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作者： Renjie Hu University of Iowa

学位级别：博士

This dissertation explores Random Neural Networks (RNNs) in several aspects and their applications. First, Novel RNNs have been proposed for dimensionality reduction and visualization. Based on Extreme Learning Machines (ELMs) and Self-Organizing Maps (SOMs) a new method is created to identify the important variables and visualize the data. This technique reduces the curse of dimensionality and improves furthermore the interpretability of the visualization and is tested on real nursing survey datasets. ELM-SOM+ is an autoencoder created to preserves the intrinsic quality of SOM and also brings continuity to the projection using two ELMs. This new methodology shows considerable improvement over SOM on real datasets. Second, as a Supervised Learning method, ELMs has been applied to the hierarchical multiscale method to bridge the the molecular dynamics to continua. The method is tested on simulation data and proven to be efficient for passing the information from one scale to another. Lastly, the regularization of ELMs has been studied and a new regularization algorithm for ELMs is created using a modified lanczos algorithm. The lanczos ELM on average divide computational time by 20 and reduce the Normalized MSE by 14% comparing with regular ELMs.

关键词： Data Visualization Dimensionality Reduction Feature Selection lanczos algorithm Random Neural Networks Regularization

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Solving topological lattice models on coprocessors

Solving topological lattice models on coprocessors

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作者： Topi Siro Aalto University

学位级别：博士

Understanding how the various electronic properties of matter emerge from the motion and interaction of electrons has been an important goal of physics since the early 1900s. One important tool has been the study of quantum lattice models, which can be considered as simpliﬁed depictions of solids. Provided that the system is small enough, it is possible to solve the low energy spectrum accurately with numerical methods. Like most problems in modern physics, studying lattice models requires extensive numerical computation. Traditionally, computer programs have been written to run on the central processing unit, but in recent years, various new parallel computing coprocessors have been introduced. Graphics processing units, which were originally added to render images on the computer screen, can now also be used for general purpose computation. Another new platform is the Xeon Phi coprocessor, speciﬁcally designed to accelerate parallel programs. Both of these coprocessors are parallel systems, where there are hundreds or thousands of computational threads running concurrently. This poses challenges in designing and implementing algorithms that beneﬁt from the parallelism. In this Thesis, we implement the exact diagonalization method on graphics processors and the Xeon Phi. We apply it on topological lattice models, which have been under intense study recently. They feature topological phases that cannot be explained with Landau's symmetry- breaking theory, but instead require studying the topological properties of the ground state. One key quantity in identifying the phases is the Chern number that is related to the transverse conductance in quantum Hall phases. In the so called checkerboard model, We show that the topological ground state can withstand strong local impurities. With increasing impurity density, we observe transitions to a metallic state and an insulating state. In another model, the Haldane-Hubbard model, we study the phase diagram with changing on-s

关键词： lattice model GPU CUDA Xeon Phi lanczos algorithm

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Rigorous Guarantees for Randomized Diagonalization algorithms

Rigorous Guarantees for Randomized Diagonalization Algorithm...

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作者： Garza-Vargas, Jorge University of California Berkeley

