Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 406-435, March 2024. Abstract. Coarse-graining or model reduction is a term describing a range of approaches used to extend the timescale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analyzed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics, such as the mean-squared displacement, display systematic error, even when a system exhibits large timescale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.
{"title":"Dynamical Properties of Coarse-Grained Linear SDEs","authors":"Thomas Hudson, Xingjie Helen Li","doi":"10.1137/23m1549249","DOIUrl":"https://doi.org/10.1137/23m1549249","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 406-435, March 2024. <br/> Abstract. Coarse-graining or model reduction is a term describing a range of approaches used to extend the timescale of molecular simulations by reducing the number of degrees of freedom. In the context of molecular simulation, standard coarse-graining approaches approximate the potential of mean force and use this to drive an effective Markovian model. To gain insight into this process, the simple case of a quadratic energy is studied in an overdamped setting. A hierarchy of reduced models is derived and analyzed, and the merits of these different coarse-graining approaches are discussed. In particular, while standard recipes for model reduction accurately capture static equilibrium statistics, it is shown that dynamical statistics, such as the mean-squared displacement, display systematic error, even when a system exhibits large timescale separation. In the linear setting studied, it is demonstrated both analytically and numerically that such models can be augmented in a simple way to better capture dynamical statistics.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"179 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007046","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 369-405, March 2024. Abstract. There are two main approaches to modelling interfaces within nonequilibrium thermodynamics, the so-called sharp and diffuse interface models. Both of them are based on the local equilibrium assumption (LEA) in the bulk, but the latter additionally assumes the validity of this concept also within the interface itself (as the thermodynamic description is available and smoothly varying even within the interface), that is, on a finer length scale, which we call super-LEA. Instead of testing the two approaches against molecular dynamic simulations, we explore the mutual compatibility of these two descriptions of an interface in a nonequilibrium situation. Based on the level of detail in the two frameworks, one naturally cannot reconstruct a diffuse interface model from a sharp interface counterpart. One can test, however, whether diffuse interface models are indeed a more detailed description of the interface. Namely, assuming that both approaches are valid, we use the diffuse interface model (van der Waals entropy together with the Cahn–Hilliard type energy with the mass density as the order parameter) and its sharp interface counterpart (with the additional set of interfacial state variables subjected to known thermodynamic constraints) to test their mutual compatibility and indirectly verify the correctness of the additional super-LEA of the diffuse models. That is, thanks to super-LEA, we define five interfacial temperatures that should be equal. However, when we analyze diffuse interface results like experimental or simulation data in terms of sharp interfaces, we show that, contrary to molecular simulation data, they do not yield equal interfacial temperatures. We argue that the culprit is the super-LEA which is most prominently expressed in the accessibility of the entropy density profile. Nevertheless, it is observed that there is an inconsistency between diffuse and sharp interface descriptions; they cannot both be correct. The sharp interface framework has been recently tested against molecular dynamics and the obtained results suggest that super-LEA is the potential weakness of the diffuse framework. In this sense, sharp interfaces are found to be superior to diffuse interfaces in their general ability to model physical systems with interfaces.
