Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 811-835, June 2024. Abstract. Numerical homogenization of multiscale equations typically requires taking an average of the solution to a microscale problem. Both the boundary conditions and domain size of the microscale problem play an important role in the accuracy of the homogenization procedure. In particular, imposing naive boundary conditions leads to an [math] error in the computation, where [math] is the characteristic size of the microscopic fluctuations in the heterogeneous media, and [math] is the size of the microscopic domain. This so-called boundary or “cell resonance” error can dominate discretization error and pollute the entire homogenization scheme. There exist several techniques in the literature to reduce the error. Most strategies involve modifying the form of the microscale cell problem. Below we present an alternative procedure based on the observation that the resonance error itself is an oscillatory function of domain size [math]. After rigorously characterizing the oscillatory behavior for one-dimensional and quasi-one-dimensional microscale domains, we present a novel strategy to reduce the resonance error. Rather than modifying the form of the cell problem, the original problem is solved for a sequence of domain sizes, and the results are averaged against kernels satisfying certain moment conditions and regularity properties. Numerical examples in one and two dimensions illustrate the utility of the approach.
{"title":"On the Nature of the Boundary Resonance Error in Numerical Homogenization and Its Reduction","authors":"Sean P. Carney, Milica Dussinger, Björn Engquist","doi":"10.1137/23m1594492","DOIUrl":"https://doi.org/10.1137/23m1594492","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 811-835, June 2024. <br/> Abstract. Numerical homogenization of multiscale equations typically requires taking an average of the solution to a microscale problem. Both the boundary conditions and domain size of the microscale problem play an important role in the accuracy of the homogenization procedure. In particular, imposing naive boundary conditions leads to an [math] error in the computation, where [math] is the characteristic size of the microscopic fluctuations in the heterogeneous media, and [math] is the size of the microscopic domain. This so-called boundary or “cell resonance” error can dominate discretization error and pollute the entire homogenization scheme. There exist several techniques in the literature to reduce the error. Most strategies involve modifying the form of the microscale cell problem. Below we present an alternative procedure based on the observation that the resonance error itself is an oscillatory function of domain size [math]. After rigorously characterizing the oscillatory behavior for one-dimensional and quasi-one-dimensional microscale domains, we present a novel strategy to reduce the resonance error. Rather than modifying the form of the cell problem, the original problem is solved for a sequence of domain sizes, and the results are averaged against kernels satisfying certain moment conditions and regularity properties. Numerical examples in one and two dimensions illustrate the utility of the approach.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 784-810, June 2024. Abstract. A wide variety of physical, chemical, and biological processes involve diffusive particles interacting with surfaces containing reactive patches. The theory of boundary homogenization seeks to encapsulate the effective reactivity of such a patchy surface by a single trapping rate parameter. In this paper, we derive the trapping rate for partially reactive patches occupying a small fraction of a surface. We use matched asymptotic analysis, double perturbation expansions, and homogenization theory to derive formulas for the trapping rate in terms of the far-field behavior of solutions to certain partial differential equations (PDEs). We then develop kinetic Monte Carlo (KMC) algorithms to rapidly compute these far-field behaviors. These KMC algorithms depend on probabilistic representations of PDE solutions, including using the theory of Brownian local time. We confirm our results by comparing to KMC simulations of the full stochastic system. We further compare our results to prior heuristic approximations.
