This paper concerns the derivation of radiative transfer equations for acoustic waves propagating in a randomly fluctuating half-space in the weak-scattering regime, and the study of boundary effects through an asymptotic analysis of the Wigner transform of the wave solution. These radiative transfer equations allow one to model the transport of wave energy density, taking into account the scattering by random heterogeneities. The approach builds on the method of images, where the half-space problem is extended to a full-space, with two symmetric sources and an even map of mechanical properties. Two contributions to the total energy density are then identified: one similar to the energy density propagation in a full-space, for which the resulting lack of statistical stationarity of the medium properties has no leading-order effect; and one supported within one wavelength of the boundary, which describes interference effects between the waves produced by the two symmetric sources. In the case of a homogeneous Neumann boundary conditions, this boundary effect yields a doubling of the intensity, and in the case of homogeneous Dirichlet boundary conditions, a canceling of that intensity.
{"title":"Boundary Effects in Radiative Transfer of Acoustic Waves in a Randomly Fluctuating Half-space","authors":"Adel Messaoudi, Regis Cottereau, Christophe Gomez","doi":"10.1137/22m1537795","DOIUrl":"https://doi.org/10.1137/22m1537795","url":null,"abstract":"This paper concerns the derivation of radiative transfer equations for acoustic waves propagating in a randomly fluctuating half-space in the weak-scattering regime, and the study of boundary effects through an asymptotic analysis of the Wigner transform of the wave solution. These radiative transfer equations allow one to model the transport of wave energy density, taking into account the scattering by random heterogeneities. The approach builds on the method of images, where the half-space problem is extended to a full-space, with two symmetric sources and an even map of mechanical properties. Two contributions to the total energy density are then identified: one similar to the energy density propagation in a full-space, for which the resulting lack of statistical stationarity of the medium properties has no leading-order effect; and one supported within one wavelength of the boundary, which describes interference effects between the waves produced by the two symmetric sources. In the case of a homogeneous Neumann boundary conditions, this boundary effect yields a doubling of the intensity, and in the case of homogeneous Dirichlet boundary conditions, a canceling of that intensity.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135768757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we analyze the narrow capture problem for a single Brownian particle diffusing in a three-dimensional (3D) bounded domain containing a set of small, spherical traps. The boundary surface of each trap is taken to be a semipermeable membrane. That is, the continuous flux across the interface is proportional to an associated jump discontinuity in the probability density. The constant of proportionality is identified with the permeability . In addition, we allow for discontinuities in the diffusivity and chemical potential across each interface; the latter introduces a directional bias. We also assume that the particle can be absorbed (captured) within the interior of each trap at some Poisson rate . In the small-trap limit, we use matched asymptotics and Green’s function methods to calculate the splitting probabilities and unconditional mean first passage time (MFPT) to be absorbed by one of the traps. However, the details of the analysis depend on how various parameters scale with the characteristic trap radius . Under the scalings and , we show that the semipermeable membrane reduces the effective capacitance of each spherical trap compared to the standard example of totally absorbing traps. The latter case is recovered in the dual limits and , with equal to the intrinsic capacitance of a sphere, namely, the radius. We also illustrate how the asymptotic expansions are modified when (slow absorption) or (low permeability). Finally, we consider the unidirectional limit in which each interface only allows particles to flow into a trap. The traps then act as partially absorbing surfaces with a constant reaction rate . Combining asymptotic analysis with the encounter-based formulation of partially reactive surfaces, we show how a generalized surface absorption mechanism (non-Markovian) can be analyzed in terms of the capacitances . We thus establish that a wide range of narrow capture problems can be characterized in terms of the effective capacitances of the traps.