{"title":"On the Compatibility of Sharp and Diffuse Interfaces Out of Equilibrium","authors":"Václav Klika, Hans Christian Öttinger","doi":"10.1137/22m1529294","DOIUrl":"https://doi.org/10.1137/22m1529294","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 369-405, March 2024. <br/> Abstract. There are two main approaches to modelling interfaces within nonequilibrium thermodynamics, the so-called sharp and diffuse interface models. Both of them are based on the local equilibrium assumption (LEA) in the bulk, but the latter additionally assumes the validity of this concept also within the interface itself (as the thermodynamic description is available and smoothly varying even within the interface), that is, on a finer length scale, which we call super-LEA. Instead of testing the two approaches against molecular dynamic simulations, we explore the mutual compatibility of these two descriptions of an interface in a nonequilibrium situation. Based on the level of detail in the two frameworks, one naturally cannot reconstruct a diffuse interface model from a sharp interface counterpart. One can test, however, whether diffuse interface models are indeed a more detailed description of the interface. Namely, assuming that both approaches are valid, we use the diffuse interface model (van der Waals entropy together with the Cahn–Hilliard type energy with the mass density as the order parameter) and its sharp interface counterpart (with the additional set of interfacial state variables subjected to known thermodynamic constraints) to test their mutual compatibility and indirectly verify the correctness of the additional super-LEA of the diffuse models. That is, thanks to super-LEA, we define five interfacial temperatures that should be equal. However, when we analyze diffuse interface results like experimental or simulation data in terms of sharp interfaces, we show that, contrary to molecular simulation data, they do not yield equal interfacial temperatures. We argue that the culprit is the super-LEA which is most prominently expressed in the accessibility of the entropy density profile. Nevertheless, it is observed that there is an inconsistency between diffuse and sharp interface descriptions; they cannot both be correct. The sharp interface framework has been recently tested against molecular dynamics and the obtained results suggest that super-LEA is the potential weakness of the diffuse framework. In this sense, sharp interfaces are found to be superior to diffuse interfaces in their general ability to model physical systems with interfaces.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969004","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 334-368, March 2024. Abstract. In this work, we consider a nonlinear strain-limiting elastic model in heterogeneous domains. We investigate heterogeneous material with soft and stiff inclusions and perforations that are important to understand an elastic solid’s behavior and crack-tip fields. Numerical solutions of problems in computational domains with inclusions and perforations require the construction of a sufficiently fine grid that resolves heterogeneity on the grid level. Approximations on such grids lead to a large system of equations with large computational costs. To reduce the size of the system and provide an accurate solution, we present a generalized multiscale finite element approximation on the coarse grid. In this method, we construct multiscale basis functions in each local domain associated with the coarse-grid cell and based on the construction of the snapshot space and solution of the local spectral problem reduce the size of the snapshot space. Two types of multiscale basis function construction are presented. The first type is a general case that can handle any boundary conditions on the global boundary of the heterogeneous domain. The considered problem requires an accurate approximation of the crack-surface boundary. In the second type of multiscale basis functions, we incorporate global boundary conditions in the basis construction process which provide an accurate approximation of the stress and strain on the crack boundary. We present numerical results for three cases of heterogeneity: soft inclusions, stiff inclusions, and perforations. A numerical investigation is presented for the two examples of loading on the domain with and without crack boundary conditions. The presented generalized multiscale finite element solver provides an accurate solution with a large reduction of the discrete system size. Our results illustrate the significant error reduction on the crack surface when we use the second type of basis functions.
{"title":"Generalized Multiscale Finite Element Treatment of a Heterogeneous Nonlinear Strain-limiting Elastic Model","authors":"Maria Vasilyeva, S. M. Mallikarjunaiah","doi":"10.1137/22m1514179","DOIUrl":"https://doi.org/10.1137/22m1514179","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 334-368, March 2024. <br/> Abstract. In this work, we consider a nonlinear strain-limiting elastic model in heterogeneous domains. We investigate heterogeneous material with soft and stiff inclusions and perforations that are important to understand an elastic solid’s behavior and crack-tip fields. Numerical solutions of problems in computational domains with inclusions and perforations require the construction of a sufficiently fine grid that resolves heterogeneity on the grid level. Approximations on such grids lead to a large system of equations with large computational costs. To reduce the size of the system and provide an accurate solution, we present a generalized multiscale finite element approximation on the coarse grid. In this method, we construct multiscale basis functions in each local domain associated with the coarse-grid cell and based on the construction of the snapshot space and solution of the local spectral problem reduce the size of the snapshot space. Two types of multiscale basis function construction are presented. The first type is a general case that can handle any boundary conditions on the global boundary of the heterogeneous domain. The considered problem requires an accurate approximation of the crack-surface boundary. In the second type of multiscale basis functions, we incorporate global boundary conditions in the basis construction process which provide an accurate approximation of the stress and strain on the crack boundary. We present numerical results for three cases of heterogeneity: soft inclusions, stiff inclusions, and perforations. A numerical investigation is presented for the two examples of loading on the domain with and without crack boundary conditions. The presented generalized multiscale finite element solver provides an accurate solution with a large reduction of the discrete system size. Our results illustrate the significant error reduction on the crack surface when we use the second type of basis functions.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139754959","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}
David C. Freeman, Dimitrios Giannakis, Joanna Slawinska
Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 283-333, March 2024. Abstract. We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes’ law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems and show how this approach preserves important statistical and qualitative properties of the underlying chaotic dynamics.