学位级别：Ph.D., Doctor of Philosophy

We rigorously analyze numerical methods for (approximately) computing the eigenvalues and eigenvectors of a matrix. Our main results are the following:• Spectral bisection. We show that a backward-stable solution to the eigenvalue problem can be obtained by a randomized algorithm in nearly matrix-multiplication time on any input. More specifically, we show that for every δ > 0, there is a randomized version of the spectral bisection algorithm that, on any input A ∈ C n×n , with probability 1 − O(1/n), finds matrices V, D ∈ C n×n with V invertible and D diagonal such that ∥A − V DV −1∥ ≤ δ∥A∥, using O(TMM(n)log2 (n/δ)) arithmetic operations, in finite arithmetic with O(log4 (n/δ)log n) bits of precision. Here TMM(n) is the number of arithmetic operations required to multiply two n × n complex matrices numerically stably, known to satisfy TMM(n) = O(n ω+η ) for every η > 0 where ω is the exponent of matrix multiplication.• The Hessenberg shifted QR algorithm. We introduce a randomized shifting strategy for the Hessenberg QR algorithm, for which we prove (in the finite precision arithmetic model) rapid global convergence. It follows from our results, that for any δ, ϕ > 0, a randomized version of the shifted Hessenberg QR algorithm can be used to compute, on any input A ∈ C n×n , with probability at least 1 − ϕ, matrices U, T ∈ C n×n with U unitary and T upper triangular, such that ∥A − UT U∗∥ ≤ δ∥A∥, using O(n3log(n/(δϕ))2 log log(n/(δϕ))) arithmetic operations performed using O(log2 (n/(δϕ))log log(n/(δϕ))) bits of precision. This provides the first rapidly globally convergent shifting strategy in the nonsymmetric case, which was an open problem even under the assumption of exact arithmetic.• The lanczos algorithm. We analyze the lanczos algorithm where the initial vector u is sampled uniformly at random from the unit sphere Sn−1 , and show that when run on a Hermitian input A ∈ Cn×n for c1 log n iterations (where c1 depends on some coarse global property of the spec

关键词： algorithmic analysis Diagonalization lanczos algorithm QR algorithm Spectral bisection algorithm

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Efficient Finite Element Electromagnetic Analysis for High-Frequency/High-Speed Circuits And Multiconductor Transmission Lines

Efficient Finite Element Electromagnetic Analysis for High-F...

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作者： Lee, Shih-hao University of Illinois – Urbana-Champaign

学位级别：doctor

This dissertation comprises the following four components. (1) Development of a robust and efficient 3-D finite element electromagnetic field solver with high-order vector elements for high-frequency and high-speed circuit simulations. The solver supports wave port and lumped port excitations as well as the incorporation of lumped networks and circuit models in a distributed finite element model. An adaptive multipoint model order reduction method is developed for fast broadband analysis. (2) Development of a fast and accurate multiconductor transmission line simulator and parameter extractor with improved model order reduction techniques. A methodology is further proposed for a combined quasi-TEM and full-wave transmission line analysis, which possesses their respective advantages and ensures full-wave accuracy from DC to very high frequencies. The transmission line analysis also takes into account the frequency dependence of dielectric materials. (3) Study of the low-frequency instability problem in the 3-D full-wave finite element simulation. The tree-cotree splitting is combined with several other techniques to improve the matrix conditioning and extend full-wave solutions down to very low frequencies for a more robust broadband characterization of high-speed digital circuits. (4) A combined domain decomposition–model order reduction (DD–MOR) method for efficient full-wave analysis of interconnections in multilayer printed circuit boards. The method not only brings a significant enhancement to computational efficiency while maintaining full-wave accuracy, but also provides great flexibility in the finite element mesh generation

关键词： electromagnetics microwave finite element method (FEM) nodal elements vector elements edge elements higher-order elements triangular elements tetrahedral elements finite element analysis full-wave analysis impedance boundary condition port boundary condition wave port lumped port deembedding thin-wire approximation thin wire modeling internal impedance lumped elements lumped circuits field-circuit simulation EM-circuit simulation stamping reduced-order modeling model order reduction (MOR) fast frequency sweep solution space projection (SSP) quasi-static analysis quasi-TEM analysis generalized eigenproblem lanczos algorithm modal analysis eigenanalysis frequency-dependent media anisotropic media Debye model multiconductor transmission lines transmission line parameters frequency-dependent losses conductor loss dielectric loss substrate loss characteristic impedance skin effect proximity effect parameter extraction resistance inductance capacitance conductance per-unit-length composite conductors waveguide filter coplanar waveguide (CPW) striplines microstrip lines high-frequency circuits RF circuits high-speed circuits tree-cotree splitting low-frequency breakdown low-frequency instability preconditioning bonding wire printed circuit board multilayer printed circuit board (PCB) via-holes signal integrity electromagnetic coupling interconnects domain decomposition Approximate Modal Interface (AMI) Approximate Modal Interface–Solution Space Projection (AMI–SSP) domain decomposition–model order reduction (DD–MOR) computer-aided design (CAD)

