arXiv subjects

Mirjeta Pasha

Publications and source records attributed to Mirjeta Pasha.

At least 19 recordsLinked to original sources

arXiv2026

Randomized Tucker-Sketched GMRES

We address the problem of solving large-scale tensor-structured linear systems in the Tucker format. In this setting, standard iterative solvers such as GMRES face a fundamental bottleneck: the multilinear ranks of the Krylov basis vectors grow with the iteration count, leading to rapidly increasing tensor operation costs and memory requirements. To overcome these challenges, we propose two randomized algorithms within the sketched GMRES framework that replace full Arnoldi orthogonalization with short recurrences. The first, RHOSVD-Tucker sGMRES, uses randomized HOSVD with per-iteration rank selection, providing robustness across a wide range of problems. The second method, MLN-Tucker sGMRES, leverages the multilinear Nystr\"om approximation with a fixed rank, enabling streaming computations; the streamability of the approximation further allows, at no additional cost, a memory-efficient reconstruction of the solution from a compact sketched representation of the Krylov basis. Both methods outperform standard low-rank Tucker solvers in symmetric and non-symmetric settings. Applied to inverse problems, the low-rank Tucker constraint acts as an implicit regularizer; combined with adaptive projected Tikhonov penalization and automatic regularization parameter selection, the methods yield stable reconstructions.

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arXiv2026

A Sketched Generalized Krylov Subspace Method for Large-Scale Regularization

The generalized Krylov subspace (GKS) method is an effective projection technique for large-scale Tikhonov regularization with a general regularization matrix. As the subspace expands, however, two computational bottlenecks limit scalability: the thin QR factorizations of the tall projected matrices formed by the forward operator and the regularization matrix applied to the basis, and the full reorthogonalization of each new basis vector against all previous columns. We propose a sketched variant, named sGKS, that addresses both bottlenecks. The QR factorizations are performed on compressed matrices of much smaller row dimension, maintained incrementally via rank-one updates. Moreover, we observe that explicit reorthogonalization can be skipped entirely without compromising the quality of the approximation subspace, since no step of GKS relies intrinsically on the orthogonality of the basis. The resulting algorithm is independent of the choice of sketching operator and preserves the approximation quality of the original method: we show that, in the absence of sketching in the projected solve, sGKS produces iterates identical to those of standard GKS, and that the sketched projected solve delivers quasi-optimal residual norms controlled by the embedding quality. For more challenging problems where the loss of basis orthogonality becomes significant, we show that incorporating a small number of iterative refinement steps in the projected solve restores the spectral properties of the basis and recovers the full accuracy of the unsketched method. Numerical experiments on image deblurring, X-ray computerized tomography, seismic travel-time tomography, and dynamic computerized tomography demonstrate that sGKS matches the reconstruction quality of standard GKS while significantly reducing per-iteration costs and overall wall-clock time.

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arXiv2026

Nonlinear RMM-GKS for Large-Scale Dynamic and Streaming Inverse Problems with Uncertain Forward Operators

Many practical imaging systems suffer from uncertainty in acquisition geometry -- such as projection angles in computed tomography or sensor positions in photoacoustic tomography -- leading to nonlinear inverse problems that require joint estimation of both the image and the forward model parameters. Standard approaches that assume a known linear forward operator fail to account for these uncertainties, resulting in significant reconstruction artifacts. We propose a nonlinear recycled majorization-minimization generalized Krylov subspace (NL-RMM-GKS) framework for large-scale inverse problems with uncertain forward operators. The method extends MM-GKS to nonlinear settings by combining majorization-minimization for nonsmooth regularization with Krylov subspace projection and subspace recycling, ensuring bounded memory usage. Two complementary formulations are developed: an alternating minimization approach that alternates between image updates and Gauss-Newton parameter estimation, and a variable projection approach that eliminates the image variable and optimizes directly over the parameters using inexact inner solves. We further introduce streaming variants that process data sequentially, enabling reconstruction from large or dynamically acquired datasets without storing the full operator. For dynamic problems, we incorporate two temporal regularization strategies -- optical flow and anisotropic total variation -- as plug-in choices within the framework. We carry out rigorous numerical experiments in fan-beam computed tomography and photoacoustic tomography to demonstrate that our proposed framework achieves high-quality reconstructions with bounded memory requirements, making it suitable for large-scale dynamic imaging problems.

