macroscopic model via a homogenization argument. Microscopic models are often inherently stochastic, while the diffusion limit model may represent distributions...

We consider linear discrete ill-posed problems within the Bayesian framework, as-suming a Gaussian additive noise model and a Gaussian prior whose covariance matrices may be known modulo multiplicative scaling factors. In that context, we propose a new pointwise estimator for the posterior density, the priorconditioned CGLS-based quasi-MAP (qMAP) a...

Identifying feasible steady state solutions of a brain energy metabolism model is an inverse problem that allows infinitely many solutions. The characterization of the non-uniqueness, or the uncertainty quantification of the flux balance analysis, is tantamount to identifying the degrees of freedom of the solution. The degrees of freedom of multi-c...

The LMM PF is a methodology for solving the state and parameter estimation problem for ODEs system, rooted into a Bayesian framework and for it we consider a test problem with known solution to investigate in depth the error sources and how they depend on the choice of LMM used for the numerical integration. We conclude by looking at the effect on...

The Linear Multistep Method Particle Filter (LMM PF) is a method for predicting the evolution in time of a evolutionary system governed by a system of differential equations. If some of the parameters of the governing equations are unknowns, it is possible to organize the calculations so as to estimate them while following the evolution of the syst...

The time course of extracellular surface pH (pH S ), recorded by a blunt pH microelectrode, is the key physiological data to assess membrane permeability to CO 2 ( P M,CO2 ) when exposing an oocyte to a solution containing CO 2 /HCO 3 ⁻ . In an effort to extract P M,CO2 values from pH S measurements, we previously developed a reaction‐diffusion mat...

The inverse problem of MEG aims at estimating electromagnetic cerebral activity from measurements of the magnetic fields outside the head. After formulating the problem within the Bayesian framework, a hierarchical conditionally Gaussian prior model is introduced, including a physiologically inspired prior model that takes into account the preferre...

We show that a Schur form of a real orthogonal matrix can be obtained from a full CS decomposition. Based on this fact a CS decomposition-based orthogonal eigenvalue method is developed. We also describe an algorithm for orthogonal similarity transformation of an orthogonal matrix to a condensed product form, and an algorithm for full CS decomposit...

Artificial boundary conditions have long been an active research topic in numerical approximation of scattering waves: The truncation of the computational domain and the assignment of the conditions along the ficti- tious boundary must be done so that no spurious reflections occur. In inverse boundary value problems, a similar problem appears when...

In [3] (Part 1), the authors discussed the electrical impedance tomography (EIT) problem, in which the computational domain with an unknown conductivity distribution comprises only a portion of the whole conducting body, and a boundary condition along the artificial boundary needs to be set so as to minimally disturbs the estimate in the domain of...

This article discusses some of the basic principles of mathematical modeling in life sciences, and in particular the special features that make the modeling task fundamentally different from the traditional reductive modeling. The intricacies of the modeling in living systems are elucidated by simple and tractable examples that underline the proble...

This article develops a three-dimensional spatially distributed model of brain cellular metabolism and investigates how the locus of the synaptic activity in reference to the capillaries and diffusion affects the behavior of the model, a type of analysis which is impossible to carry out in spatially lumped models, which are shown to be consistent s...

Muscle forces can be selected from a space of muscle recruitment strategies that produce stable motion and variable muscle and joint forces. However, current optimization methods provide only a single muscle recruitment strategy. We modelled the spectrum of muscle recruitment strategies while walking. The equilibrium equations at the joints, muscle...

The solution of linear inverse problems when the unknown parameters outnumber data requires addressing the problem of a nontrivial null space. After restating the problem within the Bayesian framework, a priori information about the unknown can be utilized for determining the null space contribution to the solution. More specifically, if the soluti...

The degrees of freedom of multi-compartment mathematical models for energy metabolism of a neuron-astrocyte complex may offer a key to understand the different ways in which the energetic needs of the brain are met. In this paper we address the problem within a steady state framework and we use the techniques of linear algebra to identify the degre...

We address the problem of estimating the unknown parameters of a model of tracer kinetics from sequences of positron emission tomography (PET) scan data using a statistical sequential algorithm for the inference of magnitudes of dynamic parameters. The method, based on Bayesian statistical inference, is a modification of a recently proposed particl...

