检索结果-内蒙古大学图书馆

Doubly-stochastic normalization of the Gaussian kernel is robust to heteroskedastic noise

学校读者我要写书评

暂无评论

arXiv 2020年

作者： Landa, Boris Coifman, Ronald R. Kluger, Yuval Program in Applied Mathematics Yale University Interdepartmental Program in Computational Biology and Bioinformatics Yale University Department of Pathology Yale University School of Medicine

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal (i.e. no self loops) is robust to heteroskedastic noise. That is, the doubly-stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly-stochastic) noisy affinity matrix converges to its clean counterpart with rate m−1/2, where m is the ambient dimension. We demonstrate this result numerically, and show that in contrast, the popular row-stochastic and symmetric normalizations behave unfavourably under heteroskedastic noise. Furthermore, we provide a prototypical example of simulated single-cell RNA sequence data with strong intrinsic heteroskedasticity, where the advantage of the doubly-stochastic normalization for exploratory analysis is evident. Copyright © 2020, The Authors. All rights reserved.

关键词： Stochastic systems

From Dyson models to many-body quantum chaos

学校读者我要写书评

暂无评论

Physical Review B 2025年第3期111卷 035147-035147页

作者： Alexei Andreanov Matteo Carrega Jeff Murugan Jan Olle Dario Rosa Ruth Shir Center for Theoretical Physics of Complex Systems Institute for Basic Science (IBS) Daejeon 34126 Korea Basic Science Program Korea University of Science and Technology (UST) Daejeon 34113 Korea CNR-SPIN Via Dodecaneso 33 16146 Genova Italy The Laboratory for Quantum Gravity & Strings Department of Mathematics and Applied Mathematics University of Cape Town Cape Town 7701 South Africa The National Institute for Theoretical and Computational Sciences Private Bag X1 Matieland 7602 South Africa Max Planck Institute for the Science of Light 91058 Erlangen Germany ICTP South American Institute for Fundamental Research Instituto de Física Teórica UNESP—University of Estadual Paulista Rua Dr. Bento Teobaldo Ferraz 271 01140-070 São Paulo SP Brazil Department of Physics and Materials Science University of Luxembourg L-1511 Luxembourg Luxembourg

A deep understanding of the mechanisms underlying many-body quantum chaos is one of the big challenges in contemporary theoretical physics. We tackle this problem in the context of a set of perturbed quadratic Sachdev-Ye-Kitaev (SYK) Hamiltonians defined on graphs. This allows us to disentangle the geometrical properties of the underlying single-particle problem and the importance of the interaction terms, showing that the former is the dominant feature ensuring the single-particle to many-body chaotic transition. Our results are verified numerically with state-of-the-art numerical techniques, capable of extracting eigenvalues in a desired energy window of very large Hamiltonians. Our approach essentially provides a new way of viewing many-body chaos from a single-particle perspective.

关键词： Quantum chaos

Unbiased Estimation using a Class of Diffusion Processes

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Ruzayqat, Hamza Beskos, Alexandros Crisan, Dan Jasra, Ajay Kantas, Nikolas Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 KSA Saudi Arabia Department of Statistical Science University College London LondonWC1E 6BT United Kingdom Department of Mathematics Imperial College London LondonSW7 2AZ United Kingdom

We study the problem of unbiased estimation of expectations with respect to (w.r.t.) π a given, general probability measure on (d, B(d)) that is absolutely continuous with respect to a standard Gaussian measure. We focus on simulation associated to a particular class of diffusion processes, sometimes termed the Schrödinger-Föllmer Sampler, which is a simulation technique that approximates the law of a particular diffusion bridge process {Xt}tϵ[0,1]on d, d ϵ 0. This latter process is constructed such that, starting at X0= 0, one has X1∼ π. Typically, the drift of the diffusion is intractable and, even if it were not, exact sampling of the associated diffusion is not possible. As a result, [10, 16] consider a stochastic Euler-Maruyama scheme that allows the development of biased estimators for expectations w.r.t. π. We show that for this methodology to achieve a mean square error of O(ϵ2), for arbitrary ϵ > 0, the associated cost is O(ϵ-5). We then introduce an alternative approach that provides unbiased estimates of expectations w.r.t. π, that is, it does not suffer from the time discretization bias or the bias related with the approximation of the drift function. We prove that to achieve a mean square error of O(ϵ2), the associated cost (which is random) is, with high probability, O(ϵ-2| log(ϵ)|2+δ), for any δ > 0. We implement our method on several examples including Bayesian inverse *** Codes 60J60, 62D05, 65C40 © 2022, CC BY.

