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Existence and Regularity of Strong Solutions to a Nonhomogeneous Kelvin-Voigt-Cahn-Hilliard System

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SSRN

SSRN 2023年

作者： Giorgini, A. Ngana, A. Ndongmo Medjo, T. Tachim Temam, R. Politecnico di Milano Dipartimento di Matematica Via E. Bonardi 9 Milano20133 Italy School of Mathematical and Statistical Sciences North West University South Africa Department of Mathematics and Statistics Florida International University MMC MiamiFL33199 United States Institute for Scientific Computing and Applied Mathematics Indiana University BloomingtonIN47405 United States

A well-known diffuse interface model consists of the Navier-Stokes equations nonlinearly coupled with the convective Cahn-Hilliard equation. This system describes the evolution of an incompressible isothermal Newtonian mixture of binary fluids. In this article we investigate a variant of this model, which consists of the nonhomogeneous Kelvin-Voigt equations coupled with the Cahn-Hilliard equations. We prove the existence of global weak and strong solutions in a two and three dimensions. Furthermore, we prove some regularity results for the strong solutions and show their uniqueness in both two or three-dimensional bounded domains. Lastly, we also solve a problem of uniqueness of regular solutions that was left open by Giorgini and Temam in [J. Math. Pures Appl. 144 (2020), 194-249.]. © 2023, The Authors. All rights reserved.

关键词： Navier Stokes equations

NON-CONCENTRATION PHENOMENON FOR ONE DIMENSIONAL REACTION-DIFFUSION SYSTEMS WITH MASS DISSIPATION

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arXiv 2022年

作者： Yang, Juan Kostianko, Anna Sun, Chunyou Tang, Bao Quoc Zelik, Sergey School of Mathematics and Statistics Lanzhou University Lanzhou730000 China Institute of Mathematics and Scientific Computing University of Graz Heinrichstrasse 36 Graz8010 Austria Imperial College LondonSW7 2AZ United Kingdom National Research University Higher School Of Economics Russia University of Surrey Department of Mathematics GuildfordGU2 7XH United Kingdom Keldysh Institute of Applied Mathematics Moscow Russia

Reaction-diffusion systems with mass dissipation are known to possess blow-up solutions in high dimensions when the nonlinearities have super quadratic growth rates. In dimension one, it has been shown recently that one can have global existence of bounded solutions if nonlinearities are at most cubic. For the cubic intermediate sum condition, i.e. nonlinearities might have arbitrarily high growth rates, an additional entropy inequality had to be imposed. In this article, we remove this extra entropy assumption completely and obtain global boundedness for reaction-diffusion systems with cubic intermediate sum condition. The novel idea is to show a non-concentration phenomenon for mass dissipating systems, that is the mass dissipation implies a dissipation in a Morrey space M1,δ(Ω) for some δ > 0. As far as we are concerned, it is the first time such a bound is derived for mass dissipating reaction-diffusion systems. The results are then applied to obtain global existence and boundedness of solutions to an oscillatory Belousov-Zhabotinsky system, which satisfies cubic intermediate sum condition but does not fulfill the entropy assumption. Extensions include global existence mass controlled systems with slightly-super cubic intermediate sum condition. © 2022, CC0.

关键词： Entropy

Approximation methods for piecewise deterministic Markov processes and their costs

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arXiv 2017年

作者： Kritzer, Peter Leobacher, Gunther Szölgyenyi, Michaela Thonhauser, Stefan Austrian Academy of Sciences Altenbergerstraße 69 Linz4040 Austria Institute for Mathematics and Scientific Computing University of Graz Heinrichstraße 36 Graz8010 Austria Institute for Statistics University of Klagenfurt Universitätsstraße 65-67 Klagenfurt9020 Austria Seminar for Applied Mathematics and RiskLab Switzerland ETH Zurich Rämistrasse 101 Zurich8092 Switzerland Institute for Statistics Graz University of Technology Kopernikusgasse 24/III Graz8010 Austria

In this paper, we analyse piecewise deterministic Markov processes, as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of piecewise deterministic Markov processes. In this context, one is interested in computing certain quantities of interest such as the probability of ruin of an insurance company, or the insurance company's value, defined as the expected discounted future dividend payments until the time of ruin. Instead of explicitly solving the integro-(partial) differential equation related to the quantity of interest considered (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules;this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. On the analytical side, we prove a convergence result for our PDMP approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and provide a comparative study of deterministic and Monte Carlo integration.60J25, 91G60, 65D32 Copyright © 2017, The Authors. All rights reserved.

