arXiv subjects

Daniela Calvetti

Publications and source records attributed to Daniela Calvetti.

At least 19 recordsLinked to original sources

arXiv2026

Spotlight, priorsketching and Bayesian approximation error paradigms

A way to lower computational cost in large scale inverse problems and problems depending on poorly known model parameters is to replace the detailed model by an approximate one. Inverse problems are typically ill-posed, and the model discrepancy introduced by using approximate models often shows up in the computed solutions as disturbing artifacts or blurring. In this article, we consider two methods of addressing certain types of modeling errors, the Bayesian approximation error (BAE) method and linear algebraic spotlight inversion to suppress clutter in the computational model by orthogonal projections. Through the process of analyzing the two approaches, we show that they turn out to be closely related but not equivalent, and we highlight a connection to sketching schemes in randomized linear algebra. The similarities between the methods and their successful suppression of most of the clutter effects is elucidated with two computed examples, one addressing of X-ray tomography and the other electrical impedance tomography.

math.NA↗

arXiv2026

Sparse Dictionary-Based Solution of Dynamic Inverse Problems

In ill-posed dynamic inverse problems expected spatial features and temporal correlation between frames can be leveraged to improve the quality of the computed solution, in particular when the available data are limited and the dimensionality of the unknown is large. One way to take advantage of the spatial and temporal traits believed to characterize the solution is to encode them into the entries of a dictionary, and to seek the solution as a sparse linear combination of the dictionary atoms. To promote a vector of coefficients with mostly vanishing entries, we consider a stochastic extension of the dictionary coding problem model with a random hierarchical sparsity promoting prior. We compute the Maximum A Posteriori (MAP) estimate of the coefficient vector using the Iterative Alternating Sequential Algorithm (IAS), which has been demonstrated to efficiently solve inverse problems with minimal need for parameter tuning. The proposed methodology is tested on real-world dynamic Computed Tomography and MRI datasets, where it is compared to the popular Alternating Direction Method of Minimizers (ADMM). The computed examples show the that proposed methodology is competitive with the ADMM for compressed sensing, with a significantly lower sensitivity to hyper-parameter selection.

math.NA↗

arXiv2026

Discretization-free Bayesian inverse problems in distribution spaces

The Bayesian approach to inverse problems provides a practical way to solve ill-posed problems by augmenting the observation model with prior information. Due to its measure-theoretic underpinnings, the approach has raised theoretical interest, leading to a rather comprehensive description in infinite-dimensional function spaces. The goal of this article is to bridge the infinite-dimensional theory for linear inverse problems in distribution spaces and associated computational inverse problems without resorting to a discrete approximation of the forward model. We show that the discretization of the unknown of interest is not necessary for the numerical treatment of the problem, the only approximations required being numerical quadratures that are independent of any discrete representation of the unknown. To demonstrate the viability of the approach, an analysis of X-ray tomography inverse problem is given in the proposed framework, and an analysis of the connection between the proposed approach and a discretization-based one is also provided.

math.NA↗

arXiv2025

Dictionary learning methods for brain activity mapping with MEG data

A central goal in many brain studies is the identification of those brain regions that are activated during an observation window that may correspond to a motor task, a stimulus, or simply a resting state. While functional MRI is currently the most commonly employed modality for such task, methods based on the electromagnetic activity of the brain are valuable alternatives because of their excellent time resolution and of the fact that the measured signals are directly related to brain activation and not to a secondary effect such as the hemodynamic response. In this work we focus on the MEG modality, investigating the performance of a recently proposed Bayesian dictionary learning (BDL) algorithm for brain region identification. The partitioning of the source space into the 148 regions of interest (ROI) corresponding to parcellation of the Destrieux atlas provides a natural determination of the subdictionaries necessary for the BDL algorithm. We design a simulation protocol where a small randomly selected patch in each ROI is activated, the MEG signal is computed and the inverse problem of active brain region identification is solved using the BDL algorithm. The BDL algorithm consists of two phases, the first one comprising dictionary compression and Bayesian compression error analysis, and the second one performing dictionary coding with a deflated dictionary built on the output of the first phase, both steps relying on Bayesian sparsity promoting computations. For assessing the performance, we give a probabilistic interpretation of the confusion matrix, and consider different impurity measures for a multi-class classifier.