关键词： Diffusion

Machine learning based non-Newtonian fluid model with molecular fidelity

学校读者我要写书评

暂无评论

arXiv 2020年

作者： Lei, Huan Wu, Lei Weinan, E. Department of Computational Mathematics Science & Engineering Department of Statistics & Probability Michigan State University MI48824 United States Department of Mathematics and Program in Applied and Computational Mathematics Princeton University NJ08544 United States

We introduce a machine-learning-based framework for constructing continuum non-Newtonian fluid dynamics model directly from a micro-scale description. Polymer solution is used as an example to demonstrate the essential ideas. To faithfully retain molecular fidelity, we establish a micro-macro correspondence via a set of encoders for the micro-scale polymer configurations and their macro-scale counterparts, a set of nonlinear conformation tensors. The dynamics of these conformation tensors can be derived from the micro-scale model and the relevant terms can be parametrized using machine learning. The final model, named the deep non-Newtonian model (DeePN2), takes the form of conventional non-Newtonian fluid dynamics models, with a new form of the objective tensor derivative. Numerical results demonstrate the accuracy of DeePN2. Copyright © 2020, The Authors. All rights reserved.

关键词： Viscous flow

A sharp interface Lagrangian-Eulerian method for flexible-body fluid-structure interaction

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Kolahdouz, Ebrahim M. Wells, David R. Rossi, Simone Aycock, Kenneth I. Craven, Brent A. Griffith, Boyce E. Center for Computational Biology Flatiron Institute Simons Foundation New YorkNY United States Department of Mathematics University of North Carolina Chapel HillNC United States Division of Applied Mechanics Office of Science and Engineering Laboratories Center for Devices and Radiological Health United States Food and Drug Administration Silver SpringMD United States Departments of Mathematics and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

This paper introduces a sharp-interface approach to simulating fluid-structure interaction (FSI) involving flexible bodies described by general nonlinear material models and across a broad range of mass density ratios. This new flexible-body immersed Lagrangian-Eulerian (ILE) scheme extends our prior work on integrating partitioned and immersed approaches to rigid-body FSI. Our numerical approach incorporates the geometrical and domain solution flexibility of the immersed boundary (IB) method with an accuracy comparable to body-fitted approaches that sharply resolve flows and stresses up to the fluid-structure interface. Unlike many IB methods, our ILE formulation uses distinct momentum equations for the fluid and solid subregions with a Dirichlet-Neumann coupling strategy that connects fluid and solid subproblems through simple interface conditions. As in earlier work, we use approximate Lagrange multiplier forces to treat the kinematic interface conditions along the fluid-structure interface. This penalty approach simplifies the linear solvers needed by our formulation by introducing two representations of the fluid-structure interface, one that moves with the fluid and another that moves with the structure, that are connected by stiff springs. This approach also enables the use of multi-rate time stepping, which allows us to use different time step sizes for the fluid and structure subproblems. Our fluid solver relies on an immersed interface method (IIM) for discrete surfaces to impose stress jump conditions along complex interfaces while enabling the use of fast structured-grid solvers for the incompressible Navier-Stokes equations. The dynamics of the volumetric structural mesh are determined using a standard finite element approach to large-deformation nonlinear elasticity via a nearly incompressible solid mechanics formulation. This formulation also readily accommodates compressible structures with a constant total volume, and it can handle fully compressibl

关键词： Fluid structure interaction

A Nodal Immersed Finite Element-Finite Difference Method

学校读者我要写书评

暂无评论

SSRN

SSRN 2022年

作者： Wells, David Vadala-Roth, Ben Lee, Jae Ho Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States U.S. Army Corps of Engineers Engineer Research and Development Center Coastal and Hydraulic Laboratory VicksburgMS United States Department of Mechanical Engineering Institute for Computational Medicine Johns Hopkins University BaltimoreMD United States Departments of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