关键词： Iterative methods

Characterizing the nonlinear structure of shared variability in cortical neuron populations using latent variable models

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arXiv 2019年

作者： Whiteway, Matthew R. Socha, Karolina Bonin, Vincent Butts, Daniel A. Program in Applied Mathematics and Statistics and Scientific Computation University of Maryland College ParkMD United States Neuro-Electronics Research Flanders Leuven Belgium Department of Biology Leuven Brain Institute KU Leuven Leuven Belgium VIB Leuven Belgium Department of Biology and Program in Neuroscience and Cognitive Sciences University of Maryland College ParkMD United States

Sensory neurons often have variable responses to repeated presentations of the same stimulus, which can significantly degrade the stimulus information contained in those responses. This information can in principle be preserved if variability is shared across many neurons, but depends on the structure of the shared variability and its relationship to sensory encoding at the population level. The structure of this shared variability in neural activity can be characterized by latent variable models, although they have thus far typically been used under restrictive mathematical assumptions, such as assuming linear transformations between the latent variables and neural activity. Here we introduce two nonlinear latent variable models for analyzing large-scale neural *** first present a general nonlinear latent variable model that is agnostic to the stimulus tuning properties of the individual neurons, and is hence well suited for exploring neural populations whose tuning properties are not well characterized. This motivates a second class of model, the Generalized Affine Model, which simultaneously determines each neuron's stimulus selectivity and a set of latent variables that modulate these stimulus-driven responses both additively and multiplicatively. While these approaches can detect very general nonlinear relationships in shared neural variability, we find that neural activity recorded in anesthetized primary visual cortex (V1) is best described by a single additive and single multiplicative latent variable, i.e., an "affine model". In contrast, application of the same models to recordings in awake macaque prefrontal cortex discover more general nonlinearities to compactly describe the population response variability. These results thus demonstrate how nonlinear latent variable models can be used to describe population variability, and suggest that a range of methods is necessary to study different brain regions under different experimental conditions. C

关键词： Neurons

The Exponential Transformation Based Lattice Boltzmann Model for Convection-Diffusion Equation

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SSRN

SSRN 2022年

作者： Zhang, Ting Cui, Shuqi Hong, Ning Shi, Baochang College of Sciences Wuhan University of Science and Technology Wuhan430081 China School of General Education Wuchang University of Technology Wuhan430223 China Institute of Interdisciplinary Research for Mathematics and Applied Science School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China

In this paper, an exponential transformation based lattice Boltzmann (LB) model for solving the n -dimensional ( n D) convection-diffusion equation(CDE) is developed. Firstly, a class of exponential transformation is proposed to convert the n D CDE into a diffusion equation. Then, the converted diffusion equation is solved by the LB model. So, compared to the available LB models for CDE, the present LB model can eliminate the difficulty in treating the convection term. With the direct Taylor expansion method, it is shown that the CDE can be exactly derived from the exponential transformation based LB model. Finally, a variety of numerical tests have been conducted to validate the present LB model. It can be found that the numerical results agree well with the analytical solutions. Moreover, we also find that the present LB model has a second-order convergence rate in space, and it is more effective and more stable than the previous LB model for the CDE. © 2022, The Authors. All rights reserved.

关键词： Boltzmann equation

PARALLEL ELEMENT-BASED ALGEBRAIC MULTIGRID FOR H(curl) AND H(div) PROBLEMS USING THE PARELAG LIBRARY

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arXiv 2021年

作者： Kalchev, Delyan Z. Vassilevski, Panayot S. Villa, Umberto Center for Applied Scientific Computing Lawrence Livermore National Laboratory P.O. Box 808 L-561 LivermoreCA94551 United States Department of Mathematics and Statistics Portland State University PortlandOR97207 United States Oden Institute for Computational Engineering & Sciences The University of Texas at Austin AustinTX78712 United States

This paper presents the use of element-based algebraic multigrid (AMGe) hierarchies, implemented in the ParELAG (Parallel Element Agglomeration Algebraic Multigrid Upscaling and Solvers) library, to produce multilevel preconditioners and solvers for H(curl) and H(div) formulations. ParELAG constructs hierarchies of compatible nested spaces, forming an exact de Rham sequence on each level. This allows the application of hybrid smoothers on all levels and AMS (Auxiliary-space Maxwell Solver) or ADS (Auxiliary-space Divergence Solver) on the coarsest levels, obtaining complete multigrid cycles. Numerical results are presented, showing the parallel performance of the proposed methods. As a part of the exposition, this paper demonstrates some of the capabilities of ParELAG and outlines some of the components and procedures within the library. © 2021, CC BY.