math.NA↗

arXiv2025

Spotlight inversion by orthogonal projections

Many computational problems involve solving a linear system of equations, although only a subset of the entries of the solution are needed. In inverse problems, where the goal is to estimate unknown parameters from indirect noisy observations, it is not uncommon that the forward model linking the observed variables to the unknowns depends on variables that are not of primary interest, often referred to as nuisance parameters. In this article, we consider linear problems, and propose a novel projection technique to eliminate, or at least mitigate, the contribution of the nuisance parameters in the model. We refer to this approach as spotlight inversion, as it allows to focus on only the portion of primary interest of the unknown parameter vector, leaving the uninteresting part in the shadow. The viability of the approach is illustrated with two computed examples, one where it works as model reduction for a finite element approximation of an elliptic PDE, the other amounting to local fanbeam X-ray tomography, spotlighting the region of interest that is part of the full target.

math.NA↗

arXiv2025

Bayesian dictionary learning estimation of cell membrane permeability from surface pH data

Gas transport across cell membrane is a very important process in biochemistry which is essential for many crucial tasks, including cell respiration pH regulation in the cell. In the late 1990's, the suggestion that gasses are transported via preferred gas channels embedded into the cell membrane challenged the century old Overton's theory that gases pass through the lipid cell membrane by diffusing across the concentration gradient. Since experimental evidence alone does not provide enough evidence to favor one of the proposed mechanisms, mathematical models have been introduced to provide a context for the interpretation of laboratory measurement. Following up on previous work where the membrane permeability was estimated using particle filter, in this article we propose an algorithm based on dictionary learning for estimating cell membrane permeability. Computed examples illustrate that the novel approach, which can be applied when the properties of the membrane do not change in the course of the data collection process, is computationally much more efficient than particle filter.

math.NA↗

arXiv2025

Subspace Splitting Fast Sampling from Gaussian Posterior Distributions of Linear Inverse Problems

It is well-known that the posterior density of linear inverse problems with Gaussian prior and Gaussian likelihood is also Gaussian, hence completely described by its covariance and expectation. Sampling from a Gaussian posterior may be important in the analysis of various non-Gaussian inverse problems in which a estimates from a Gaussian posterior distribution constitute an intermediate stage in a Bayesian workflow. Sampling from a Gaussian distribution is straightforward if the Cholesky factorization of the covariance matrix or its inverse is available, however when the unknown is high dimensional, the computation of the posterior covariance maybe unfeasible. If the linear inverse problem is underdetermined, it is possible to exploit the orthogonality of the fundamental subspaces associated with the coefficient matrix together with the idea behind the Randomize-Then-Optimize approach to design a low complexity posterior sampler that does not require the posterior covariance to be formed. The performance of the proposed sampler is illustrated with a few computed examples, including non-Gaussian problems with non-linear forward model, and hierarchical models comprising a conditionally Gaussian submodel.

math.NA↗

arXiv2024

An efficient hierarchical Bayesian method for the Kuopio tomography challenge 2023

The aim of Electrical Impedance Tomography (EIT) is to determine the electrical conductivity distribution inside a domain by applying currents and measuring voltages on its boundary. Mathematically, the EIT reconstruction task can be formulated as a non-linear inverse problem. The Bayesian inverse problems framework has been applied expensively to solutions of the EIT inverse problem, in particular in the cases when the unknown conductivity is believed to be blocky. Recently, the Sparsity Promoting Iterative Alternating Sequential (PS-IAS) algorithm, originally proposed for the solution of linear inverse problems, has been adapted for the non linear case of EIT reconstruction in a computationally efficient manner. Here we introduce a hybrid version of the SP-IAS algorithms for the nonlinear EIT inverse problem, providing a detailed description of the implementation details, with a specific focus on parameters selection. The method is applied to the 2023 Kuopio Tomography Challenge dataset, with a comprehensive report of the running times for the different cases and parameter selections.

math.NA↗

arXiv2024

Sparsity-promoting hierarchical Bayesian model for EIT with a blocky target

The electrical impedance tomography (EIT) problem of estimating the unknown conductivity distribution inside a domain from boundary current or voltage measurements requires the solution of a nonlinear inverse problem. Sparsity promoting hierarchical Bayesian models have been shown to be very effective in the recovery of almost piecewise constant solutions in linear inverse problems. We demonstrate that by exploiting linear algebraic considerations it is possible to organize the calculation for the Bayesian solution of the nonlinear EIT inverse problem via finite element methods with sparsity promoting priors in a computationally efficient manner. The proposed approach uses the Iterative Alternating Sequential (IAS) algorithm for the solution of the linearized problems. Within the IAS algorithm, a substantial reduction in computational complexity is attained by exploiting the low dimensionality of the data space and an adjoint formulation of the Tikhonov regularized solution that constitutes part of the iterative updating scheme. Numerical tests illustrate the computational efficiency of the proposed algorithm. The paper sheds light also on the convexity properties of the objective function of the maximum a posteriori (MAP) estimation problem.