{"title":"Boundary Homogenization for Partially Reactive Patches","authors":"Claire E. Plunkett, Sean D. Lawley","doi":"10.1137/23m1573422","DOIUrl":"https://doi.org/10.1137/23m1573422","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 784-810, June 2024. <br/> Abstract. A wide variety of physical, chemical, and biological processes involve diffusive particles interacting with surfaces containing reactive patches. The theory of boundary homogenization seeks to encapsulate the effective reactivity of such a patchy surface by a single trapping rate parameter. In this paper, we derive the trapping rate for partially reactive patches occupying a small fraction of a surface. We use matched asymptotic analysis, double perturbation expansions, and homogenization theory to derive formulas for the trapping rate in terms of the far-field behavior of solutions to certain partial differential equations (PDEs). We then develop kinetic Monte Carlo (KMC) algorithms to rapidly compute these far-field behaviors. These KMC algorithms depend on probabilistic representations of PDE solutions, including using the theory of Brownian local time. We confirm our results by comparing to KMC simulations of the full stochastic system. We further compare our results to prior heuristic approximations.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550894","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 752-783, June 2024. Abstract. We investigate the effective heat transfer in complex systems involving porous media and surrounding fluid layers in the context of mathematical homogenization. We differentiate between two fundamentally different cases: Case (a), where the solid part of the porous media consists of disconnected inclusions, and Case (b), where the solid matrix is connected. For both scenarios, we consider a heat equation with convection where a small scale parameter [math] characterizes the heterogeneity of the porous medium and conducts a limit process [math] via two-scale convergence for the solutions of the [math]-problems. In Case (a), we arrive at a one-temperature problem exhibiting a memory term and in Case (b) at a two-phase mixture model. We compare and discuss these two limit models with several simulation studies both with and without convection.
{"title":"Effective Heat Transfer Between a Porous Medium and a Fluid Layer: Homogenization and Simulation","authors":"Michael Eden, Tom Freudenberg","doi":"10.1137/22m1541794","DOIUrl":"https://doi.org/10.1137/22m1541794","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 752-783, June 2024. <br/> Abstract. We investigate the effective heat transfer in complex systems involving porous media and surrounding fluid layers in the context of mathematical homogenization. We differentiate between two fundamentally different cases: Case (a), where the solid part of the porous media consists of disconnected inclusions, and Case (b), where the solid matrix is connected. For both scenarios, we consider a heat equation with convection where a small scale parameter [math] characterizes the heterogeneity of the porous medium and conducts a limit process [math] via two-scale convergence for the solutions of the [math]-problems. In Case (a), we arrive at a one-temperature problem exhibiting a memory term and in Case (b) at a two-phase mixture model. We compare and discuss these two limit models with several simulation studies both with and without convection.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140812933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 713-751, June 2024. Abstract. This paper is devoted to establishing the resonant modal expansion of the low-frequency part of the scattered field for acoustic bubbles embedded in elastic materials. Due to the nanobubble with damping, the Minnaert resonance can be induced at certain discrete resonant frequencies, which forms the fundamental basis of effectively constructing elastic metamaterials via the composite material theory. There are two major contributions in this work. First, we ansatz a special form of the density, approximate the incident field with a finite number of modes, and then obtain an expansion with a finite number of modes for the acoustic-elastic wave scattering in the time-harmonic regime. Second, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the modal expansion with sharp error estimates. Interestingly, we find that the 0th mode is the main contribution to reconstruct the information of the low-frequency part of the scattered field.
{"title":"Resonant Modal Approximation of Time-Domain Elastic Scattering from Nano-Bubbles in Elastic Materials","authors":"Bochao Chen, Yixian Gao, Yong Li, Hongyu Liu","doi":"10.1137/23m1574774","DOIUrl":"https://doi.org/10.1137/23m1574774","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 2, Page 713-751, June 2024. <br/> Abstract. This paper is devoted to establishing the resonant modal expansion of the low-frequency part of the scattered field for acoustic bubbles embedded in elastic materials. Due to the nanobubble with damping, the Minnaert resonance can be induced at certain discrete resonant frequencies, which forms the fundamental basis of effectively constructing elastic metamaterials via the composite material theory. There are two major contributions in this work. First, we ansatz a special form of the density, approximate the incident field with a finite number of modes, and then obtain an expansion with a finite number of modes for the acoustic-elastic wave scattering in the time-harmonic regime. Second, we show that the low-frequency part of the scattered field in the time domain can be well approximated by using the modal expansion with sharp error estimates. Interestingly, we find that the 0th mode is the main contribution to reconstruct the information of the low-frequency part of the scattered field.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"2016 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140583382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Zhichao Peng, Yanlai Chen, Yingda Cheng, Fengyan Li
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 639-666, March 2024. Abstract.Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high-dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there has been a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work toward that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [Z. Peng, Y. Chen, Y. Cheng, and F. Li, J. Sci. Comput., 91 (2022), pp. 1–27] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed RBM (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: (1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and (2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux. We perform benchmark test problems in 2D2V, and the numerical tests show that the MMD-RBM can capture the low-rank structure effectively when it exists. A careful study in the computational cost shows that the offline stage of the MMD-RBM is more efficient than the proper orthogonal decomposition method, and in the low-rank case, it even outperforms a standard full-order solve. Therefore, the proposed MMD-RBM can be seen both as a surrogate builder and a low-rank solver at the same time. Furthermore, it can be readily incorporated into multiquery scenarios to accelerate problems arising from uncertainty quantification, control, inverse problems, and optimization.
