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Quantum spectral methods for differential equations

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arXiv 2019年

作者： Childs, Andrew M. Liu, Jin-Peng Joint Center for Quantum Information and Computer Science Department of Computer Science Institute for Advanced Computer Studies Applied Mathematics Statistics and Scientific Computation

Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity poly(log d). While several of these algorithms approximate the solution to within o with complexity poly(log(1/o)), no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity poly(log d, log(1/o)). Copyright © 2019, The Authors. All rights reserved.

关键词： Quantum theory

Partial Resampling to Approximate Covering Integer programs 16

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Partial Resampling to Approximate Covering Integer Programs

Annual ACM-Society for Industrial and applied Mathmatics Symposium on Discrete Algorithms

作者： Antares Chen David G. Harris Aravind Srinivasan University of California Berkeley Department of Applied Mathematics University of Maryland Department of Computer Science and Institute for Advanced Computer Studies University of Maryland

ISBN: (纸本)9781510819672

We consider positive covering integer programs, which generalize set cover and which have attracted a long line of research developing (randomized) approximation algorithms. Srinivasan (2006) gave a rounding algorithm based on the FKG inequality for systems which are "column-sparse." This algorithm may return an integer solution in which the variables get assigned large (integral) values; Kolliopoulos & Young (2005) modified this algorithm to limit the solution size, at the cost of a worse approximation ratio. We develop a new rounding scheme based on the Partial Resampling variant of the Lovasz Local Lemma developed by Harris & Srinivasan (2013). This achieves an approximation ratio of 1 + (ln(Δ_1+1))/a_(min) + O({the square root of}(log(Δ_1+1))/a_(min))), where a_(min) is the minimum covering constraint and Δ_1 is the maximum l_1-norm of any column of the covering matrix (whose entries are scaled to lie in [0, 1]); we also show nearlymatching inapproximability and integrality-gap lower bounds. Our approach improves asymptotically, in several different ways, over known results. First, it replaces Δ_0, the maximum number of nonzeroes in any column (from the result of Srinivasan) by Δ_1 which is always - and can be muc h - smaller than Δ_0; this is the first such result in this context. Second, our algorithm automatically handles multi-criteria programs; we achieve improved approximation ratios compared to the algorithm of Srinivasan, and give, for the first time when the number of objective functions is large, polynomial-time algorithms with good multi-criteria approximations. We also significantly improve upon the upper-bounds of Kolliopoulos & Young when the integer variables are required to be within (1 + ε) of some given upper-bounds, and show nearly-matching inapproximability.

关键词： Integer algorithms Resampling rounding Objective function

Multilevel Monte Carlo Methods for the Grad-Shafranov Free Boundary Problem

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SSRN

SSRN 2023年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Applied Mathematics & Statistics and Scientific Computing Program University of Maryland College Park United States Department of Mathematics The University of Arizona United States

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costsby performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductionstypically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the model. © 2023, The Authors. All rights reserved.

关键词： Monte Carlo methods

Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem

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arXiv 2023年

The equilibrium configuration of a plasma in an axially symmetric reactor is described mathematically by a free boundary problem associated with the celebrated Grad-Shafranov equation. The presence of uncertainty in the model parameters introduces the need to quantify the variability in the predictions. This is often done by computing a large number of model solutions on a computational grid for an ensemble of parameter values and then obtaining estimates for the statistical properties of solutions. In this study, we explore the savings that can be obtained using multilevel Monte Carlo methods, which reduce costs by performing the bulk of the computations on a sequence of spatial grids that are coarser than the one that would typically be used for a simple Monte Carlo simulation. We examine this approach using both a set of uniformly refined grids and a set of adaptively refined grids guided by a discrete error estimator. Numerical experiments show that multilevel methods dramatically reduce the cost of simulation, with cost reductions typically on the order of 60 or more and possibly as large as 200. Adaptive gridding results in more accurate computation of geometric quantities such as x-points associated with the *** Codes 65Z05, 65C05, 62P35, 35R35, 35R60 © 2023, CC BY.

关键词： Monte Carlo methods

Cerebral and muscle tissue oxygenation during exercise in healthy adults: A systematic review

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Journal of Sport and Health Science 2024年第4期13卷 459-471,F0003页

作者： Diego Orcioli-Silva Victor Spiandor Beretta Paulo Cezar Rocha Santos Felipe Marroni Rasteiro Anita Brum Marostegan Rodrigo Vitorio Claudio Alexandre Gobatto Fulvia Barros Manchado-Gobatto Laboratory of Applied Sport Physiology(LAFAE) School of Applied Sciences(FCA)University of Campinas(UNICAMP)Limeira 13484-350Brazil Posture and Gait Studies Laboratory(LEPLO) Institute of BiosciencesSao Paulo State University(UNESP)Rio Claro 13506-900Brazil Physical Education Department School of Technology and SciencesSao Paulo State University(UNESP)Presidente Prudente 19060-900Brazil Department of Computer Science&Applied Mathematics Weizmann Institute of ScienceRehovot 7610001Israel Center of Advanced Technologies in Rehabilitation Sheba Medical CenterRamat Gan 5265601Israel Department of Sport Exercise and RehabilitationNorthumbria UniversityNewcastle upon Tyne NE18STUK

