检索结果-内蒙古大学图书馆

Multi-Frequency Progressive Refinement for Learned Inverse Scattering

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Melia, Owen Tsang, Olivia Charisopoulos, Vasileios Khoo, Yuehaw Hoskins, Jeremy Willett, Rebecca Department of Computer Science University of Chicago United States Data Science Institute University of Chicago United States Computational and Applied Mathematics Department of Statistics University of Chicago United States

Interpreting scattered acoustic and electromagnetic wave patterns is a computational task that enables remote imaging in a number of important applications, including medical imaging, geophysical exploration, sonar and radar detection, and nondestructive testing of materials. However, accurately and stably recovering an inhomogeneous medium from far-field scattered wave measurements is a computationally difficult problem, due to the nonlinear and non-local nature of the forward scattering process. We design a neural network, called Multi-Frequency Inverse Scattering Network (MFISNet), and a training method to approximate the inverse map from far-field scattered wave measurements at multiple frequencies. We consider three variants of MFISNet, with the strongest performing variant inspired by the recursive linearization method - a commonly used technique for stably inverting scattered wavefield data - that progressively refines the estimate with higher frequency content. MFISNet outperforms past methods in regimes with high-contrast, heterogeneous large objects, and inhomogeneous unknown backgrounds. Copyright © 2024, The Authors. All rights reserved.

关键词： Forward scattering

Certified Evaluations of Hölder Continuous Functions at Roots of Polynomials 4th

学校读者我要写书评

暂无评论

Certified Evaluations of Hölder Continuous Functions at Roo...

4th Maple Conference, MC 2020

作者： Edwards, Parker B. Hauenstein, Jonathan D. Smyth, Clifford D. Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Department of Mathematics and Statistics University of North Carolina at Greensboro GreensboroNC27402 United States

ISBN: (纸本)9783030816971

Various methods can obtain certified estimates for roots of polynomials. Many applications in science and engineering additionally utilize the value of functions evaluated at roots. For example, critical values are obtained by evaluating an objective function at critical points. For analytic evaluation functions, Newton’s method naturally applies to yield certified estimates. These estimates no longer apply, however, for Hölder continuous functions, which are a generalization of Lipschitz continuous functions where continuous derivatives need not exist. This work develops and analyzes an alternative approach for certified estimates of evaluating locally Hölder continuous functions at roots of polynomials. An implementation of the method in Maple demonstrates efficacy and efficiency. © 2021, Springer Nature Switzerland AG.

关键词： Function evaluation

Comparing the p-independence number of regular graphs to the q-independence number of their line graphs

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Caro, Yair Davila, Randy Pepper, Ryan Department of Mathematics University of Haifa-Oranim Tivon36006 Israel Department of Computational Applied Mathematics & Operations Research Rice University HoustonTX77005 United States Department of Mathematics and Statistics University of Houston Downtown HoustonTX77002 United States

Let G be a simple graph and let L(G) denote the line graph of G. A p-independent set in G is a set of vertices S ⊆ V (G) such that the subgraph induced by S has maximum degree at most p. The p-independence number of G, denoted by αp(G), is the cardinality of a maximum p-independent set in G. In this paper, and motivated by the recent result that independence number is at most matching number for regular graphs [5], we investigate which values of the non-negative integers p, q, and r have the property that αp(G) ≤ αq(L(G)) for all r-regular graphs. Triples (p, q, r) having this property are called valid α-triples. Among the results we prove are: • (p, q, r) is valid α-triple for p ≥ 0, q ≥ 3, and r ≥ 2. • (p, q, r) is valid α-triple for p ≤ q 17(16p+1) o . We also show a close relation between undetermined possible valid α-triples, the Linear Aboricity Conjecture, and the Path-Cover Conjecture. © 2024, CC0.

关键词： Graph theory

Sparse grid implementation of a fixed-point fast sweeping WENO scheme for Eikonal equations

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Miksis, Zachary M. Zhang, Yong-Tao Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre DameIN46556 United States

