On the regularization of singular systems of linear algebraic equations by shifts

Морозов, В.А.; Мухамадиев, Э. М.; Назимов, А.Б.

doi:10.1134/s1064562408020245

Cited by 5 publications

(4 citation statements)

References 1 publication

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“…Existen algunos puntos clave en el análisis del funcionamiento de la metodología, pero en este trabajo nos concentramos solamente en el estudio de la selección del valor de 𝜀 en la etapa de filtrado. En este sentido, en el trabajo original de Otero y Frontini (2022) se siguió el principio de discrepancia (Morozov, 1966), una técnica de regularización donde se considera 𝜀 = 𝜀 𝑀𝑀𝐹 como el error asociado al ruido en las mediciones 𝜎 𝑦 .…”

Section: Desarrollounclassified

Aprendizaje Basado en Problemas: Estimación Óptima para evaluar una técnica de inversión basada en Monte Carlo

Otero,

Managó

2024

Preprint

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In this work, we propose, design, and implement a solution to a problem of evaluating a numerical method. Thisproblem was conceived with the intention of applying the methodology of problem-based learning (PBL) inpostgraduate students of a course called "Applied Mathematics to Indirect Measurements". The main objective is forthe student to understand and relate several fundamental concepts that appear in the problem at hand, withoutadding additional complications in the development of the code. Therefore, a linear toy model with a single randomvariable is considered. The specific problem is the study of the performance of a sequential Monte Carlo (SMC)method, developed by the authors in the field of inverse problems, through the estimation of the method's tuningparameter. To do this, a supervised classification is proposed that use as training data those generated by the optimalestimation (OE) method. The idea of the work is that students, through the developed code, can analyze how thetopics of two very important areas of applied mathematics today, namely, inverse problems (IP) and machinelearning (ML), are integrated in practice through a Bayesian approach, such as the OE method.

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Section: Desarrollounclassified

Aprendizaje Basado en Problemas: Estimación Óptima para evaluar una técnica de inversión basada en Monte Carlo

Otero,

Managó

2024

Preprint

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“…The solution obtained by a Tikhonov-type method is therefore an approximate solution to the ill-posed problem, stable with respect to the data but dependent on the regularization parameter α used. There are several techniques for choosing the regularization parameter, such as the Morozov criterion [ 41 ], the cross-validation criterion [ 42 ] or the L-curve technique [ 43 ]. For low values of the hyperparameter α , the sensitivity to changes in admittivity is higher than for large values [ 39 ], but the regularizing effect is reduced.…”

Section: Tools For Robust Imagingmentioning

confidence: 99%

Robust imaging using electrical impedance tomography: review of current tools

2022

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Electrical impedance tomography (EIT) is a medical imaging technique with many advantages and great potential for development in the coming years. Currently, some limitations of EIT are related to the ill-posed nature of the problem. These limitations are translated on a practical level by a lack of genericity of the developed tools. In this paper, the main robust data acquisition and processing tools for EIT proposed in the scientific literature are presented. Their relevance and potential to improve the robustness of EIT are analysed, in order to conclude on the feasibility of a robust EIT tool capable of providing resistivity or difference of resistivity mapping in a wide range of applications. In particular, it is shown that certain measurement acquisition tools and algorithms, such as faulty electrode detection algorithm or particular electrode designs, can ensure the quality of the acquisition in many circumstances. Many algorithms, aiming at processing acquired data, are also described and allow to overcome certain difficulties such as an error in the knowledge of the position of the boundaries or the poor conditioning of the inverse problem. They have a strong potential to faithfully reconstruct a quality image in the presence of disturbances such as noise or boundary modelling error.

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“…Markov chain Monte Carlo, in which n must be large to be successful, n can be moderate or small for inverse problems, depending on the noise η in (2). In the following theorem we show this is possible via Morozov's discrepancy principle [68]. To avoid over-fitting the noise, from (1) one seeks a MAP point u such that |d j − w (x j ; u )| ≈ σ, i.e.…”

Section: Lemmamentioning

confidence: 99%

A data-scalable randomized misfit approach for solving large-scale PDE-constrained inverse problems

Myers

Bui-Thanh

et al. 2017

Inverse Problems

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Abstract. A randomized misfit approach is presented for the efficient solution of large-scale PDE-constrained inverse problems with high-dimensional data. The purpose of this paper is to offer a theory-based framework for random projections in this inverse problem setting. The stochastic approximation to the misfit is analyzed using random projection theory. By expanding beyond mean estimator convergence, a practical characterization of randomized misfit convergence can be achieved. The theoretical results developed hold with any valid random projection in the literature. The class of feasible distributions is broad yet simple to characterize compared to previous stochastic misfit methods. This class includes very sparse random projections which provide additional computational benefit. A different proof for a variant of the Johnson-Lindenstrauss lemma is also provided. This leads to a different intuition for the O(ε −2 ) factor in bounds for Johnson-Lindenstrauss results. The main contribution of this paper is a theoretical result showing the method guarantees a valid solution for small reduced misfit dimensions. The interplay between Johnson-Lindenstrauss theory and Morozov's discrepancy principle is shown to be essential to the result. The computational cost savings for large-scale PDE-constrained problems with highdimensional data is discussed. Numerical verification of the developed theory is presented for model problems of estimating a distributed parameter in an elliptic partial differential equation. Results with different random projections are presented to demonstrate the viability and accuracy of the proposed approach.

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On the regularization of singular systems of linear algebraic equations by shifts

Cited by 5 publications

References 1 publication

Aprendizaje Basado en Problemas: Estimación Óptima para evaluar una técnica de inversión basada en Monte Carlo

Aprendizaje Basado en Problemas: Estimación Óptima para evaluar una técnica de inversión basada en Monte Carlo

Robust imaging using electrical impedance tomography: review of current tools

A data-scalable randomized misfit approach for solving large-scale PDE-constrained inverse problems

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