Regularization parameter determination for discrete ill-posed problems

Straightforward solution of discrete ill-posed linear systems of equations or least-squares problems with error contaminated data does not, in general, give meaningful results, because the propagated error destroys the computed solution. The problems have to be modified to reduce their sensitivity to the error in the data. The amount of modification is determined by a regularization parameter. It can be difficult to determine a suitable value of the regularization parameter when no knowledge of the norm of error in the data is available. This paper proposes a new simple technique for determining a value of the regularization parameter that can be applied in this situation. It is based on comparing computed solutions determined by Tikhonov regularization and truncated singular value decomposition. Analogous comparisons are proposed for large-scale problems. The technique for determining the regularization parameter implicity provides an estimate for the norm of the error in the data. Keywords: Ill-posed problem; Regularization; Noise level estimation; TSVD; Tikhonov regularization; Heuristic parameter choice rul

Multidirectional Subspace Expansion for One-Parameter and Multiparameter Tikhonov Regularization

Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results

The Euro Diffusion Project

From 1st January 2002 we have the unique possibility to follow the spread of national euro coins over the different European countries. We model and analyse this movement and estimate the time it will take before on average half the coins in our wallet will be foreign

The UPS Prototype: An Experimental End-User Service Across E-Print Archives

A meeting was held in Santa Fe, New Mexico, October 21-22, 1999, to generate discussion and consensus about interoperability of publicly available scholarly information archives. The invitees represented several well known e-print and report archive initiatives, as well as organizations with interests in digital libraries and the transformation of scholarly communication. The central goal of the meeting was to agree on recommendations that would make the creation of end-user services -- such as scientific search engines and linking systems -- for data originating from distributed and dissimilar archives easier. The Universal Preprint Service (UPS) Prototype was developed in preparation for this meeting. As a proof-of-concept of a multi-discipline digital library of publicly available scholarly material, the Prototype harvested nearly 200,000 records from several different archives and created an attractive end-user environment. This paper describes the results of the project. This is done in two ways. On the one hand, the experimental end-user service that was created during the project is illustrated. On the other hand, the lessons that the project team drew from the experience of creating the Prototype are presented

On the Convergence of Ritz Pairs and Refined Ritz Vectors for Quadratic Eigenvalue Problems

For a given subspace, the Rayleigh-Ritz method projects the large quadratic eigenvalue problem (QEP) onto it and produces a small sized dense QEP. Similar to the Rayleigh-Ritz method for the linear eigenvalue problem, the Rayleigh-Ritz method defines the Ritz values and the Ritz vectors of the QEP with respect to the projection subspace. We analyze the convergence of the method when the angle between the subspace and the desired eigenvector converges to zero. We prove that there is a Ritz value that converges to the desired eigenvalue unconditionally but the Ritz vector converges conditionally and may fail to converge. To remedy the drawback of possible non-convergence of the Ritz vector, we propose a refined Ritz vector that is mathematically different from the Ritz vector and is proved to converge unconditionally. We construct examples to illustrate our theory.Comment: 20 page

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.This work was supported by the Spanish Ministerio de Ciencia e Innovacion under project TIN2009-07519. Author's addresses: E. Romero, Institut I3M, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain), and J. E. Roman, Departament de Sistemes Informatics i Computacio, Universitat Politecnica de Valencia, Cami de Vera s/n, 46022 Valencia, Spain; email: [email protected] Alcalde, E.; Román Moltó, JE. (2014). A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc. ACM Transactions on Mathematical Software. 40(2):13:01-13:29. https://doi.org/10.1145/2543696S13:0113:29402P. Arbenz, M. Becka, R. Geus, U. Hetmaniuk, and T. Mengotti. 2006. On a parallel multilevel preconditioned Maxwell eigensolver. Parallel Comput. 32, 2, 157--165.Z. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst, Eds. 2000. 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Prediction of setup times for an advanced upper limb functional electrical stimulation system

Introduction: Rehabilitation devices take time to don, and longer or unpredictable setup time impacts on usage. This paper reports on the development of a model to predict setup time for upper limb functional electrical stimulation. Methods: Participants’ level of impairment (Fugl Meyer-Upper Extremity Scale), function (Action Research Arm Test) and mental status (Mini Mental Scale) were measured. Setup times for each stage of the setup process and total setup times were recorded. A predictive model of setup time was devised using upper limb impairment and task complexity. Results: Six participants with stroke were recruited, mean age 60 (�17) years and mean time since stroke 9.8 (�9.6) years. Mean Fugl Meyer-Upper Extremity score was 31.1 (�6), Action Research Arm Test 10.4 (�7.9) and Mini Mental Scale 26.1 (�2.7). Linear regression analysis showed that upper limb impairment and task complexity most effectively predicted setup time (51% as compared with 39%) (F(2,21) ¼ 12.782, adjusted R2 ¼ 0.506; p<.05). Conclusions: A model to predict setup time based on upper limb impairment and task complexity accounted for 51% of the variation in setup time. Further studies are required to test the model in real-world settings and to identify other contributing factors

Fast Approximated POD for a Flat Plate Benchmark with a Time Varying Angle of Attack

An approximate POD algorithm provides an empirical Galerkin approximation with guaranteed a priori lower bound on the required resolution. The snapshot ensemble is partitioned into several sub-ensembles. Cross correlations between these sub-ensembles are approximated in terms of a far smaller correlation matrix. Computational speedup is nearly linear in the number of partitions, up to a saturation that can be estimated a priori. The algorithm is particularly suitable for analyzing long transient trajectories of high dimensional simulations, but can be applied also for spatial partitioning and parallel processing of very high spatial dimension data. The algorithm is demonstrated using transient data from two simulations. First, a two dimensional simulation of the flow over a flat plate, as it transitions from AOA = 30° to a horizontal position and back. Second, a three dimensional simulation of a flat plate with aspect ratio two as it transitions from a horizontal position to AOA = 30°