math.NA↗

arXiv2024

Distributed Tikhonov regularization for ill-posed inverse problems from a Bayesian perspective

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. In the standard formulation, Tikhonov regularization compensates for the inherent ill-conditioning of linear inverse problems by augmenting the data fidelity term measuring the mismatch between the data and the model output with a scaled penalty functional. The selection of the scaling is the core problem in Tikhonov regularization. If an estimate of the amount of noise in the data is available, a popular way is to use the Morozov discrepancy principle, stating that the scaling parameter should be chosen so as to guarantee that the norm of the data fitting error is approximately equal to the norm of the noise in the data. A too small value of the regularization parameter would yield a solution that fits to the noise while a too large value would lead to an excessive penalization of the solution. In many applications, it would be preferable to apply distributed regularization, replacing the regularization scalar by a vector valued parameter, allowing different regularization for different components of the unknown, or for groups of them. A distributed Tikhonov-inspired regularization is particularly well suited when the data have significantly different sensitivity to different components, or to promote sparsity of the solution. The numerical scheme that we propose, while exploiting the Bayesian interpretation of the inverse problem and identifying the Tikhonov regularization with the Maximum A Posteriori (MAP) estimation, requires no statistical tools. A combination of numerical linear algebra and optimization tools makes the scheme computationally efficient and suitable for problems where the matrix is not explicitly available.

math.NA↗

arXiv2023

Adaptive anisotropic Bayesian meshing for inverse problems

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 the errors arising from the discretization can be detrimental for ill-posed inverse problems, as discretization error behaves as correlated noise. While this problem can be avoided with a discretization fine enough to suppress the modeling error level below that of the exogenous noise that is addressed, e.g., by regularization, the computational resources needed to deal with the additional degrees of freedom may require high performance computing environment. Following an earlier idea, we advocate the notion that the discretization is one of the unknowns of the inverse problem, and is updated iteratively together with the solution. In this approach, the discretization, defined in terms of an underlying metric, is refined selectively only where the representation power of the current mesh is insufficient. In this paper we allow the metrics and meshes to be anisotropic, and we show that this leads to significant reduction of memory allocation and computing time.

math.NA↗

arXiv2023

Bayesian sparsity and class sparsity priors for dictionary learning and coding

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 search engines. Sparse coding techniques are used to guarantee that the dictionary matching identifies only few of the dictionary entries, and dictionary compression methods are used to reduce the complexity of the matching problem. In this article, we propose a work flow to facilitate the dictionary matching process. First, the full dictionary is divided into subdictionaries that are separately compressed. The error introduced by the dictionary compression is handled in the Bayesian framework as a modeling error. Furthermore, we propose a new Bayesian data-driven group sparsity coding method to help identify subdictionaries that are not relevant for the dictionary matching. After discarding irrelevant subdictionaries, the dictionary matching is addressed as a deflated problem using sparse coding. The compression and deflation steps can lead to substantial decreases of the computational complexity. The effectiveness of compensating for the dictionary compression error and using the novel group sparsity promotion to deflate the original dictionary are illustrated by applying the methodology to real world problems, the glitch detection in the LIGO experiment and hyperspectral remote sensing.

stat.ML↗

arXiv2023

Computationally efficient sampling methods for sparsity promoting hierarchical Bayesian models

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 generalized gamma distributions. Most of the work in the literature has concentrated on numerical approximation of the maximum a posteriori (MAP) estimates, and less attention has been paid on sampling methods or other means for uncertainty quantification. Sampling from the hierarchical models is challenging mainly for two reasons: The hierarchical models are typically high-dimensional, thus suffering from the curse of dimensionality, and the strong correlation between the unknown of interest and its variance can make sampling rather inefficient. This work addresses mainly the first one of these obstacles. By using a novel reparametrization, it is shown how the posterior distribution can be transformed into one dominated by a Gaussian white noise, allowing sampling by using the preconditioned Crank-Nicholson (pCN) scheme that has been shown to be efficient for sampling from distributions dominated by a Gaussian component. Furthermore, a novel idea for speeding up the pCN in a special case is developed, and the question of how strongly the hierarchical models are concentrated on sparse solutions is addressed in light of a computed example.

math.NA↗

arXiv2022

On the Fast Track: Rapid construction of stellar stream paths

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 reconstruct a stream's path, propagating measurement errors and data sparsity into the stream path uncertainty. The technique is Galactic-model independent, non-parametric, and works on phase-wrapped streams. With this technique, we can uniformly analyze and compare data with simulations, enabling both comparison of simulation techniques and ensemble analysis with stream tracks of many stellar streams. Our method is implemented in the public Python package TrackStream, available at https://github.com/nstarman/trackstream.

