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A decision-theoretic approach to evaluate radiation detection algorithms

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A decision-theoretic approach to evaluate radiation detectio...

IEEE Conference on Technologies for Homeland Security

作者： Mallory A. Nobles Landon H. Sego Scott K. Cooley Luke J. Gosink Richard M. Anderson Spencer E. Hays Mark F. Tardiff Applied Statistics and Computational Modeling Group Pacific Northwest National Laboratory Richland WA Puget Sound Institute University of Washington Tacoma Tacoma WA Department of Statistical Sciences and Operations Research Virginia Commonwealth University Richmond VA

There are a variety of sensor systems deployed at border crossings and ports of entry throughout the world that scan for illicit nuclear material. These systems employ detection algorithms that interpret the output of the scans and determine whether additional investigation is warranted. In this work, we demonstrate an approach for comparing the performance of such detection algorithms. We optimize each algorithm by minimizing risk, which considers the probability distribution of threat sources and the consequence of detection errors. Our method is flexible and is easily adapted to many different assumptions regarding the probability of a conveyance containing illicit material and the relative consequences of false positive and false negative errors. This approach can help developers and decision makers identify optimal settings for these algorithms. We illustrate the method by comparing the risk from two families of detection algorithms and discuss the generalizability of the method.

关键词： Algorithm design and analysis Prediction algorithms Materials Radiation detectors Detection algorithms Sensitivity Educational institutions

Riccati conditioning and sensitivity for a MinMax controlled cable-mass system

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Riccati conditioning and sensitivity for a MinMax controlled...

IEEE Conference on Decision and Control

作者： Lizette Zietsman Katie A. Evans J. Teye Brown Richard A. Idowu Faculty of the Interdisciplinary Center for Applied Mathematics Virginia Technology Blacksburg VA USA Faculty of Mathematics and Statistics Louisiana Tech University Ruston LA USA Computational Analysis and Modeling Program Louisiana Tech University Ruston LA USA

ISBN: (纸本)9781424431236

In this paper we present a numerical study that investigates the relationship between the parameter q, used in the design of the MinMax controller, and the conditioning of the approximate algebraic Riccati equations, the sensitivity of the eigenvalues of I-¿ 2 P¿ to ¿ as well as the effect of q on the stability radia and the stability margin of the system. In order to guarantee accurate numerical solutions to the approximate Riccati equations, the Riccati equations must remain well-conditioned for the values of ¿ that are considered. This condition number reflects the combined sensitivity of the Riccati equations to the system inputs A, B, R, C and ¿. In addition, we also consider the sensitivity of the eigenvalues of I-¿ 2 P¿ to ¿. We study the possibility of these sensitivities serving as an indication of the largest value of ¿ for which I-¿ 2 P¿ remains positive definite. This sensitivity could also serve as an indication of the accuracy of the computation of I-¿ 2 P¿. Lastly, in order to design efficient low order controllers, it is important to ensure the robustness of the design. Stability radius and stability margin serve as measures of the robustness of the controller. A one-dimensional nonlinear cable mass system is considered to illustrate these ideas and numerical results are presented.

关键词： Riccati equations Minimax techniques Control systems Eigenvalues and eigenfunctions Robust control Robust stability Optimal control Contracts Mathematics Military computing

Sampled Tikhonov regularization for large linear inverse problems

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arXiv 2018年

作者： Tanner Slagel, J. Chung, Julianne Chung, Matthias Kozak, David Tenorio, Luis Department of Mathematics Virginia Tech BlacksburgVA United States Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States