Fixed-point fast sweeping methods are a class of explicit iterative methods developed in the literature to efficiently solve steady state solutions of hyperbolic partial differential equations (PDEs). As other types of fast sweeping schemes, fixed-point fast sweeping methods use the Gauss-Seidel iterations and alternating sweeping strategy to cover characteristics of hyperbolic PDEs in a certain direction simultaneously in each sweeping order. The resulting iterative schemes have fast convergence rate to steady state solutions. Moreover, an advantage of fixed-point fast sweeping methods over other types of fast sweeping methods is that they are explicit and do not involve inverse operation of any nonlinear local system. Hence they are robust and flexible, and have been combined with high order accurate weighted essentially non-oscillatory (WENO) schemes to solve various hyperbolic PDEs in the literature. For multidimensional nonlinear problems, high order fixed-point fast sweeping WENO methods still require quite large amount of computational costs. In this technical note, we apply sparse-grid techniques, an effective approximation tool for multidimensional problems, to fixed-point fast sweeping WENO method for reducing its computational costs. Here we focus on a robust Runge-Kutta (RK) type fixed-point fast sweeping WENO scheme with third order accuracy (Zhang et al. 2006 [33]), for solving Eikonal equations, an important class of static Hamilton-Jacobi (H-J) equations. Numerical experiments on solving multidimensional Eikonal equations and a more general static H-J equation are performed to show that the sparse grid computations of the fixed-point fast sweeping WENO scheme achieve large savings of CPU times on refined meshes, and at the same time maintain comparable accuracy and resolution with those on corresponding regular single *** Codes 65N06 © 2022, CC BY.

关键词： Differential equations

A theory of non-linear feature learning with one gradient step in two-layer neural networks 24

学校读者我要写书评

暂无评论

A theory of non-linear feature learning with one gradient st...

Proceedings of the 41st International Conference on Machine Learning

作者： Behrad Moniri Donghwan Lee Hamed Hassani Edgar Dobriban Department of Electrical and Systems Engineering University of Pennsylvania PA Graduate Group in Applied Mathematics and Computational Science University of Pennsylvania PA Department of Statistics and Data Science University of Pennsylvania PA

Feature learning is thought to be one of the fundamental reasons for the success of deep neural networks. It is rigorously known that in two-layer fully-connected neural networks under certain conditions, one step of gradient descent on the first layer can lead to feature learning; characterized by the appearance of a separated rank-one component--spike--in the spectrum of the feature matrix. However, with a constant gradient descent step size, this spike only carries information from the linear component of the target function and therefore learning non-linear components is impossible. We show that with a learning rate that grows with the sample size, such training in fact introduces multiple rank-one components, each corresponding to a specific polynomial feature. We further prove that the limiting largedimensional and large sample training and test errors of the updated neural networks are fully characterized by these spikes. By precisely analyzing the improvement in the training and test errors, we demonstrate that these non-linear features can enhance learning.

关键词：

Comparing the $P$-Independence Number of Regular Graphs to the $Q$-Independence Number of Their Line Graphs

学校读者我要写书评

暂无评论

SSRN

SSRN 2024年

作者： Davila, Randy Caro, Yair Pepper, Ryan Department of Computational Applied Mathematics & Operations Research Rice University HoustonTX77005 United States Department of Mathematics University of Haifa Oranim Tivon 36006 Israel Department of Mathematics and Statistics University of Houston–Downtown HoustonTX77002 United States

Let $G$ be a simple graph, and let $p$, $q$, and $r$ be non-negative integers. A \emph{$p$-independent} set in $G$ is a set of vertices $S \subseteq V(G)$ such that the subgraph induced by $S$ has maximum degree at most $p$. The \emph{$p$-independence number} of $G$, denoted by $\alpha_p(G)$, is the cardinality of a maximum $p$-independent set in $G$. Letting $L(G)$ denote the line graph of $G$. In this paper, motivated by the result $\alpha_0(G) \leq \alpha_0(L(G))$, for every $r$-regular graph with $r \geq 1$ (see~\cite{CaDaPe2020}), we investigate which values of $p$, $q$, and $r$ have the property that $\alpha_p(G) \leq \alpha_q(L(G))$ for all $r$-regular graphs $G$. Triples $(p, q, r)$ having this property are called \emph{valid $\alpha$-triples}. Among the results we prove are :\begin{itemize} \item $(p, q, r)$ is valid $\alpha$-triple for $p \geq 0$, $q \geq 3$ , and $r\geq 2$. \item $(p, q, r)$ is valid $\alpha$-triple for $p \leq q © 2024, The Authors. All rights reserved.