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Fast load-flow calculations based on spanning trees

Fast load-flow calculations based on spanning trees

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2023 IEEE PES Innovative Smart Grid Technologies Europe, ISGT EUROPE 2023

作者： Guichard, Pierrick Retière, Nicolas Mayou, Didier Univ. Grenoble Alpes CNRS Institut NEEL GrenobleF-38042 France Univ. Grenoble Alpes CNRS Grenoble INP G2Elab GrenobleF-38000 France

ISBN: (纸本)9798350396782

In this article a method is implemented for computing load flows that does not rely on the full resolution of a linear system. Instead, we solve very quickly many local load flows corresponding to dipoles (the electrical power is injected at two nodes) with an algorithm based on a Hamiltonian formalism, as it is usually done in quantum physics. The computational burden is quadratic in the size of the network when the dipoles are defined by the lines of a spanning tree of the network. This leads to a considerable speedup when compared to traditional methods relying on the full system inversion as well as methods involving other subproblems. © 2023 European Union.

关键词： Hamiltonian formalism lanczos algorithm Load flow Local approach Spanning trees

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PairDiag: An exact diagonalization program for solving general pairing Hamiltonians

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COMPUTER PHYSICS COMMUNICATIONS 2021年 259卷 107349-107349页

作者： Liu, Xiao-Yu Qi, Chong Royal Inst Technol Dept Phys S-10405 Stockholm Sweden Chinese Acad Sci Inst Modern Phys Lanzhou 730000 Peoples R China Univ Chinese Acad Sci Beijing 100049 Peoples R China

We present a program for solving exactly the general pairing Hamiltonian based on diagonalization. The program generates the seniority-zero shell-model-like basis vectors via the '01' inversion algorithm. The Hamiltonian matrix is constructed in this seniority-zero space. The program evaluates all non-zero elements of the Hamiltonian matrix "on the fly" using the scattering operator and a search algorithm. The matrix is diagonalized by using the iterative lanczos algorithm. The OpenMP parallel program thus developed, PairDiag, can efficiently calculate the ground-state eigenvalue and eigenvector of the general pairing Hamiltonian for both the even-mass and the odd-mass system. The program is packaged in a Fortran module, which makes it easy to use the program to replace the BCS approximation in standard self-consistent mean field calculations. For systems with dimension around 10(8), the calculation can be done within hours on standard desktop computers. (C) 2020 The Authors. Published by Elsevier B.V.

关键词： General pairing Hamiltonian Diagonalization lanczos algorithm

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Computational study of the rovibrational spectrum of H2O-HF

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JOURNAL OF MOLECULAR SPECTROSCOPY 2022年 384卷 111587-111587页

作者： Viglaskan, Dominika Wang, Xiao-Gang Carrington, Tucker, Jr. Tew, David P. UFR Sci UMR CNRS 7331 Grp Spectrometrie Mol & Atmospher BP 1039 F-51687 Reims 2 France Queens Univ Chem Dept Kingston ON K7L 3N6 Canada Univ Oxford Phys & Theoret Chem Lab South Parks Rd Oxford OX1 3QZ England

In this paper we report rovibrational energy levels, transition frequencies, and intensities computed for H2O-HF using a new ab initio potential energy surface and compare with available experimental data. We use the rigid monomer approximation. A G(4) symmetry-adapted lanczos algorithm and an uncoupled product basis are employed. The rovibrational levels are computed up to J = 4. The new analytic 9-D potential is fit to 39 771 counterpoise corrected CCSD(T)(F12*)/aug-cc-pVTZ energies and reduces to the sum of uncoupled H2O and HF potentials in the dissociation limit. On the new potential better agreement with experiment is obtained by re-assigning the R(1) transitions of two vibrational states.