astro-ph.GA↗

arXiv2022

A Spatially Distributed Model of Brain Metabolism Highlights the Role of Diffusion in Brain Energy Metabolism

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 and the cerebral blood flow. A comprehensive mathematical model of brain metabolism should account not only for the biochemical processes and the interaction of neurons and astrocytes, but also the diffusion of metabolites. In the present article, we present a computational methodology based on a multidomain model of the brain tissue and a homogenization argument for the diffusion processes. In our spatially distributed compartment model, communication between compartments occur both through local transport fluxes, as is the case within local astrocyte-neuron complexes, and through diffusion of some substances in some of the compartments. The model assumes that diffusion takes place in the extracellular space (ECS) and in the astrocyte compartment. In the astrocyte compartment, the diffusion across the syncytium network is implemented as a function of gap junction strength. The diffusion process is implemented numerically by means of a finite element method (FEM) based spatial discretization, and robust stiff solvers are used to time integrate the resulting large system. Computed experiments show the effects of ECS tortuosity, gap junction strength and spatial anisotropy in the astrocyte network on the brain energy metabolism.

q-bio.TO↗

arXiv2020

Overcomplete representation in a hierarchical Bayesian framework

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 appropriate numerical scheme to accomplish the task. A computationally efficient alternative for finding sparse solutions to linear inverse problems is provided by Bayesian hierarchical models, in which the sparsity is encoded by defining a conditionally Gaussian prior model with the prior parameter obeying a generalized gamma distribution. An iterative alternating sequential (IAS) algorithm has been demonstrated to lead to a computationally efficient scheme, and combined with Krylov subspace iterations with an early termination condition, the approach is particularly well suited for large scale problems. Here the Bayesian approach to sparsity is extended to problems whose solution allows a sparse coding in an overcomplete system such as composite frames. It is shown that among the multiple possible representations of the unknown, the IAS algorithm, and in particular, a hybrid version of it, is effectively identifying the most sparse solution. Computed examples show that the method is particularly well suited not only for traditional imaging applications but also for dictionary learning problems in the framework of machine learning.

math.NA↗

arXiv2020

Metapopulation network models for understanding, predicting and managing the coronavirus disease COVID-19

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 the population. To address this problem, we propose to account for the differences between rural and urban settings using network-based, distributed models where the spread of the pandemic is described in distinct local cohorts with nested SEIR models. The setting of the model parameters takes into account the fact that SARS-CoV-2 transmission occurs mostly via human-to-human contact, and that the frequency of contact among individuals differs between urban and rural areas, and may change over time. Moreover, the probability that the virus spreads into an uninfected community is associated with influx of individuals from other communities where the infection is present. To account for these important aspects, each node of the network is characterized by the frequency of contact between its members and by its level of connectivity with other nodes. Census and cell phone data can be used to set up the adjacency matrix of the network, which can, in turn, be modified to account for different levels of mitigation measures. In order to make the network SEIR model that we propose easy to customize, it is formulated in terms of easily interpretable parameters that can be estimated from available community level data. The models parameters are estimated with Bayesian techniques using COVID-19 data for the states of Ohio and Michigan. The network model also gives rise to a geographically distributed computational model that explains the geographic dynamics of the contagion, e.g., in larger cities surrounded by suburban and rural areas.

q-bio.PE↗

arXiv2020

Bayesian dynamical estimation of the parameters of an SE(A)IR COVID-19 spread model

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 documented infectious individuals, the model includes a cohort of asymptomatic or oligosymptomatic infectious individuals, not accounted for in the data of new daily counts of infections. A Bayesian particle filtering algorithm is used to update dynamically the relevant cohort and simultaneously estimate the transmission rate as the new data on the number of new infections and disease related death become available. The underlying assumption of the model is that the infectivity rate is dynamically changing during the epidemics, either because of a mutation of the pathogen or in response to mitigation and containment measures. The sequential Bayesian framework naturally provides a quantification of the uncertainty in the estimate of the model parameters, including the reproduction number, and of the size of the different cohorts. Moreover, we introduce a dimensionless quantity, which is the equilibrium ratio between asymptomatic and symptomatic cohort sizes, and propose a simple formula to estimate the quantity. This ratio leads naturally to another dimensionless quantity that plays the role of the basic reproduction number $R_0$ of the model. When we apply the model and particle filter algorithm to COVID-19 infection data from several counties in Northeastern Ohio and Southeastern Michigan we found the proposed reproduction number $R_0$ to have a consistent dynamic behavior within both states, thus proving to be a reliable summary of the success of the mitigation measures.

q-bio.PE↗