In this paper, we investigate iterative methods that are based on sampling of the data for computing Tikhonov-regularized solutions. We focus on very large inverse problems where access to the entire data set is not possible all at once (e.g., for problems with streaming or massive datasets). Row-access methods provide an ideal framework for solving such problems, since they only require access to "blocks" of the data at any given time. However, when using these iterative sampling methods to solve inverse problems, the main challenges include a proper choice of the regularization parameter, appropriate sampling strategies, and a convergence analysis. To address these challenges, we first describe a family of sampled iterative methods that can incorporate data as they become available (e.g., randomly sampled). We consider two sampled iterative methods, where the iterates can be characterized as solutions to a sequence of approximate Tikhonov problems. The first method requires the regularization parameter to be fixed a priori and converges asymptotically to an unregularized solution for randomly sampled data. This is undesirable for inverse problems. Thus, we focus on the second method where the main benefits are that the regularization parameter can be updated during the iterative process and the iterates converge asymptotically to a Tikhonov-regularized solution. We describe adaptive approaches to update the regularization parameter that are based on sampled residuals, and we describe a limited-memory variant for larger problems. Numerical examples, including a large-scale super-resolution imaging example, demonstrate the potential for these *** Codes 65F22, 65F10, 15A29 Copyright © 2018, The Authors. All rights reserved.

关键词： Iterative methods

Sampled limited memory methods for massive linear inverse problems

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arXiv 2019年

作者： Chung, Julianne Chung, Matthias Tanner Slagel, J. Tenorio, Luis Department of Mathematics Computational Modeling and Data Analytics Division Academy of Integrated Science Virginia Tech BlacksburgVA United States Department of Mathematics Virginia Tech BlacksburgVA United States Department of Applied Mathematics and Statistics Colorado School of Mines GoldenCO United States

In many modern imaging applications the desire to reconstruct high resolution images, coupled with the abundance of data from acquisition using ultra-fast detectors, have led to new challenges in image reconstruction. A main challenge is that the resulting linear inverse problems are massive. The size of the forward model matrix exceeds the storage capabilities of computer memory, or the observational dataset is enormous and not available all at once. Row-action methods that iterate over samples of rows can be used to approximate the solution while avoiding memory and data availability constraints. However, their overall convergence can be slow. In this paper, we introduce a sampled limited memory row-action method for linear least squares problems, where an approximation of the global curvature of the underlying least squares problem is used to speed up the initial convergence and to improve the accuracy of iterates. We show that this limited memory method is a generalization of the damped block Kaczmarz method, and we prove linear convergence of the expectation of the iterates and of the error norm up to a convergence horizon. Numerical experiments demonstrate the benefits of these sampled limited memory row-action methods for massive 2D and 3D inverse problems in tomography applications. Copyright © 2019, The Authors. All rights reserved.

关键词： Tomography

POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS I: LOW DIMENSIONAL CASE

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arXiv 2024年

作者： Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, for any odd n and any integer m ≥ 1 with n tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ , x ∈ n, where V is a real-valued potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m + V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying (Equation presented), where the constants C, h > 0, and h can be specified by m, n and the zero energy resonances of H. A similar result for smoothing operators like (Equation presented) is also given. One of the main novelties lies in a new and unified approach to obtain the asymptotic expansions of perturbed resolvents under various types of zero energy resonances. © 2024, CC0.

关键词： Schrodinger equation

POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS II: HIGH DIMENSIONAL CASE

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arXiv 2024年

In this paper, for any odd n and any integer m ≥ 1 with n > 4m, we study the fundamental solution of the higher order Schrödinger equation i∂tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ R, x ∈ Rn, where V is a real-valued C n+21 −2m potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m+V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying n(m−1) |K(t, x, y)| ≤ C(1 + |t|)−(2nm −σ)(1 + |t|− 2nm ) (1 + |t|− 21m |x − y|)− 2m−1 , t , 0, x, y ∈ Rn, where σ = 2 if 0 is an eigenvalue of H, and σ = 0 otherwise. A similar result for smoothing operators H 2αm e−itHPac(H) is also given. The regularity condition V ∈ C n+21 −2m is optimal in the second order case, and it also seems optimal when m > 1. © 2024, CC0.