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arXiv2026

An Efficient Cumulative Edge-Detection Method for Image Reconstruction

When reconstructing images from noisy measurements, such as in medical scans or scientific imaging, we face an inverse problem: recovering an unknown image from indirect, corrupted observations. These problems are typically ill-posed, meaning small amounts of noise can lead to inaccurate reconstructions. Regularization techniques address this by incorporating prior assumptions about the solution, such as smoothness or sparsity. However, standard methods often blur sharp edges--the boundaries between tissues or structures--losing critical detail. A powerful strategy for edge preservation is iterative reweighting, which solves a sequence of weighted subproblems with adaptively updated weights. Non-cumulative schemes derive weights from the current iterate alone and can be solved efficiently using the Recycled Majorization-Minimization Generalized Krylov Subspace method (RMM-GKS). The cumulative approach of Gazzola et al. progressively accumulates edge information across iterations, achieving superior edge preservation but at high computational cost. This work introduces CR-$\ell_q$-RMM-GKS, which combines cumulative edge detection with computational efficiency. We integrate Gazzola's cumulative weighting with RMM-GKS, which handles general $\ell_q$ penalties ($0 < q \le 2$), automatically selects regularization parameters, and recycles Krylov subspaces between iterations, reducing the nested structure to two levels. Numerical experiments in signal deblurring and tomography demonstrate that CR-$\ell_q$-RMM-GKS produces significantly sharper edge reconstructions than standard non-cumulative methods. In particular, CR-$\ell_1$-RMM-GKS outperforms both standard $\ell_1$ methods and CR-$\ell_2$-RMM-GKS, indicating that cumulative weighting and $\ell_1$ penalties are highly complementary.

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arXiv2026

Efficient Sketching-Based Summation of Tucker Tensors

We present efficient, sketching-based methods for the summation of tensors in Tucker format. Leveraging the algebraic structure of Khatri-Rao and Kronecker products, our approach enables compressed arithmetic on Tucker tensors while controlling rank growth and computational cost. The proposed sketching framework avoids the explicit formation of large intermediate tensors, instead operating directly on the factor matrices and core tensors to produce accurate low-rank approximations of tensor sums. Furthermore, we analyze the computational complexity and the theoretical approximation properties of the proposed methodology. Numerical experiments demonstrate the effectiveness of our approach on four problems: two synthetic test cases, a parameter-dependent elliptic equation (commonly referred to as the cookie problem) solved via GMRES, and a one-dimensional linear transport problem discretized via high-order discontinuous Galerkin methods, where repeated tensor summation arises as a core computational bottleneck. Across these examples, the sketching-based summation achieves substantial computational savings while preserving high accuracy relative to direct summation and re-compression.

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arXiv2026

Hybrid ABBA-GMRES for Unmatched Backprojectors in Large Scale X-Ray Computerized Tomography

In large-scale X-ray computed tomography (CT), matrix-free iterative methods are essential due to the prohibitive cost of explicitly forming the system matrix. In practice, forward projectors and backprojectors are often implemented with different discretizations or accelerations, leading to unmatched projector pairs. This mismatch violates the adjointness assumptions underlying classical least-squares solvers, so the resulting iterations no longer correspond to a true least-squares problem and can exhibit non-symmetric or inconsistent behavior. Prior work has explored Krylov subspace solvers such as AB-GMRES and BA-GMRES to handle unmatched projector pairs, where these methods exhibit semi-convergent regularizing behavior. Under matched conditions, AB-GMRES and BA-GMRES reduce to LSQR and LSMR, respectively. However, in the presence of unmatched projectors, AB- and BA-GMRES have been observed to yield improved reconstruction quality compared to classical least-squares solvers. In this paper, we develop hybrid AB- and BA-GMRES methods that incorporate Tikhonov regularization directly into the Krylov subspace iterations. We also examine the relationship between the proposed methods and hybrid variants of LSQR and LSMR, considering both matched and unmatched backprojectors. We propose automatic strategies for selecting regularization parameters, including approaches based on the L-curve and generalized cross validation (GCV), and analyze their effect on convergence behavior and image quality. Numerical experiments on two-dimensional CT problems using GPU-accelerated projectors demonstrate that the proposed hybrid AB- and BA-GMRES methods mitigate semi-convergence, produce higher-quality reconstructions, and exhibit more stable stopping behavior than their non-hybrid counterparts.