关键词： Finite element method

Scaling laws and spatial effects of Brazilian health regions: a research protocol

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Procedia Computer Science 2023年 219卷 1325-1332页

作者： Giovane Thomazini Soares Diego Bettiol Yamada Filipe Andrade Bernardi Mariane Barros Neiva Luis Pedro Lombardi Junior André Luiz Teixeira Vinci Ana Clara de Andrade Mioto Domingos Alves Ribeirão Preto Medical School University of Sao Paulo Av Bandeirantes 3900 14049900 Ribeirao Preto SP Brazil Bioengineering Postgraduate Program Av Trabalhador Sao-Carlense 400 13566590 Sao Carlos SP Brazil Institute of Mathematics and Computer Sciences Av Trabalhador Sao-Carlense 400 13566590 Sao Carlos SP Brazil Institute of Mathematics Statistics and Scientific Computing State University of Campinas Rua Sérgio Buarque de Holanda 651 13083859 Campinas SP Brazil

The literature has already consolidated the importance of health regions for Brazilian public health. Complexity properties strongly mark such regions. In this context, there are abundant indications that health regions should be analyzed with approaches linked to the sciences of complexity. One of these approaches, the estimation of scaling laws, can describe important properties of socio-spatial elements. However, no studies estimate the scaling laws of Brazilian health regions. This research protocol can remedy this limitation, proposing the estimation of scaling laws of the previously mentioned regions, mainly considering variables relevant to Brazilian public health. Still, this paper can substantially mitigate other relevant limitations of usual research that estimate scaling laws of socio-spatial elements. These mitigations, which provide advances in the literature on estimating scaling laws, are given by the proposal of modeling (if necessary) spatial effects and estimating scaling laws for the entire population of the socio-spatial elements. According to the theory, the expected results are non-linear scaling laws, which will likely vary with space and time and coexist with relevant spatial effects. From such laws and effects, it will be possible to accurately characterize the performance of each health region through Spatial and Scale Adjusted Metropolitan Indicators and unravel spatio-temporal properties, stabilities, and instabilities of sets composed of health regions. The expected findings of this paper can help rearrange health regions and improve the quality of information used in Brazilian public health planning.

关键词： Scaling laws Spatial effects Research protocol

Learning interface conditions in domain decomposition solvers 22

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Learning interface conditions in domain decomposition solver...

Proceedings of the 36th International Conference on Neural Information Processing Systems

作者： Ali Taghibakhshi Nicolas Nytko Tareq Zaman Scott MacLachlan Luke Olson Matthew West Mechanical Science and Engineering University of Illinois Urbana-Champaign Urbana IL Computer Science University of Illinois Urbana-Champaign Urbana IL Scientific Computing Program Memorial University of Newfoundland and Labrador St. John's NL Canada Mathematics and Statistics Memorial University of Newfoundland and Labrador St. John's NL Canada

ISBN: (纸本)9781713871088

Domain decomposition methods are widely used and effective in the approximation of solutions to partial differential equations. Yet the optimal construction of these methods requires tedious analysis and is often available only in simplified, structured-grid settings, limiting their use for more complex problems. In this work, we generalize optimized Schwarz domain decomposition methods to unstructured-grid problems, using Graph Convolutional Neural Networks (GCNNs) and unsupervised learning to learn optimal modifications at subdomain interfaces. A key ingredient in our approach is an improved loss function, enabling effective training on relatively small problems, but robust performance on arbitrarily large problems, with computational cost linear in problem size. The performance of the learned linear solvers is compared with both classical and optimized domain decomposition algorithms, for both structured- and unstructured-grid problems.

关键词：

CONVERGENCE ANALYSIS FOR MINIMUM ACTION METHODS COUPLED WITH A FINITE DIFFERENCE METHOD

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arXiv 2021年

作者： Hong, Jialin Jin, Diancong Sheng, Derui Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The minimum action method (MAM) is an effective approach to numerically solving minimums and minimizers of Freidlin–Wentzell (F-W) action functionals, which is used to study the most probable transition path and probability of the occurrence of transitions for stochastic differential equations (SDEs) with small noise. In this paper, we focus on MAMs based on a finite difference method, and present the convergence analysis of minimums and minimizers of the discrete F-W action functional. The main result shows that the convergence orders of the minimum of the discrete F-W action functional in the cases of multiplicative noises and additive noises are 1/2 and 1, respectively. Our main result also reveals the convergence of the stochastic θ-method for SDEs with small noise in terms of large deviations. Copyright © 2021, The Authors. All rights reserved.

关键词： Finite difference method