多尺度建模与仿真》（Multiscale Modeling &Simulation ），第 22 卷第 1 期，第 639-666 页，2024 年 3 月。 摘要.动力学输运方程因其复杂的多尺度行为和需要数值解析高维概率密度函数而众所周知地难以模拟。过去的文献主要通过分析方法建立降阶模型（ROM）。近年来，人们对使用数据驱动或计算工具来开发 ROM 产生了浓厚的兴趣，因为这些工具具有更强的适用性和灵活性。本文正是朝着这一方向努力的成果。受我们之前利用角变量诱导的解流形的低秩结构为静态辐射传递方程设计 ROM 的工作[Z. Peng, Y. Chen, Y. Cheng, and F. Li, J. Sci. Comput.特别是，我们采用了著名的还原基方法（RBM），并提出了一种新颖的微宏观分解 RBM（MMD-RBM）。MMD-RBM 是以一种贪婪的方式，利用微观和宏观解流形在角度和时间变量方面的低秩结构而构建的。我们的降阶代理由降阶子空间的降阶基和角度空间的降阶正交规则组成。所提出的 MMD-RBM 有几个结构保持组件：(1) 一种尊重平衡的策略，用于构建能更好地利用分解系统结构的还原阶子空间；以及 (2) 一种保持正交权重正向性的方法，从而保持底层还原求解器的稳定性。由此产生的 ROM 可用于快速在线求解训练集以外角度方向的角通量以及角通量的任意阶矩。我们在 2D2V 中执行了基准测试问题，数值测试表明 MMD-RBM 可以有效捕捉存在的低阶结构。对计算成本的仔细研究表明，MMD-RBM 的离线阶段比适当的正交分解方法更有效，在低阶情况下，它甚至优于标准的全阶求解。因此，所提出的 MMD-RBM 可同时被视为代建器和低阶求解器。此外，它还可以很容易地融入多查询场景，以加速不确定性量化、控制、逆问题和优化等问题的解决。
{"title":"A Micro-Macro Decomposed Reduced Basis Method for the Time-Dependent Radiative Transfer Equation","authors":"Zhichao Peng, Yanlai Chen, Yingda Cheng, Fengyan Li","doi":"10.1137/22m1533487","DOIUrl":"https://doi.org/10.1137/22m1533487","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 639-666, March 2024. <br/> Abstract.Kinetic transport equations are notoriously difficult to simulate because of their complex multiscale behaviors and the need to numerically resolve a high-dimensional probability density function. Past literature has focused on building reduced order models (ROM) by analytical methods. In recent years, there has been a surge of interest in developing ROM using data-driven or computational tools that offer more applicability and flexibility. This paper is a work toward that direction. Motivated by our previous work of designing ROM for the stationary radiative transfer equation in [Z. Peng, Y. Chen, Y. Cheng, and F. Li, J. Sci. Comput., 91 (2022), pp. 1–27] by leveraging the low-rank structure of the solution manifold induced by the angular variable, we here further advance the methodology to the time-dependent model. Particularly, we take the celebrated reduced basis method (RBM) approach and propose a novel micro-macro decomposed RBM (MMD-RBM). The MMD-RBM is constructed by exploiting, in a greedy fashion, the low-rank structures of both the micro- and macro-solution manifolds with respect to the angular and temporal variables. Our reduced order surrogate consists of reduced bases for reduced order subspaces and a reduced quadrature rule in the angular space. The proposed MMD-RBM features several structure-preserving components: (1) an equilibrium-respecting strategy to construct reduced order subspaces which better utilize the structure of the decomposed system, and (2) a recipe for preserving positivity of the quadrature weights thus to maintain the stability of the underlying reduced solver. The resulting ROM can be used to achieve a fast online solve for the angular flux in angular directions outside the training set and for arbitrary order moment of the angular flux. We perform benchmark test problems in 2D2V, and the numerical tests show that the MMD-RBM can capture the low-rank structure effectively when it exists. A careful study in the computational cost shows that the offline stage of the MMD-RBM is more efficient than the proper orthogonal decomposition method, and in the low-rank case, it even outperforms a standard full-order solve. Therefore, the proposed MMD-RBM can be seen both as a surrogate builder and a low-rank solver at the same time. Furthermore, it can be readily incorporated into multiquery scenarios to accelerate problems arising from uncertainty quantification, control, inverse problems, and optimization.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sabine Haberland, Patrick Jaap, Stefan Neukamm, Oliver Sander, Mario Varga