关键词： Schrodinger equation

Explicit representation for a class of Type 2 constacyclic codes over the ring F2m[u]/(u2λ) with even length

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arXiv 2019年

作者： Cao, Yuan Cao, Yonglin Dinh, Hai Q. Sriboonchitta, Songsak Wang, Guidong School of Mathematics and Statistics Shandong University of Technology Zibo Shandong255091 China Hubei Key Laboratory of Applied Mathematics Faculty of Mathematics and Statistics Hubei University Wuhan430062 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and Technology Changsha Hunan410114 China Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc Thang University Ho Chi Minh City Viet Nam Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City Viet Nam Faculty of Economics Chiang Mai University Chiang Mai 52000 Thailand

Let F2m be a finite field of cardinality 2m, λ and k be integers satisfying λ, k ≥ 2 and denote R = F2m[u]/hu2λ. Let δ, α ∞ F2m. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ+αu2)-constacyclic codes over R of length 2kn, and provide a clear formula to count the number of all these codes. As a corollary, we conclude that every (δ + αu2)-constacyclic code over R of length 2kn is an ideal generated by at most 2 polynomials in the residue class ring R[x]/(x2kn-(δ + αu2). Copyright © 2019, The Authors. All rights reserved.

关键词： Codes (symbols)

Construction and enumeration for self-dual cyclic codes of even length over F2m + uF2m

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arXiv 2019年

作者： Caoa, Yuan Cao, Yonglin Dinh, Hai Q. Fu, Fang-Wei Ma, Fanghui School of Mathematics and Statistics Shandong University of Technology Zibo Shandong255091 China Hubei Key Laboratory of Applied Mathematics Faculty of Mathematics and Statistics Hubei University Wuhan430062 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and Technology Changsha Hunan410114 China Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc Thang University Ho Chi Minh City Viet Nam Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City Viet Nam Chern Institute of Mathematics and LPMC Nankai University Tianjin300071 China

Let F2m be a finite field of cardinality 2m, R = F2m + uF2m (u2 = 0) and s, n be positive integers such that n is odd. In this paper, we give an explicit representation for every self-dual cyclic code over the finite chain ring R of length 2sn and provide a calculation method to obtain all distinct codes. Moreover, we obtain a clear formula to count the number of all these self-dual cyclic codes. As an application, self-dual and 2-quasi-cyclic codes over F2m of length 2s+1n can be obtained from self-dual cyclic code over R of length 2sn and by a Gray map preserving orthogonality and distances from R onto F22m. Copyright © 2019, The Authors. All rights reserved.

关键词：

An efficient method to construct self-dual cyclic codes of length ps over Fpm + uFpm

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arXiv 2019年

作者： Cao, Yuan Cao, Yonglin Dinh, Hai Q. Jitman, Somphong School of Mathematics and Statistics Shandong University of Technology Zibo Shandong255091 China Hubei Key Laboratory of Applied Mathematics Faculty of Mathematics and Statistics Hubei University Wuhan430062 China Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering Changsha University of Science and Technology Changsha Hunan410114 China Division of Computational Mathematics and Engineering Institute for Computational Science Ton Duc Thang University Ho Chi Minh City Viet Nam Faculty of Mathematics and Statistics Ton Duc Thang University Ho Chi Minh City Viet Nam Department of Mathematics Faculty of Science Silpakorn University Nakhon Pathom73000 Thailand

Let p be an odd prime number, Fpm be a finite field of cardinality pm and s a positive integer. Using some combinatorial identities, we obtain certain properties for Kronecker product of matrices over Fp with a specific type. On that basis, we give an explicit representation and enumeration for all distinct self-dual cyclic codes of length ps over the finite chain ring Fpm+uFpm (u2 = 0). Moreover, We provide an efficient method to construct every self-dual cyclic code of length ps over Fpm + uFpm precisely. Copyright © 2019, The Authors. All rights reserved.

关键词： Matrix algebra