Particle filter (PF) sequential Monte Carlo (SMC) methods are very attractive for the estimation of parameters of time dependent systems where the data is either not all available at once, or the range of time constants is wide enough to create problems in the numerical time propagation of the states. The need to evolve a large number of particles...

Articles published in the special issue of the Inverse Problems journal reveal that the Bayesian methodology has gained a lot of popularity as a framework for considering inverse problems and has integrated successfully with many traditional inverse problems ideas and techniques, providing novel ways to interpret and implement traditional procedure...

This article addresses the problem of compensating for discretization errors in inverse problems based on partial differential equation models. Multidimensional inverse problems are by nature computationally intensive, and a key challenge in practical applications is to reduce the computing time. In particular, a reduction by coarse discretization...

A commonly encountered problem in numerous areas of applications is to estimate the unknown coefficients of a dynamical system from direct or indirect observations at discrete times of some of the components of the state vector. A related problem is to estimate unobserved components of the state. An egregious example of such a problem is provided b...

In this article we discuss ill-posed inverse problems, with an emphasis on hierarchical variable order regularization. Traditionally, smoothness penalties in Tikhonov regularization assume a fixed degree of regularity of the unknown over the whole domain. Using a Bayesian framework with hierarchical priors, we derive a prior model, formally represe...

A common approach to understand and analyze complex biological systems is to describe the dynamics in terms of a system of ordinary differential equations (ODE) depending on numerous biologically meaningful and descriptive parameters that are estimated using observed data. The ODE models are often based on the implicit assumption of well-mixed dyna...

The modeling of glutamate/GABA-glutamine cycling in the brain tissue involving astrocytes, glutamatergic and GABAergic neurons leads to a complex compartmentalized metabolic network that comprises neurotransmitter synthesis, shuttling, and degradation. Without advanced computational tools, it is difficult to quantitatively track possible scenarios...

Numerical integration is the main bottleneck in particle filter methodologies for dynamic inverse problems to estimate model parameters, initial values, and non-observable components of an ordinary differential equation (ODE) system from partial, noisy observations, because proposals may result in stiff systems which first slow down or paralyze the...

We consider the computational problem arising in magnetoencephalography (MEG), where the goal is to estimate the electric activity within the brain noninvasively from extracranial measurements of the magnetic field components. The problem is severely ill-posed due to the intrinsic nonuniqueness of the solution, and suffers further from the challeng...

Mathematical modeling of the energy metabolism of brain cells plays a central role in understanding data collected with different imaging modalities, and in making predictions based on them. During the last decade, several sophisticated brain metabolism models have appeared. Unfortunately, the picture of the metabolic details that emerges from them...

Electrical impedance spectroscopy (EIS) is a noninvasive modality that can be used to determine the electrical admittivity inside a body given a discrete set of current/voltage measurements made on the surface. Of particular interest is the use of EIS in the diagnosis of breast cancer, as the admittivity spectra of malignant and benign tumors diffe...

We present a new hybrid stochastic-deterministic, spatially distributed computational model to simulate growth competition assays on a relatively immobile monolayer of peripheral blood mononuclear cells (PBMCs), commonly used for determining ex vivo fitness of human immunodeficiency virus type-1 (HIV-1). The novel features of our approach include i...

We have developed and implemented a novel mathematical model for simulating transients in surface pH (pH(S)) and intracellular pH (pH(i)) caused by the influx of carbon dioxide (CO(2)) into a Xenopus oocyte. These transients are important tools for studying gas channels. We assume that the oocyte is a sphere surrounded by a thin layer of unstirred...

A common problem in computational inverse problems is to find an efficient way of solving linear or nonlinear least-squares problems. For large-scale problems, iterative solvers are the method of choice for solving the associated linear systems, and for nonlinear problems, an additional effective local linearization method is required. In this pape...

This work is a computational study based on a new detailed metabolic network model comprising well-mixed compartments representing separate cytosol and mitochondria of astrocytes, glutamatergic and gamma aminobutyric acid (GABA)ergic neurons, communicating through an extracellular space compartment and fed by arterial blood flow. Our steady-state a...