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 588-638, March 2024. Abstract. We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary [math]-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl–Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.
{"title":"Representative Volume Element Approximations in Elastoplastic Spring Networks","authors":"Sabine Haberland, Patrick Jaap, Stefan Neukamm, Oliver Sander, Mario Varga","doi":"10.1137/23m156656x","DOIUrl":"https://doi.org/10.1137/23m156656x","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 588-638, March 2024. <br/> Abstract. We study the large-scale behavior of a small-strain lattice model for a network composed of elastoplastic springs with random material properties. We formulate the model as an evolutionary rate independent system. In an earlier work we derived a homogenized continuum model, which has the form of linearized elastoplasticity, as an evolutionary [math]-limit as the lattice parameter tends to zero. In the present paper we introduce a periodic representative volume element (RVE) approximation for the homogenized system. As a main result we prove convergence of the RVE approximation as the size of the RVE tends to infinity. We also show that the hysteretic stress-strain relation of the effective system can be described with the help of a generalized Prandtl–Ishlinskii operator, and we prove convergence of a periodic RVE approximation for that operator. We combine the RVE approximation with a numerical scheme for rate-independent systems and obtain a computational scheme that we use to numerically investigate the homogenized system in the specific case when the original network is given by a two-dimensional lattice model. We simulate the response of the system to cyclic and uniaxial, monotonic loading, and numerically investigate the convergence rate of the periodic RVE approximation. In particular, our simulations show that the RVE error decays with the same rate as the RVE error in the static case of linear elasticity.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"159 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196331","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 561-587, March 2024. Abstract. The noisy leaky integrate-and-fire (NLIF) model describes the voltage configurations of neuron networks with an interacting many-particles system at a microscopic level. When simulating neuron networks of large sizes, computing a coarse-grained mean-field Fokker–Planck equation solving the voltage densities of the networks at a macroscopic level practically serves as a feasible alternative in its high efficiency and credible accuracy when the interaction within the network remains relatively low. However, the macroscopic model fails to yield valid results of the networks when simulating considerably synchronous networks with active firing events. In this paper, we propose a multiscale solver for the NLIF networks, inheriting the macroscopic solver’s low cost and the microscopic solver’s high reliability. For each temporal step, the multiscale solver uses the macroscopic solver when the firing rate of the simulated network is low, while it switches to the microscopic solver when the firing rate tends to blow up. Moreover, the macroscopic and microscopic solvers are integrated with a high-precision switching algorithm to ensure the accuracy of the multiscale solver. The validity of the multiscale solver is analyzed from two perspectives: first, we provide practically sufficient conditions that guarantee the mean-field approximation of the macroscopic model and present rigorous numerical analysis on simulation errors when coupling the two solvers; second, the numerical performance of the multiscale solver is validated through simulating several large neuron networks, including networks with either instantaneous or periodic input currents which prompt active firing events over some time.