{"title":"The 3D Narrow Capture Problem for Traps with Semipermeable Interfaces","authors":"Paul C. Bressloff","doi":"10.1137/22m1535462","DOIUrl":"https://doi.org/10.1137/22m1535462","url":null,"abstract":"In this paper we analyze the narrow capture problem for a single Brownian particle diffusing in a three-dimensional (3D) bounded domain containing a set of small, spherical traps. The boundary surface of each trap is taken to be a semipermeable membrane. That is, the continuous flux across the interface is proportional to an associated jump discontinuity in the probability density. The constant of proportionality is identified with the permeability . In addition, we allow for discontinuities in the diffusivity and chemical potential across each interface; the latter introduces a directional bias. We also assume that the particle can be absorbed (captured) within the interior of each trap at some Poisson rate . In the small-trap limit, we use matched asymptotics and Green’s function methods to calculate the splitting probabilities and unconditional mean first passage time (MFPT) to be absorbed by one of the traps. However, the details of the analysis depend on how various parameters scale with the characteristic trap radius . Under the scalings and , we show that the semipermeable membrane reduces the effective capacitance of each spherical trap compared to the standard example of totally absorbing traps. The latter case is recovered in the dual limits and , with equal to the intrinsic capacitance of a sphere, namely, the radius. We also illustrate how the asymptotic expansions are modified when (slow absorption) or (low permeability). Finally, we consider the unidirectional limit in which each interface only allows particles to flow into a trap. The traps then act as partially absorbing surfaces with a constant reaction rate . Combining asymptotic analysis with the encounter-based formulation of partially reactive surfaces, we show how a generalized surface absorption mechanism (non-Markovian) can be analyzed in terms of the capacitances . We thus establish that a wide range of narrow capture problems can be characterized in terms of the effective capacitances of the traps.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136060861","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This work is devoted to the construction of multiscale numerical schemes efficient in the finite Larmor radius approximation of the collisional Vlasov equation. Following the paper of Bostan and Finot [Commun. Contemp. Math., 22 (2020), 1950047], the system involves two different regimes, a highly oscillatory and a dissipative regime, whose asymptotic limits do not commute. In this work, we consider a Particle-in-Cell discretization of the collisional Vlasov system which enables us to deal with the multiscale characteristics equations. Different multiscale time integrators are then constructed and analyzed. We prove asymptotic properties of these schemes in the highly oscillatory regime and in the collisional regime. In particular, the asymptotic preserving property towards the modified equilibrium of the averaged collision operator is recovered. Numerical experiments are then shown to illustrate the properties of the numerical schemes.
本文研究了碰撞Vlasov方程的有限Larmor半径近似的多尺度数值格式的构造。继Bostan and Finot (common)的文件之后。一栏。数学。[j]， 22(2020)， 1950047]，系统涉及两个不同的状态，一个高振荡状态和一个耗散状态，其渐近极限不交换。在这项工作中，我们考虑了碰撞Vlasov系统的单元内粒子离散化，使我们能够处理多尺度特征方程。然后构造并分析了不同的多尺度时间积分器。我们证明了这些格式在高振荡区和碰撞区的渐近性质。特别地，恢复了平均碰撞算子对修正平衡的渐近保持性质。数值实验说明了数值格式的性质。
{"title":"Multiscale Numerical Schemes for the Collisional Vlasov Equation in the Finite Larmor Radius Approximation Regime","authors":"Anaïs Crestetto, Nicolas Crouseilles, Damien Prel","doi":"10.1137/22m1496839","DOIUrl":"https://doi.org/10.1137/22m1496839","url":null,"abstract":"This work is devoted to the construction of multiscale numerical schemes efficient in the finite Larmor radius approximation of the collisional Vlasov equation. Following the paper of Bostan and Finot [Commun. Contemp. Math., 22 (2020), 1950047], the system involves two different regimes, a highly oscillatory and a dissipative regime, whose asymptotic limits do not commute. In this work, we consider a Particle-in-Cell discretization of the collisional Vlasov system which enables us to deal with the multiscale characteristics equations. Different multiscale time integrators are then constructed and analyzed. We prove asymptotic properties of these schemes in the highly oscillatory regime and in the collisional regime. In particular, the asymptotic preserving property towards the modified equilibrium of the averaged collision operator is recovered. Numerical experiments are then shown to illustrate the properties of the numerical schemes.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135488356","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ellis R. Crabtree, Juan M. Bello-Rivas, Andrew L. Ferguson, Ioannis G. Kevrekidis