关键词： Van der Waals molecule lanczos algorithm Ro-vibrational spectroscopy

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Application of a Krylov subspace method for an efficient solution of acoustic transfer functions

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2021年 148卷 107135-107135页

作者： Sittl, Christopher Marburg, Steffen Wagner, Marcus OTH Regensburg Lab Finite Element Method Fac Mech Engn D-93053 Regensburg Germany Tech Univ Munich Dept Mech Engn Chair Vibroacoust Vehicles & Machines D-85748 Garching Germany

Solving acoustic radiation problems, arising from systems including fluid-structure interaction, is of interest in many engineering applications. Computing frequency response functions over a large frequency range is a concern in such applications. A method which solves the Helmholtz equation for multiple frequencies in one step is the matrix-Pade-via-lanczos connection for unsymmetric systems, as presented by Wagner et al. [1]. The present work is based on Ref. [1] and presents a method for efficiently computing frequency responses over a frequency range for coupled structural-acoustic problems, where the structure and the acoustic near field are discretized with finite elements and an analytical Dirichlet-to-Neumann map approximates the far field. The method is based on a Krylov-subspace projection technique which derives a matrix-valued Pade approximation for a restricted area in the near field and the pressure field on a spherical boundary. On the spherical boundary, where the finite domain is truncated, the non-local modified Dirichlet-to-Neumann operator is applied as a low-rank update matrix. The present contribution extends this method and incorporates new techniques for a more stable model reduction through the lanczos algorithm and a novel weighted adaptive windowing technique. Further, structural damping is incorporated, for computing the acoustic radiation of a harmonically excited plate. These computed results are compared with acoustic measurements in an anechoic chamber and verified with computational results obtained with a commercial code that uses the perfectly matched layer method. (C) 2020 Elsevier Ltd. All rights reserved.

关键词： lanczos algorithm Krylov-subspace projection Dirichlet-to-Neumann map Pade approximation Fluid-structure interaction Acoustics

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EIT Complex Admitance Reconstruction through Intervalar Evaluation

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IFAC-PapersOnLine 2024年第24期58卷 157-162页

作者： Thiago C. Martins Marcos S.G. Tsuzuki Escola Politécnica da Universidade de São Paulo Computational Geometry Laboratory Brazil

Electrical Impedance Tomography (EIT) stands out as a widely utilized imaging method. It becomes crucial to recover the intricate admittance in EIT for various purposes, particularly when the interest extends beyond mere visualization of fixed conductivity (or resistivity) patterns within a region. This study addresses a complex EIT inverse problem, which involves reconstructing the complex distribution of admittance based on surface potential measurements and corresponding injected currents. A Bayesian approach leads to Maximum a Posteriori Reconstructions using non-convex optimization. This paper adapts non-convex optimization approaches to EIT to handle the non-Hermitian FEM models that arise from complex admittance. Finally, a synthetic phantom experiment was conducted, demonstrating the successful reconstruction of the overall shape of the admittance distribution.

关键词： Electrical Impedance Tomography Inverse Problems Simulated Annealing lanczos algorithm Preconditioning

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Delocalization for the 3D discrete random Schrodinger operator at weak disorder

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2014年第30期47卷 305202-305202页

作者： Liaw, Constanze King, Westin Kirby, Robert C. Baylor Univ Dept Math Waco TX 76798 USA

We apply a recently developed approach (Liaw 2013 J. Stat. Phys. 153 1022-38) to study the existence of extended states for the three-dimensional discrete random Schrodinger operator at small disorder. The conclusion of delocalization at small disorder agrees with other numerical and experimental observations (see e.g. (Lagendijk et al 2009 Phys. Today 82 24-29)). Further the work furnishes a verification of the numerical approach and its implementation. Not being based on scaling theory, this method eliminates problems due to boundary conditions, common to previous numerical methods in the field. At the same time, as with any numerical experiment, one cannot exclude finite-size effects with complete certainty. Our work can be thought of as a new and quite different use of lanczos' algorithm;a posteriori tests how that the orthogonality loss is very small. We numerically track the 'bulk distribution' (here: the distribution of where we most likely find an electron) of a wave packet initially located at the origin, after iterative application of the discrete random Schrodinger operator.

关键词： 3D Anderson model extended states lanczos algorithm

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