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arXiv2026

A Scalable Sequential Framework for Dynamic Inverse Problems via Model Parameter Estimation

Large-scale dynamic inverse problems are often ill-posed due to model complexity and the high dimensionality of the unknown parameters. Regularization is commonly employed to mitigate ill-posedness by incorporating prior information and structural constraints. However, classical regularization formulations are frequently infeasible in this setting due to prohibitive memory requirements, necessitating sequential methods that process data and state information online, eliminating the need to form the full space-time problem. In this work, we propose a memory-efficient framework for reconstructing dynamic sequences of undersampled images from computerized tomography data that requires minimal hyperparameter tuning. The approach is based on a prior-informed, dimension-reduced Kalman filter with smoothing. While well suited for dynamic image reconstruction, practical deployment is challenging when the state transition model and covariance parameters must be initialized without prior knowledge and estimated in a single pass. To address these limitations, we integrate regularized motion models with expectation-maximization strategies for the estimation of state transition dynamics and error covariances within the Kalman filtering framework. We demonstrate the effectiveness of the proposed method through numerical experiments on limited-angle and single-shot computerized tomography problems, highlighting improvements in reconstruction accuracy, memory efficiency, and computational cost.

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arXiv2025

A Nodal Discontinuous Galerkin Method with Rank-Adaptive Velocity Space Representation for the Multiscale BGK Model

A novel hybrid algorithm is presented for the Boltzmann-BGK equation, in which a rank-adaptive decomposition is applied solely in the velocity subspace, while a full-rank representation is maintained in the physical (position) space. This approach establishes a foundation for extending modern rank-adaptive techniques to solve the Boltzmann equation in realistic settings, particularly where structured representations, such as conformal geometries, may not be feasible in practical engineering applications. A nodal discontinuous Galerkin method is employed for spatial discretization, coupled with a rank-adaptive decomposition over the velocity grid, as well as implicit-explicit Runge-Kutta methods for time integration. To handle the limit of vanishing collision time, a multiscale implicit integrator based on an auxiliary moment equation is utilized. The algorithm's order of accuracy, reduced computational complexity, and robustness are demonstrated on a suite of canonical gas kinetics problems with increasing complexity.

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arXiv2025

Priorconditioned Sparsity-Promoting Projection Methods for Deterministic and Bayesian Linear Inverse Problems

High-quality reconstructions of signals and images with sharp edges are needed in a wide range of applications. To overcome the large dimensionality of the parameter space and the complexity of the regularization functional, {sparisty-promoting} techniques for both deterministic and hierarchical Bayesian regularization rely on solving a sequence of high-dimensional iteratively reweighted least squares (IRLS) problems on a lower-dimensional subspace. Generalized Krylov subspace (GKS) methods are a particularly potent class of hybrid Krylov schemes that efficiently solve sequences of IRLS problems by projecting large-scale problems into a relatively small subspace and successively enlarging it. We refer to methods that promote sparsity and use GKS as S-GKS. A disadvantage of S-GKS methods is their slow convergence. In this work, we propose techniques that improve the convergence of S-GKS methods by combining them with priorconditioning, which we refer to as PS-GKS. Specifically, integrating the PS-GKS method into the IAS algorithm allows us to automatically select the shape/rate parameter of the involved generalized gamma hyper-prior, which is often fine-tuned otherwise. Furthermore, we proposed and investigated variations of the proposed PS-GKS method, including restarting and recycling (resPS-GKS and recPS-GKS). These respectively leverage restarted and recycled subspaces to overcome situations when memory limitations of storing the basis vectors are a concern. We provide a thorough theoretical analysis showing the benefits of priorconditioning for sparsity-promoting inverse problems. Numerical experiment are used to illustrate that the proposed PS-GKS method and its variants are competitive with or outperform other existing hybrid Krylov methods.

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arXiv2025

Efficient Dynamic Image Reconstruction with motion estimation

Dynamic inverse problems are challenging to solve due to the need to identify and incorporate appropriate regularization in both space and time. Moreover, the very large scale nature of such problems in practice presents an enormous computational challenge. In this work, in addition to the use of edge-enhancing regularization of spatial features, we propose a new regularization method that incorporates a temporal model that estimates the motion of objects in time. In particular, we consider the optical flow model that simultaneously estimates the motion and provides an approximation for the desired image, and we incorporate this information into the cost functional as an additional form of temporal regularization. We propose a computationally efficient algorithm to solve the jointly regularized problem that leverages a generalized Krylov subspace method. We illustrate the effectiveness of the prescribed approach on a wide range of numerical experiments, including limited angle and single-shot computerized tomography.