The immersed finite element-finite difference (IFED) method is a computational approach to modeling interactions between a fluid and an immersed structure. The IFED method uses a finite element (FE) method to approximate the stresses and forces on a structural mesh and a finite difference (FD) method to approximate the momentum of the entire fluid-structure system on a Cartesian grid. The fundamental approach used by this method follows the immersed boundary framework for modeling fluid-structure interaction (FSI), in which a force spreading operator prolongs structural forces to a Cartesian grid, and a velocity interpolation operator restricts a velocity field defined on that grid back onto the structural mesh. With an FE structural mechanics framework, force spreading first requires that the force itself be projected onto the finite element space. Similarly, velocity interpolation requires projecting velocity data onto the FE basis functions. Consequently, evaluating either coupling operator requires solving a matrix equation at every time step. Mass lumping, in which the projection matrices are replaced by diagonal approximations, has the potential to accelerate this method considerably. This paper provides both numerical and computational analyses of the effects of this replacement for evaluating the force projection and for the IFED coupling operators. Constructing the coupling operators also requires determining the locations on the structure mesh where the forces and velocities are sampled. Here we show that sampling the forces and velocities at the nodes of the structural mesh is equivalent to using lumped mass matrices in the IFED coupling operators. A key theoretical result of our analysis is that if both of these approaches are used together, the IFED method permits the use of lumped mass matrices derived from nodal quadrature rules for any standard interpolatory element. This is different from standard FE methods, which require specialized treatments to

关键词： Fluid structure interaction

An immersed peridynamics model of fluid-structure interaction accounting for material damage and failure

学校读者我要写书评

暂无评论

arXiv 2022年

作者： Kim, Keon Ho Bhalla, Amneet P.S. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Mechanical Engineering San Diego State University San DiegoCA United States Departments of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary methods. The major difference between our approach and conventional immersed boundary methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well-characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations. © 2022, CC BY-SA.

关键词： Continuum mechanics

Correction: Unbiased and multilevel methods for a class of diffusions partially observed via marked point processes

学校读者我要写书评

暂无评论

Statistics and Computing 2024年第6期34卷 1-1页

作者： Alvarez, Miguel Jasra, Ajay Ruzayqat, Hamza Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal Saudi Arabia School of Data Science The Chinese University of Hong Kong Shenzhen China

An Immersed Peridynamics Model of Fluid-Structure Interaction Accounting for Material Damage and Failure

学校读者我要写书评

暂无评论

SSRN

SSRN 2022年

作者： Kim, Keon Ho Bhalla, Amneet Pal Singh Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Mechanical Engineering San Diego State University San DiegoCA United States Departments of Mathematics Applied Physical Sciences and Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina Chapel HillNC United States McAllister Heart Institute University of North Carolina Chapel HillNC United States

This paper develops and benchmarks an immersed peridynamics method to simulate the deformation, damage, and failure of hyperelastic materials within a fluid-structure interaction framework. The immersed peridynamics method describes an incompressible structure immersed in a viscous incompressible fluid. It expresses the momentum equation and incompressibility constraint in Eulerian form, and it describes the structural motion and resultant forces in Lagrangian form. Coupling between Eulerian and Lagrangian variables is achieved by integral transforms with Dirac delta function kernels, as in standard immersed boundary (IB) methods. The major difference between our approach and conventional IB methods is that we use peridynamics, instead of classical continuum mechanics, to determine the structural forces. We focus on non-ordinary state-based peridynamic material descriptions that allow us to use a constitutive correspondence framework that can leverage well characterized nonlinear constitutive models of soft materials. The convergence and accuracy of our approach are compared to both conventional and immersed finite element methods using widely used benchmark problems of nonlinear incompressible elasticity. We demonstrate that the immersed peridynamics method yields comparable accuracy with similar numbers of structural degrees of freedom for several choices of the size of the peridynamic horizon. We also demonstrate that the method can generate grid-converged simulations of fluid-driven material damage growth, crack formation and propagation, and rupture under large deformations. © 2022, The Authors. All rights reserved.

关键词： Fluid structure interaction