关键词： Graph theory

OPTIMAL SOLUTIONS OF WELL-POSED LINEAR SYSTEMS VIA LOW-PRECISION RIGHT-PRECONDITIONED GMRES WITH FORWARD AND BACKWARD STABILIZATION

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Jiao, Xiangmin Department of Applied Mathematics & Statistics Institute for Advanced Computational Science Stony Brook University Stony BrookNY11794 United States

In scientific applications, linear systems are typically well-posed, and yet the coefficient matrices may be nearly singular in that the condition number κ(A) may be close to 1/Εw, where Εw denotes the unit roundoff of the working precision. Accurate and efficient solutions to such systems pose daunting challenges. It is well known that iterative refinement (IR) can make the forward error independent of κ(A) if κ(A) is sufficiently smaller than 1/Εw and the residual is computed in higher precision. Recently, Carson and Higham [SISC, 39(6), 2017] proposed a variant of IR called GMRES-IR, which replaced the triangular solves in IR with left-preconditioned GMRES using the LU factorization of A as the preconditioner. GMRES-IR relaxed the requirement on κ(A) in IR, but it requires triangular solves to be evaluated in higher precision, complicating its application to large-scale sparse systems. We propose a new iterative method, called Forward-and-Backward Stabilized Minimal Residual or FBSMR, by conceptually hybridizing right-preconditioned GMRES (RP-GMRES) with quasi-minimization. We develop FBSMR based on a new theoretical framework of essential-forward-and-backward stability (EFBS), which extends the backward error analysis to consider the intrinsic condition number of a well-posed problem. We stabilize the forward and backward errors in RP-GMRES to achieve EFBS by evaluating a small portion of the algorithm in higher precision while evaluating the preconditioner in lower precision. FBSMR can achieve optimal accuracy in terms of both forward and backward errors for well-posed problems with unpolluted matrices, independently of κ(A). With low-precision preconditioning, FBSMR can reduce the computational, memory, and energy requirements over direct methods with or without IR. FBSMR can also leverage parallelization-friendly classical Gram-Schmidt in Arnoldi iterations without compromising EFBS. We demonstrate the effectiveness of FBSMR using both random and realistic li

关键词： Linear systems

HOMOGENIZATION OF HIGH-CONTRAST DIELECTRIC ELASTOMER COMPOSITES

学校读者我要写书评

暂无评论

arXiv 2024年

作者： Dang, Thuyen Gorb, Yuliya Bolaños, Silvia Jiménez Department of Statistics/Committee on Computational and Applied Mathematics University of Chicago 5747 S. Ellis Avenue ChicagoIL60637 United States National Science Foundation 2415 Eisenhower Avenue AlexandriaVA22314 United States Department of Mathematics Colgate University 13 Oak Drive HamiltonNY13346 United States

This paper focuses on the homogenization of high-contrast dielectric elastomer composites, which are materials that deform in response to electrical stimulation. The considered heterogeneous material, consisting of an ambient material with inserted particles, is described by a weakly coupled system of an electrostatic equation with an elastic equation enriched with electrostriction. It is assumed that particles gradually become rigid as the fine-scale parameter approaches zero. This study demonstrates that the effective response of this system entails a homogeneous dielectric elastomer, described by a system of PDEs whose equations are fully decoupled. The coefficients of the homogenized equations depend on various factors, including the geometry of the composite, the periodicity of the original microstructure, and the coefficients characterizing the initial heterogeneous material. In particular, these coefficients are significantly influenced by the high-contrast nature of the coefficients of the fine-scale *** Codes 35B27, 35J47, 74A40, 74F15 © 2024, CC BY.

关键词： Electrostriction

A Neural Network-Based Approach to Normality Testing for Dependent Data

学校读者我要写书评

暂无评论

arXiv 2023年

作者： Kim, Minwoo Genton, Marc G. Huser, Raphaël Castruccio, Stefano Department of Statistics Pusan National University Korea Republic of Statistics program King Abdullah University of Science and Technology Saudi Arabia Department of Applied and Computational Mathematics and Statistics University of Notre Dame United States

There is a wide availability of methods for testing normality under the assumption of independent and identically distributed data. When data are dependent in space and/or time, however, assessing and testing the marginal behavior is considerably more challenging, as the marginal behavior is impacted by the degree of dependence. We propose a new approach to assess normality for dependent data by non-linearly incorporating existing statistics from normality tests as well as sample moments such as skewness and kurtosis through a neural network. We calibrate (deep) neural networks by simulated normal and non-normal data with a wide range of dependence structures and we determine the probability of rejecting the null hypothesis. We compare several approaches for normality tests and demonstrate the superiority of our method in terms of statistical power through an extensive simulation study. A real world application to global temperature data further demonstrates how the degree of spatio-temporal aggregation affects the marginal normality in the data. © 2023, CC BY.

关键词： Higher order statistics