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arXiv2024

TRIPs-Py: Techniques for Regularization of Inverse Problems in Python

In this paper, we describe TRIPs-Py, a new Python package of linear discrete inverse problems solvers and test problems. The goal of the package is two-fold: 1) to provide tools for solving small and large-scale inverse problems, and 2) to introduce test problems arising from a wide range of applications. The solvers available in TRIPs-Py include direct regularization methods (such as truncated singular value decomposition and Tikhonov) and iterative regularization techniques (such as Krylov subspace methods and recent solvers for $\ell_p$-$\ell_q$ formulations, which enforce sparse or edge-preserving solutions and handle different noise types). All our solvers have built-in strategies to define the regularization parameter(s). Some of the test problems in TRIPs-Py arise from simulated image deblurring and computerized tomography, while other test problems model realistic problems in dynamic computerized tomography. Numerical examples are included to illustrate the usage as well as the performance of the described methods on the provided test problems. To the best of our knowledge, TRIPs-Py is the first Python software package of this kind, which may serve both research and didactical purposes.

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arXiv2024

Tensor Completion with BMD Factor Nuclear Norm Minimization

This paper is concerned with the problem of recovering third-order tensor data from limited samples. A recently proposed tensor decomposition (BMD) method has been shown to efficiently compress third-order spatiotemporal data. Using the BMD, we formulate a slicewise nuclear norm penalized algorithm to recover a third-order tensor from limited observed samples. We develop an efficient alternating direction method of multipliers (ADMM) scheme to solve the resulting minimization problem. Experimental results on real data show our method to give reconstruction comparable to those of HaLRTC (Liu et al., IEEE Trans Ptrn Anal Mchn Int, 2012), a well-known tensor completion method, in about the same number of iterations. However, our method has the advantage of smaller subproblems and higher parallelizability per iteration.

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arXiv2023

Recycling MMGKS for large-scale dynamic and streaming data

Reconstructing high-quality images with sharp edges requires the use of edge-preserving constraints in the regularized form of the inverse problem. The use of the $\ell_q$-norm on the gradient of the image is a common such constraint. For implementation purposes, the $\ell_q$-norm term is typically replaced with a sequence of $\ell_2$-norm weighted gradient terms with the weights determined from the current solution estimate. While (hybrid) Krylov subspace methods can be employed on this sequence, it would require generating a new Krylov subspace for every new two-norm regularized problem. The majorization-minimization Krylov subspace method (MM-GKS) addresses this disadvantage by combining norm reweighting with generalized Krylov subspaces (GKS). After projecting the problem using a small dimensional subspace - one that expands each iteration - the regularization parameter is selected. Basis expansion repeats until a sufficiently accurate solution is found. Unfortunately, for large-scale problems that require many expansion steps to converge, storage and the cost of repeated orthogonalizations presents overwhelming memory and computational requirements. In this paper we present a new method, recycled MM-GKS (RMM-GKS), that keeps the memory requirements bounded through recycling the solution subspace. Specifically, our method alternates between enlarging and compressing the GKS subspace, recycling directions that are deemed most important via one of our tailored compression routines. We further generalize the RMM-GKS approach to handle experiments where the data is either not all available simultaneously, or needs to be treated as such because of the extreme memory requirements. Numerical examples from dynamic photoacoustic tomography and streaming X-ray computerized tomography (CT) imaging are used to illustrate the effectiveness of the described methods.