{"title":"Quantum Mechanics for Closure of Dynamical Systems","authors":"David C. Freeman, Dimitrios Giannakis, Joanna Slawinska","doi":"10.1137/22m1514246","DOIUrl":"https://doi.org/10.1137/22m1514246","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 283-333, March 2024. <br/> Abstract. We propose a scheme for data-driven parameterization of unresolved dimensions of dynamical systems based on the mathematical framework of quantum mechanics and Koopman operator theory. Given a system in which some components of the state are unknown, this method involves defining a surrogate system in a time-dependent quantum state which determines the fluxes from the unresolved degrees of freedom at each timestep. The quantum state is a density operator on a finite-dimensional Hilbert space of classical observables and evolves over time under an action induced by the Koopman operator. The quantum state also updates with new values of the resolved variables according to a quantum Bayes’ law, implemented via an operator-valued feature map. Kernel methods are utilized to learn data-driven basis functions and represent quantum states, observables, and evolution operators as matrices. The resulting computational schemes are automatically positivity-preserving, aiding in the physical consistency of the parameterized system. We analyze the results of two different modalities of this methodology applied to the Lorenz 63 and Lorenz 96 multiscale systems and show how this approach preserves important statistical and qualitative properties of the underlying chaotic dynamics.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139755061","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 256-282, March 2024. Abstract. A generalized finite element method (FEM) is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter [math], based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size [math]. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard finite element (FE) discretizations. Exponential decay rates for local approximation errors with respect to [math] and [math] (at the discrete level with [math] denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to [math] in the standard [math] norm, and that if the oversampling size is relatively large with respect to [math] and [math] (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.
{"title":"Exponential Convergence of a Generalized FEM for Heterogeneous Reaction-Diffusion Equations","authors":"Chupeng Ma, J. M. Melenk","doi":"10.1137/22m1522231","DOIUrl":"https://doi.org/10.1137/22m1522231","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 256-282, March 2024. <br/> Abstract. A generalized finite element method (FEM) is proposed for solving a heterogeneous reaction-diffusion equation with a singular perturbation parameter [math], based on locally approximating the solution on each subdomain by solution of a local reaction-diffusion equation and eigenfunctions of a local eigenproblem. These local problems are posed on some domains slightly larger than the subdomains with oversampling size [math]. The method is formulated at the continuous level as a direct discretization of the continuous problem and at the discrete level as a coarse-space approximation for its standard finite element (FE) discretizations. Exponential decay rates for local approximation errors with respect to [math] and [math] (at the discrete level with [math] denoting the fine FE mesh size) and with the local degrees of freedom are established. In particular, it is shown that the method at the continuous level converges uniformly with respect to [math] in the standard [math] norm, and that if the oversampling size is relatively large with respect to [math] and [math] (at the discrete level), the solutions of the local reaction-diffusion equations provide good local approximations for the solution and thus the local eigenfunctions are not needed. Numerical results are provided to verify the theoretical results.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139755062","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 230-255, March 2024. Abstract. Sea ice profoundly influences the polar environment and the global climate. Traditionally, sea ice has been modeled as a continuum under Eulerian coordinates to describe its large-scale features, using, for instance, viscous-plastic rheology. Recently, Lagrangian particle models, also known as the discrete element method models, have been utilized for characterizing the motion of individual sea ice fragments (called floes) at scales of 10 km and smaller, especially in marginal ice zones. This paper develops a multiscale model that couples the particle and the continuum systems to facilitate an effective representation of the dynamical and statistical features of sea ice across different scales. The multiscale model exploits a Boltzmann-type system that links the particle movement with the continuum equations. For the small-scale dynamics, it describes the motion of each sea ice floe. Then, as the large-scale continuum component, it treats the statistical moments of mass density and linear and angular velocities. The evolution of these statistics affects the motion of individual floes, which in turn provides bulk feedback that adjusts the large-scale dynamics. Notably, the particle model characterizing the sea ice floes is localized and fully parallelized in a framework that is sometimes called superparameterization, which significantly improves computational efficiency. Numerical examples demonstrate the effective performance of the multiscale model. Additionally, the study demonstrates that the multiscale model has a linear-order approximation to the truth model.