Background:Near-infrared spectroscopy(NIRS)technology has allowed for the measurement of cerebral and skeletal muscle oxygenation simultaneously during *** this technology has been growing and is now successfully used in laboratory and sports settings,this systematic review aimed to synthesize the evidence and enhance an integrative understanding of bloodﬂow adjustments and oxygen(O_(2))changes(i.e.,the balance between O_(2) delivery and O_(2) consumption)within the cerebral and muscle systems during ***:A systematic review was conducted using PubMed,Embase,Scopus,and Web of Science databases to search for relevant studies that simultaneously investigated cerebral and muscle hemodynamic changes using the near-infrared spectroscopy system during *** review considered manuscripts written in English and available before February 9,*** step of screening involved evaluation by 2 inde-pendent authors,with disagreements resolved by a third *** Joanna Briggs institute Critical Appraisal Checklist was used to assess the methodological quality of the ***:Twenty studies were included,of which 80%had good methodological quality,and involved 290 young or middle-aged *** types of exercises were used to assess cerebral and muscle hemodynamic changes,such as cycling(n=11),treadmill(n=1),knee extension(n=5),isometric contraction of biceps brachii(n=3),and duet swim routines(n=1).The cerebral hemodynamics anal-ysis was focused on the frontal cortex(n=20),while in the muscle,the analysis involved vastus lateralis(n=18),gastrocnemius(n=3),biceps brachii(n=5),deltoid(n=1),and intercostal muscle(n=1).Overall,muscle deoxygenation increases during exercise,reaching a plateau in voluntary exhaustion,while in the brain,oxyhemoglobin concentration increases with exercise intensity,reaching a plateau or declining at the exhaustion ***:Muscle and cerebral oxygenation respond differently to exercise,with muscle

关键词： Near-infrared spectroscopy Hemodynamic Blood flow Brain Muscle metabolism

An Assessment of the Supremizer and Aggregation Methods of Stabilization for Reduced-Order Models

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arXiv 2023年

作者： Davie, Kayla D. Elman, Howard C. Applied Mathematics & Statistics Scientific Computation Program University of Maryland College ParkMD20742 United States Department of Computer Science Institute for Advanced Computer Studies University of Maryland College ParkMD20742 United States

MSC Codes 65N22, 65K10, 65F45We explore the features of two methods of stabilization, aggregation and supremizer methods, for reduced-order modeling of parametrized optimal control problems. In both methods, the reduced basis spaces are augmented to guarantee stability. For the aggregation method, the reduced basis approximation spaces for the state and adjoint variables are augmented in such a way that the spaces are identical. For the supremizer method, the reduced basis approximation space for the state-control product space is augmented with the solutions of a supremizer equation. We implement both of these methods for solving several parametrized control problems and assess their performance. Results indicate that the number of reduced basis vectors needed to approximate the solution space to some tolerance with the supremizer method is much larger, possibly double, that for aggregation. There are also some cases where the supremizer method fails to produce a converged solution. We present results to compare the accuracy, efficiency, and computational costs associated with both methods of stabilization which suggest that stabilization by aggregation is a superior stabilization method for control problems. © 2023, CC BY.

关键词： Stabilization

A Low-rank solver for the stochastic unsteady navier-stokes problem

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arXiv 2019年

作者： Elman, Howard Su, Tengfei Department of Computer Science Institute for Advanced Computer Studies University of Maryland College ParkMD20742 United States Applied Mathematics & Statistics and Scientific Computation Program University of Maryland College ParkMD20742 United States

We study a low-rank iterative solver for the unsteady Navier–Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once formulation where the algebraic systems at all the time steps are collected and solved simultaneously. The problem is linearized with Picard’s method. To efficiently solve the linear systems at each step, we use low-rank tensor representations within the Krylov subspace method, which leads to significant reductions in storage requirements and computational costs. Combined with effective mean-based preconditioners and the idea of inexact solve, we show that only a small number of linear iterations are needed at each Picard step. The proposed algorithm is tested with a model of flow in a two-dimensional symmetric step domain with different settings to demonstrate the computational efficiency. Copyright © 2019, The Authors. All rights reserved.

关键词： Navier Stokes equations

Surrogate approximation of the grad-shafranov free boundary problem via stochastic collocation on sparse grids

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arXiv 2021年

作者： Elman, Howard C. Liang, Jiaxing Sánchez-Vizuet, Tonatiuh Department of Computer Science Institute for Advanced Computer Studies University of Maryland College Park United States Applied Mathematics & Statistics and Scientific Computing Program University of Maryland College Park United States Courant Institute of Mathematical Sciences New York University Department of Mathematics The University of Arizona

In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over *** Codes 65C30, 65C05, 65N35, 65N99, 65Z05 Copyright © 2021, The Authors. All rights reserved.

关键词： Monte Carlo methods

Low-rank solution methods for stochastic eigenvalue problems

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arXiv 2018年

作者： Elman, Howard C. Su, Tengfei Department of Computer Science Institute for Advanced Computer Studies University of Maryland College ParkMD20742 United States Applied Mathematics & Statistics Scientific Computation Program University of Maryland College ParkMD20742 United States

We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation. Copyright © 2018, The Authors. All rights reserved.

关键词： Eigenvalues and eigenfunctions