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arXiv2023

Spatiotemporal Besov Priors for Bayesian Inverse Problems

Fast development in science and technology has driven the need for proper statistical tools to capture special data features such as abrupt changes or sharp contrast. Many inverse problems in data science require spatiotemporal solutions derived from a sequence of time-dependent objects with these spatial features, e.g., the dynamic reconstruction of computerized tomography (CT) images with edges. Conventional methods based on Gaussian processes (GP) often fall short in providing satisfactory solutions since they tend to offer oversmooth priors. Recently, the Besov process (BP), defined by wavelet expansions with random coefficients, has emerged as a more suitable prior for Bayesian inverse problems of this nature. While BP excels in handling spatial inhomogeneity, it does not automatically incorporate temporal correlation inherited in the dynamically changing objects. In this paper, we generalize BP to a novel spatiotemporal Besov process (STBP) by replacing the random coefficients in the series expansion with stochastic time functions as Q-exponential process (Q-EP) which governs the temporal correlation structure. We thoroughly investigate the mathematical and statistical properties of STBP. Simulations, two limited-angle CT reconstruction examples, a highly non-linear inverse problem involving Navier-Stokes equation, and a spatiotemporal temperature imputation problem are used to demonstrate the advantage of the proposed STBP compared with the classic STGP and a time-uncorrelated approach.

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arXiv2022

Bayesian Spatiotemporal Modeling for Inverse Problems

Inverse problems with spatiotemporal observations are ubiquitous in scientific studies and engineering applications. In these spatiotemporal inverse problems, observed multivariate time series are used to infer parameters of physical or biological interests. Traditional solutions for these problems often ignore the spatial or temporal correlations in the data (static model), or simply model the data summarized over time (time-averaged model). In either case, the data information that contains the spatiotemporal interactions is not fully utilized for parameter learning, which leads to insufficient modeling in these problems. In this paper, we apply Bayesian models based on spatiotemporal Gaussian processes (STGP) to the inverse problems with spatiotemporal data and show that the spatial and temporal information provides more effective parameter estimation and uncertainty quantification (UQ). We demonstrate the merit of Bayesian spatiotemporal modeling for inverse problems compared with traditional static and time-averaged approaches using a time-dependent advection-diffusion partial different equation (PDE) and three chaotic ordinary differential equations (ODE). We also provide theoretic justification for the superiority of spatiotemporal modeling to fit the trajectories even it appears cumbersome (e.g. for chaotic dynamics).

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arXiv2022

Optimal Transport for Parameter Identification of Chaotic Dynamics via Invariant Measures

We study an optimal transportation approach for recovering parameters in dynamical systems with a single smoothly varying attractor. We assume that the data is not sufficient for estimating time derivatives of state variables but enough to approximate the long-time behavior of the system through an approximation of its physical measure. Thus, we fit physical measures by taking the Wasserstein distance from optimal transportation as a misfit function between two probability distributions. In particular, we analyze the regularity of the resulting loss function for general transportation costs and derive gradient formulas. Physical measures are approximated as fixed points of suitable PDE-based Perron--Frobenius operators. Test cases discussed in the paper include common low-dimensional dynamical systems.

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arXiv2022

The Image Deblurring Problem: Matrices, Wavelets, and Multilevel Methods

The image deblurring problem consists of reconstructing images from blur and noise contaminated available data. In this AMS Notices article, we provide an overview of some well known numerical linear algebra techniques that are use for solving this problem. In particular, we start by carefully describing how to represent images, the process of blurring an image and modeling different kind of added noise. Then, we present regularization methods such as Tikhonov (on the standard and general form), Total Variation and other variations with sparse and edge preserving properties. Additionally, we briefly overview some of the main matrix structures for the blurring operator and finalize presenting multilevel methods that preserve such structures. Numerical examples are used to illustrate the techniques described.

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arXiv2021

Randomized algorithms for rounding in the Tensor-Train format

The Tensor-Train (TT) format is a highly compact low-rank representation for high-dimensional tensors. TT is particularly useful when representing approximations to the solutions of certain types of parametrized partial differential equations. For many of these problems, computing the solution explicitly would require an infeasible amount of memory and computational time. While the TT format makes these problems tractable, iterative techniques for solving the PDEs must be adapted to perform arithmetic while maintaining the implicit structure. The fundamental operation used to maintain feasible memory and computational time is called rounding, which truncates the internal ranks of a tensor already in TT format. We propose several randomized algorithms for this task that are generalizations of randomized low-rank matrix approximation algorithms and provide significant reduction in computation compared to deterministic TT-rounding algorithms. Randomization is particularly effective in the case of rounding a sum of TT-tensors (where we observe 20x speedup), which is the bottleneck computation in the adaptation of GMRES to vectors in TT format. We present the randomized algorithms and compare their empirical accuracy and computational time with deterministic alternatives.

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