{"title":"Particle-Continuum Multiscale Modeling of Sea Ice Floes","authors":"Quanling Deng, Samuel N. Stechmann, Nan Chen","doi":"10.1137/23m155904x","DOIUrl":"https://doi.org/10.1137/23m155904x","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 230-255, March 2024. <br/> Abstract. Sea ice profoundly influences the polar environment and the global climate. Traditionally, sea ice has been modeled as a continuum under Eulerian coordinates to describe its large-scale features, using, for instance, viscous-plastic rheology. Recently, Lagrangian particle models, also known as the discrete element method models, have been utilized for characterizing the motion of individual sea ice fragments (called floes) at scales of 10 km and smaller, especially in marginal ice zones. This paper develops a multiscale model that couples the particle and the continuum systems to facilitate an effective representation of the dynamical and statistical features of sea ice across different scales. The multiscale model exploits a Boltzmann-type system that links the particle movement with the continuum equations. For the small-scale dynamics, it describes the motion of each sea ice floe. Then, as the large-scale continuum component, it treats the statistical moments of mass density and linear and angular velocities. The evolution of these statistics affects the motion of individual floes, which in turn provides bulk feedback that adjusts the large-scale dynamics. Notably, the particle model characterizing the sea ice floes is localized and fully parallelized in a framework that is sometimes called superparameterization, which significantly improves computational efficiency. Numerical examples demonstrate the effective performance of the multiscale model. Additionally, the study demonstrates that the multiscale model has a linear-order approximation to the truth model.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139584447","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 204-229, March 2024. Abstract. A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse-scale representation of the elliptic operator, enriched by fine-scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.
{"title":"Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation","authors":"Annika Lang, Per Ljung, Axel Målqvist","doi":"10.1137/23m1569216","DOIUrl":"https://doi.org/10.1137/23m1569216","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 204-229, March 2024. <br/> Abstract. A multiscale method is proposed for a parabolic stochastic partial differential equation with additive noise and highly oscillatory diffusion. The framework is based on the localized orthogonal decomposition (LOD) method and computes a coarse-scale representation of the elliptic operator, enriched by fine-scale information on the diffusion. Optimal order strong convergence is derived. The LOD technique is combined with a (multilevel) Monte Carlo estimator and the weak error is analyzed. Numerical examples that confirm the theoretical findings are provided, and the computational efficiency of the method is highlighted.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496056","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 155-177, March 2024. Abstract. In this paper, we study two-grid preconditioners for three-dimensional single-phase nonlinear compressible flow in highly heterogeneous porous media arising from reservoir simulation. Our goal is to develop robust and efficient preconditioners that converge independently of the contrast of the media and types of boundary conditions and source term. This is accomplished by constructing coarse space that can capture important features of the local heterogeneous media. To detect these features, local eigenvalue problems are defined and eigenvectors are adaptively selected to form the coarse space. The coarse space just needs to be constructed only once with parallel computing, although the compressible flow is a time-dependent problem and the permeability field changes in different time steps. Smoothers such as Gauss–Seidel iteration and ILU(0) are used to remove high-frequency errors. We analyze this preconditioner by proving the smoothing property and approximation property. In particular, a new coarse interpolation operator is defined to facilitate the analysis. Extensive numerical experiments with different types of large-scale heterogeneous permeability fields and boundary conditions are provided to show the impressive performance of the proposed preconditioner.
{"title":"An Adaptive Preconditioner for Three-Dimensional Single-Phase Compressible Flow in Highly Heterogeneous Porous Media","authors":"Shubin Fu, Eric Chung, Lina Zhao","doi":"10.1137/22m1529075","DOIUrl":"https://doi.org/10.1137/22m1529075","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 155-177, March 2024. <br/> Abstract. In this paper, we study two-grid preconditioners for three-dimensional single-phase nonlinear compressible flow in highly heterogeneous porous media arising from reservoir simulation. Our goal is to develop robust and efficient preconditioners that converge independently of the contrast of the media and types of boundary conditions and source term. This is accomplished by constructing coarse space that can capture important features of the local heterogeneous media. To detect these features, local eigenvalue problems are defined and eigenvectors are adaptively selected to form the coarse space. The coarse space just needs to be constructed only once with parallel computing, although the compressible flow is a time-dependent problem and the permeability field changes in different time steps. Smoothers such as Gauss–Seidel iteration and ILU(0) are used to remove high-frequency errors. We analyze this preconditioner by proving the smoothing property and approximation property. In particular, a new coarse interpolation operator is defined to facilitate the analysis. Extensive numerical experiments with different types of large-scale heterogeneous permeability fields and boundary conditions are provided to show the impressive performance of the proposed preconditioner.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496078","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 178-203, March 2024. Abstract. The distinct timescales of synaptic plasticity and neural activity dynamics play an important role in the brain’s learning and memory systems. Activity-dependent plasticity reshapes neural circuit architecture, determining spontaneous and stimulus-encoding spatiotemporal patterns of neural activity. Neural activity bumps maintain short term memories of continuous parameter values, emerging in spatially organized models with short-range excitation and long-range inhibition. Previously, we demonstrated nonlinear Langevin equations derived using an interface method which accurately describe the dynamics of bumps in continuum neural fields with separate excitatory/inhibitory populations. Here we extend this analysis to incorporate effects of short term plasticity that dynamically modifies connectivity described by an integral kernel. Linear stability analysis adapted to these piecewise smooth models with Heaviside firing rates further indicates how plasticity shapes the bumps’ local dynamics. Facilitation (depression), which strengthens (weakens) synaptic connectivity originating from active neurons, tends to increase (decrease) stability of bumps when acting on excitatory synapses. The relationship is inverted when plasticity acts on inhibitory synapses. Multiscale approximations of the stochastic dynamics of bumps perturbed by weak noise reveal that the plasticity variables evolve to slowly diffusing and blurred versions of their stationary profiles. Nonlinear Langevin equations associated with bump positions or interfaces coupled to slowly evolving projections of plasticity variables accurately describe how these smoothed synaptic efficacy profiles can tether or repel wandering bumps.