{"title":"A Synchronization-Capturing Multiscale Solver to the Noisy Integrate-and-Fire Neuron Networks","authors":"Ziyu Du, Yantong Xie, Zhennan Zhou","doi":"10.1137/23m1573276","DOIUrl":"https://doi.org/10.1137/23m1573276","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 561-587, March 2024. <br/> Abstract. The noisy leaky integrate-and-fire (NLIF) model describes the voltage configurations of neuron networks with an interacting many-particles system at a microscopic level. When simulating neuron networks of large sizes, computing a coarse-grained mean-field Fokker–Planck equation solving the voltage densities of the networks at a macroscopic level practically serves as a feasible alternative in its high efficiency and credible accuracy when the interaction within the network remains relatively low. However, the macroscopic model fails to yield valid results of the networks when simulating considerably synchronous networks with active firing events. In this paper, we propose a multiscale solver for the NLIF networks, inheriting the macroscopic solver’s low cost and the microscopic solver’s high reliability. For each temporal step, the multiscale solver uses the macroscopic solver when the firing rate of the simulated network is low, while it switches to the microscopic solver when the firing rate tends to blow up. Moreover, the macroscopic and microscopic solvers are integrated with a high-precision switching algorithm to ensure the accuracy of the multiscale solver. The validity of the multiscale solver is analyzed from two perspectives: first, we provide practically sufficient conditions that guarantee the mean-field approximation of the macroscopic model and present rigorous numerical analysis on simulation errors when coupling the two solvers; second, the numerical performance of the multiscale solver is validated through simulating several large neuron networks, including networks with either instantaneous or periodic input currents which prompt active firing events over some time.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"2014 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140171989","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 527-560, March 2024. Abstract. Nonlinear Fokker–Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker–Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker–Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.
{"title":"Micro-Macro Stochastic Galerkin Methods for Nonlinear Fokker–Planck Equations with Random Inputs","authors":"Giacomo Dimarco, Lorenzo Pareschi, Mattia Zanella","doi":"10.1137/22m1509205","DOIUrl":"https://doi.org/10.1137/22m1509205","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 527-560, March 2024. <br/> Abstract. Nonlinear Fokker–Planck equations play a major role in modeling large systems of interacting particles with a proved effectiveness in describing real world phenomena ranging from classical fields such as fluids and plasma to social and biological dynamics. Their mathematical formulation often has to face physical forces having a significant random component or with particles living in a random environment whose characterization may be deduced through experimental data and leading consequently to uncertainty-dependent equilibrium states. In this work, to address the problem of effectively solving stochastic Fokker–Planck systems, we will construct a new equilibrium preserving scheme through a micro-macro approach based on stochastic Galerkin methods. The resulting numerical method, contrarily to the direct application of a stochastic Galerkin projection in the parameter space of the unknowns of the underlying Fokker–Planck model, leads to a highly accurate description of the uncertainty-dependent large time behavior. Several numerical tests in the context of collective behavior for social and life sciences are presented to assess the validity of the present methodology against standard ones.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"7 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140128533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 436-475, March 2024. Abstract. In this paper we upscale thermal models from the pore–scale to the Darcy scale for applications in permafrost. We incorporate thawing and freezing of water at the pore-scale and adapt rigorous homogenization theory from [A. Visintin, SIAM J. Math. Anal., 39 (2007), pp. 987–1017] to the original nonlinear multivalued relationship to derive the effective properties. To obtain agreement of the effective model with the known Darcy scale empirical models, we revisit and extend the pore-scale model to include the delicate microscale physics in small pores. We also propose a practical reduced model for the nonlinear effective conductivity. We illustrate with simulations.