Sampling the phase space of molecular systems—and, more generally, of complex systems effectively modeled by stochastic differential equations (SDEs)—is a crucial modeling step in many fields, from protein folding to materials discovery. These problems are often multiscale in nature: they can be described in terms of low-dimensional effective free energy surfaces parametrized by a small number of “slow” reaction coordinates; the remaining “fast” degrees of freedom populate an equilibrium measure conditioned on the reaction coordinate values. Sampling procedures for such problems are used to estimate effective free energy differences as well as ensemble averages with respect to the conditional equilibrium distributions; these latter averages lead to closures for effective reduced dynamic models. Over the years, enhanced sampling techniques coupled with molecular simulation have been developed; they often use knowledge of the system order parameters in order to sample the corresponding conditional equilibrium distributions, and estimate ensemble averages of observables. An intriguing analogy arises with the field of machine learning (ML), where generative adversarial networks (GANs) can produce high-dimensional samples from low-dimensional probability distributions. This sample generation is what in equation-free multiscale modeling is called a “lifting process”: it returns plausible (or realistic) high-dimensional space realizations of a model state, from information about its low-dimensional representation. In this work, we elaborate on this analogy, and we present an approach that couples physics-based simulations and biasing methods for sampling conditional distributions with ML-based conditional generative adversarial networks (cGANs) for the same task. The “coarse descriptors” on which we condition the fine scale realizations can either be known a priori or learned through nonlinear dimensionality reduction (here, using diffusion maps). We suggest that this may bring out the best features of both approaches: we demonstrate that a framework that couples cGANs with physics-based enhanced sampling techniques can improve multiscale SDE dynamical systems sampling, and even shows promise for systems of increasing complexity (here, simple molecules).
{"title":"GANs and Closures: Micro-Macro Consistency in Multiscale Modeling","authors":"Ellis R. Crabtree, Juan M. Bello-Rivas, Andrew L. Ferguson, Ioannis G. Kevrekidis","doi":"10.1137/22m1517834","DOIUrl":"https://doi.org/10.1137/22m1517834","url":null,"abstract":"Sampling the phase space of molecular systems—and, more generally, of complex systems effectively modeled by stochastic differential equations (SDEs)—is a crucial modeling step in many fields, from protein folding to materials discovery. These problems are often multiscale in nature: they can be described in terms of low-dimensional effective free energy surfaces parametrized by a small number of “slow” reaction coordinates; the remaining “fast” degrees of freedom populate an equilibrium measure conditioned on the reaction coordinate values. Sampling procedures for such problems are used to estimate effective free energy differences as well as ensemble averages with respect to the conditional equilibrium distributions; these latter averages lead to closures for effective reduced dynamic models. Over the years, enhanced sampling techniques coupled with molecular simulation have been developed; they often use knowledge of the system order parameters in order to sample the corresponding conditional equilibrium distributions, and estimate ensemble averages of observables. An intriguing analogy arises with the field of machine learning (ML), where generative adversarial networks (GANs) can produce high-dimensional samples from low-dimensional probability distributions. This sample generation is what in equation-free multiscale modeling is called a “lifting process”: it returns plausible (or realistic) high-dimensional space realizations of a model state, from information about its low-dimensional representation. In this work, we elaborate on this analogy, and we present an approach that couples physics-based simulations and biasing methods for sampling conditional distributions with ML-based conditional generative adversarial networks (cGANs) for the same task. The “coarse descriptors” on which we condition the fine scale realizations can either be known a priori or learned through nonlinear dimensionality reduction (here, using diffusion maps). We suggest that this may bring out the best features of both approaches: we demonstrate that a framework that couples cGANs with physics-based enhanced sampling techniques can improve multiscale SDE dynamical systems sampling, and even shows promise for systems of increasing complexity (here, simple molecules).","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135215678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equations. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then a constraint energy minimization performed in the oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layers is chosen properly, we can prove that the convergence rate is proportional to the coarse meshsize. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performance of our method.