{"title":"Multiscale Motion and Deformation of Bumps in Stochastic Neural Fields with Dynamic Connectivity","authors":"Heather L. Cihak, Zachary P. Kilpatrick","doi":"10.1137/23m1582655","DOIUrl":"https://doi.org/10.1137/23m1582655","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 178-203, March 2024. <br/> Abstract. The distinct timescales of synaptic plasticity and neural activity dynamics play an important role in the brain’s learning and memory systems. Activity-dependent plasticity reshapes neural circuit architecture, determining spontaneous and stimulus-encoding spatiotemporal patterns of neural activity. Neural activity bumps maintain short term memories of continuous parameter values, emerging in spatially organized models with short-range excitation and long-range inhibition. Previously, we demonstrated nonlinear Langevin equations derived using an interface method which accurately describe the dynamics of bumps in continuum neural fields with separate excitatory/inhibitory populations. Here we extend this analysis to incorporate effects of short term plasticity that dynamically modifies connectivity described by an integral kernel. Linear stability analysis adapted to these piecewise smooth models with Heaviside firing rates further indicates how plasticity shapes the bumps’ local dynamics. Facilitation (depression), which strengthens (weakens) synaptic connectivity originating from active neurons, tends to increase (decrease) stability of bumps when acting on excitatory synapses. The relationship is inverted when plasticity acts on inhibitory synapses. Multiscale approximations of the stochastic dynamics of bumps perturbed by weak noise reveal that the plasticity variables evolve to slowly diffusing and blurred versions of their stationary profiles. Nonlinear Langevin equations associated with bump positions or interfaces coupled to slowly evolving projections of plasticity variables accurately describe how these smoothed synaptic efficacy profiles can tether or repel wandering bumps.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"823 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139496059","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 125-141, March 2024. Abstract. Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows rigorous estimation of the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection.
{"title":"Metadynamics for Transition Paths in Irreversible Dynamics","authors":"Tobias Grafke, Alessandro Laio","doi":"10.1137/23m1563025","DOIUrl":"https://doi.org/10.1137/23m1563025","url":null,"abstract":"Multiscale Modeling &Simulation, Volume 22, Issue 1, Page 125-141, March 2024. <br/> Abstract. Stochastic systems often exhibit multiple viable metastable states that are long-lived. Over very long timescales, fluctuations may push the system to transition between them, drastically changing its macroscopic configuration. In realistic systems, these transitions can happen via multiple physical mechanisms, corresponding to multiple distinct transition channels for a pair of states. In this paper, we use the fact that the transition path ensemble is equivalent to the invariant measure of a gradient flow in pathspace, which can be efficiently sampled via metadynamics. We demonstrate how this pathspace metadynamics, previously restricted to reversible molecular dynamics, is in fact very generally applicable to metastable stochastic systems, including irreversible and time-dependent ones, and allows rigorous estimation of the relative probability of competing transition paths. We showcase this approach on the study of a stochastic partial differential equation describing magnetic field reversal in the presence of advection.","PeriodicalId":501053,"journal":{"name":"Multiscale Modeling and Simulation","volume":"108 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139459267","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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