{"title":"Upscaling an Extended Heterogeneous Stefan Problem from the Pore-Scale to the Darcy Scale in Permafrost","authors":"Malgorzata Peszynska, Naren Vohra, Lisa Bigler","doi":"10.1137/23m1552000","DOIUrl":"https://doi.org/10.1137/23m1552000","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 436-475, March 2024. <br/> Abstract. In this paper we upscale thermal models from the pore–scale to the Darcy scale for applications in permafrost. We incorporate thawing and freezing of water at the pore-scale and adapt rigorous homogenization theory from [A. Visintin, SIAM J. Math. Anal., 39 (2007), pp. 987–1017] to the original nonlinear multivalued relationship to derive the effective properties. To obtain agreement of the effective model with the known Darcy scale empirical models, we revisit and extend the pore-scale model to include the delicate microscale physics in small pores. We also propose a practical reduced model for the nonlinear effective conductivity. We illustrate with simulations.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"298 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140055363","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 476-526, March 2024. Abstract. We are interested in the resonant electromagnetic scattering by all-dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. In [Ammari, Li, and Zou, Trans. Amer. Math. Soc., 376 (2023), pp. 39–90], it has been shown that a single high-index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response. Recent physics experiments reveal that when the particles are arranged in certain periodic configurations, they may have different anomalous scattering effects in the macroscopic scale, compared to the single-particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of all-dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in such a periodic setting and their asymptotic expansions in terms of the refractive index of the nanoparticles. Then we show that real resonances always exist below the essential spectrum of the periodic Maxwell operator and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. By using group theory, we discuss the implications of the symmetry of the metasurface on the subwavelength band functions and their associated eigenfunctions. For the symmetric metasurfaces with the normal incidence, we use a variational method to show the existence of embedded eigenvalues (i.e., real subwavelength resonances embedded in the continuous radiation spectrum). Furthermore, we break the configuration symmetry either by introducing a small deformation of particles or by slightly deviating from the normal incidence and prove that Fano-type reflection and transmission anomalies can arise in both of these scenarios.
{"title":"Fano Resonances in All-Dielectric Electromagnetic Metasurfaces","authors":"Habib Ammari, Bowen Li, Hongjie Li, Jun Zou","doi":"10.1137/23m1554825","DOIUrl":"https://doi.org/10.1137/23m1554825","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 476-526, March 2024. <br/> Abstract. We are interested in the resonant electromagnetic scattering by all-dielectric metasurfaces made of a two-dimensional lattice of nanoparticles with high refractive indices. In [Ammari, Li, and Zou, Trans. Amer. Math. Soc., 376 (2023), pp. 39–90], it has been shown that a single high-index nanoresonator can couple with the incident wave and exhibit a strong magnetic dipole response. Recent physics experiments reveal that when the particles are arranged in certain periodic configurations, they may have different anomalous scattering effects in the macroscopic scale, compared to the single-particle case. In this work, we shall develop a rigorous mathematical framework for analyzing the resonant behaviors of all-dielectric metasurfaces. We start with the characterization of subwavelength scattering resonances in such a periodic setting and their asymptotic expansions in terms of the refractive index of the nanoparticles. Then we show that real resonances always exist below the essential spectrum of the periodic Maxwell operator and that they are the simple poles of the scattering resolvent with the exponentially decaying resonant modes. By using group theory, we discuss the implications of the symmetry of the metasurface on the subwavelength band functions and their associated eigenfunctions. For the symmetric metasurfaces with the normal incidence, we use a variational method to show the existence of embedded eigenvalues (i.e., real subwavelength resonances embedded in the continuous radiation spectrum). Furthermore, we break the configuration symmetry either by introducing a small deformation of particles or by slightly deviating from the normal incidence and prove that Fano-type reflection and transmission anomalies can arise in both of these scenarios.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140043846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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