{"title":"Constraint Energy Minimizing Generalized Multiscale Finite Element Method for Convection Diffusion Equation","authors":"Lina Zhao, Eric Chung","doi":"10.1137/22m1487655","DOIUrl":"https://doi.org/10.1137/22m1487655","url":null,"abstract":"In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equations. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then a constraint energy minimization performed in the oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layers is chosen properly, we can prove that the convergence rate is proportional to the coarse meshsize. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performance of our method.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135702953","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The scaling of the mobility of two-dimensional Langevin dynamics in a periodic potential as the friction vanishes is not well understood for nonseparable potentials. Theoretical results are lacking, and numerical calculation of the mobility in the underdamped regime is challenging because the computational cost of standard Monte Carlo methods is inversely proportional to the friction coefficient, while deterministic methods are ill-conditioned. In this work, we propose a new variance-reduction method based on control variates for efficiently estimating the mobility of Langevin-type dynamics. We provide bounds on the bias and variance of the proposed estimator and illustrate its efficacy through numerical experiments, first in simple one-dimensional settings and then for two-dimensional Langevin dynamics. Our results corroborate prior numerical evidence that the mobility scales as , with , in the low friction regime for a simple nonseparable potential.
{"title":"Mobility Estimation for Langevin Dynamics Using Control Variates","authors":"Grigorios A. Pavliotis, G. Stoltz, Urbain Vaes","doi":"10.1137/22m1504378","DOIUrl":"https://doi.org/10.1137/22m1504378","url":null,"abstract":"The scaling of the mobility of two-dimensional Langevin dynamics in a periodic potential as the friction vanishes is not well understood for nonseparable potentials. Theoretical results are lacking, and numerical calculation of the mobility in the underdamped regime is challenging because the computational cost of standard Monte Carlo methods is inversely proportional to the friction coefficient, while deterministic methods are ill-conditioned. In this work, we propose a new variance-reduction method based on control variates for efficiently estimating the mobility of Langevin-type dynamics. We provide bounds on the bias and variance of the proposed estimator and illustrate its efficacy through numerical experiments, first in simple one-dimensional settings and then for two-dimensional Langevin dynamics. Our results corroborate prior numerical evidence that the mobility scales as , with , in the low friction regime for a simple nonseparable potential.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135219533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a neural network-based approach to the homogenization of multiscale problems. The proposed method uses a derivative-free formulation of a training loss, which incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution. Compared with other network-based approaches for multiscale problems, the proposed method is free from the design of hand-crafted neural network architecture and the cell problem to calculate the homogenization coefficient. The exploration neighborhood of the Brownian walkers affects the overall learning trajectory. We determine the bounds of micro- and macro-time steps that capture the local heterogeneous and global homogeneous solution behaviors, respectively, through a neural network. The bounds imply that the computational cost of the proposed method is independent of the microscale periodic structure for the standard periodic problems. We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems with periodic and random field coefficients.
{"title":"A Neural Network Approach for Homogenization of Multiscale Problems","authors":"Jihun Han, Yoonsang Lee","doi":"10.1137/22m1500903","DOIUrl":"https://doi.org/10.1137/22m1500903","url":null,"abstract":"We propose a neural network-based approach to the homogenization of multiscale problems. The proposed method uses a derivative-free formulation of a training loss, which incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution. Compared with other network-based approaches for multiscale problems, the proposed method is free from the design of hand-crafted neural network architecture and the cell problem to calculate the homogenization coefficient. The exploration neighborhood of the Brownian walkers affects the overall learning trajectory. We determine the bounds of micro- and macro-time steps that capture the local heterogeneous and global homogeneous solution behaviors, respectively, through a neural network. The bounds imply that the computational cost of the proposed method is independent of the microscale periodic structure for the standard periodic problems. We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems with periodic and random field coefficients.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135721150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quasi-Convergence of an Implementation of Optimal Balance by Backward-Forward Nudging","authors":"Gökce Tuba Masur, Haidar Mohamad, Marcel Oliver","doi":"10.1137/22m1506018","DOIUrl":"https://doi.org/10.1137/22m1506018","url":null,"abstract":"","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135543822","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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