In this article, we show how to derive a multiphase model of Baer and Nunziato type with a simple stochastic model. Baer and Nunziato models are known to be unclosed, namely, they depend on modelin...
{"title":"Derivation and Closure of Baer and Nunziato Type Multiphase Models by Averaging a Simple Stochastic Model","authors":"V. Perrier, Enrique Gutiérrez","doi":"10.1137/19M1306609","DOIUrl":"https://doi.org/10.1137/19M1306609","url":null,"abstract":"In this article, we show how to derive a multiphase model of Baer and Nunziato type with a simple stochastic model. Baer and Nunziato models are known to be unclosed, namely, they depend on modelin...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123790714","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}
Ke-Ming Chen, Zachary Deiman, Ryan N. Goh, S. Jankovic, A. Scheel
We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including non-adiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift Hohenberg equation.
{"title":"Strain and Defects in Oblique Stripe Growth","authors":"Ke-Ming Chen, Zachary Deiman, Ryan N. Goh, S. Jankovic, A. Scheel","doi":"10.1137/21m1397210","DOIUrl":"https://doi.org/10.1137/21m1397210","url":null,"abstract":"We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including non-adiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift Hohenberg equation.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116724067","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}
Discrete-state continuous-time Markov processes are an important class of models employed broadly across the sciences. When the system size becomes large, standard approaches can become intractable to exact solution and numerical simulation. Approximations posed on a continuous state space are often more tractable and are presumed to converge in the limit as the system size tends to infinity. For example, an expansion of the master equation truncated at second order yields the Fokker--Planck equation, a widely used continuum approximation equipped with an underlying process of continuous state. Surprisingly, in [Doering textit{et. al.} Multiscale Model. Sim. 2005 3:2, p.283--299] it is shown that the Fokker--Planck approximation may exhibit exponentially large errors, even in the infinite system-size limit. Crucially, the source of this inaccuracy has not been addressed. In this paper, we focus on the family of continuous-state approximations obtained by arbitrary-order truncations. We uncover how the exponentially large error stems from the truncation by quantifying the rapid error decay with increasing truncation order. Furthermore, we explain why this discrepancy only comes to light in a subset of problems. The approximations produced by finite truncation beyond second order lack underlying stochastic processes. Nevertheless, they retain valuable information that explains the previously observed discrepancy by bridging the gap between the continuous and discrete processes. The insight conferred by this broader notion of ``continuum approximation'', where we do not require an underlying stochastic process, prompts us to revisit previously expressed doubts regarding continuum approximations. In establishing the utility of higher-order truncations, this approach also contributes to the extensive discussion in the literature regarding the second-order truncation: while recognising the appealing features of an associated stochastic process, in certain cases it may be advantageous to dispense of the process in exchange for the increased approximation accuracy guaranteed by higher-order truncations.
{"title":"On Continuum Approximations of Discrete-State Markov Processes of Large System Size","authors":"D. Lunz","doi":"10.1137/20M1332293","DOIUrl":"https://doi.org/10.1137/20M1332293","url":null,"abstract":"Discrete-state continuous-time Markov processes are an important class of models employed broadly across the sciences. When the system size becomes large, standard approaches can become intractable to exact solution and numerical simulation. Approximations posed on a continuous state space are often more tractable and are presumed to converge in the limit as the system size tends to infinity. For example, an expansion of the master equation truncated at second order yields the Fokker--Planck equation, a widely used continuum approximation equipped with an underlying process of continuous state. Surprisingly, in [Doering textit{et. al.} Multiscale Model. Sim. 2005 3:2, p.283--299] it is shown that the Fokker--Planck approximation may exhibit exponentially large errors, even in the infinite system-size limit. Crucially, the source of this inaccuracy has not been addressed. In this paper, we focus on the family of continuous-state approximations obtained by arbitrary-order truncations. We uncover how the exponentially large error stems from the truncation by quantifying the rapid error decay with increasing truncation order. Furthermore, we explain why this discrepancy only comes to light in a subset of problems. The approximations produced by finite truncation beyond second order lack underlying stochastic processes. Nevertheless, they retain valuable information that explains the previously observed discrepancy by bridging the gap between the continuous and discrete processes. The insight conferred by this broader notion of ``continuum approximation'', where we do not require an underlying stochastic process, prompts us to revisit previously expressed doubts regarding continuum approximations. In establishing the utility of higher-order truncations, this approach also contributes to the extensive discussion in the literature regarding the second-order truncation: while recognising the appealing features of an associated stochastic process, in certain cases it may be advantageous to dispense of the process in exchange for the increased approximation accuracy guaranteed by higher-order truncations.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126290427","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}
Beilei Liu, Huajie Chen, Geneviève Dusson, Jun Fang, Xingyu Gao
We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with Kohn–Sham density functional theory (DFT) and the projector augmented wave (PAW) method, illustrating the efficiency and potential of the algorithm.
{"title":"An Adaptive Planewave Method for Electronic Structure Calculations","authors":"Beilei Liu, Huajie Chen, Geneviève Dusson, Jun Fang, Xingyu Gao","doi":"10.1137/21m1396241","DOIUrl":"https://doi.org/10.1137/21m1396241","url":null,"abstract":"We propose an adaptive planewave method for eigenvalue problems in electronic structure calculations. The method combines a priori convergence rates and accurate a posteriori error estimates into an effective way of updating the energy cut-off for planewave discretizations, for both linear and nonlinear eigenvalue problems. The method is error controllable for linear eigenvalue problems in the sense that for a given required accuracy, an energy cut-off for which the solution matches the target accuracy can be reached efficiently. Further, the method is particularly promising for nonlinear eigenvalue problems in electronic structure calculations as it shall reduce the cost of early iterations in self-consistent algorithms. We present some numerical experiments for both linear and nonlinear eigenvalue problems. In particular, we provide electronic structure calculations for some insulator and metallic systems simulated with Kohn–Sham density functional theory (DFT) and the projector augmented wave (PAW) method, illustrating the efficiency and potential of the algorithm.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"222 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127640180","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}
In this paper, we propose and analyze a gradient flow based model for electronic structure calculations. First, based on an extended gradient flow proposed in this paper, we propose a Kohn--Sham gr...
{"title":"Gradient Flow Based Kohn-Sham Density Functional Theory Model","authors":"X. Dai, Qiao Wang, Aihui Zhou","doi":"10.1137/19m1276170","DOIUrl":"https://doi.org/10.1137/19m1276170","url":null,"abstract":"In this paper, we propose and analyze a gradient flow based model for electronic structure calculations. First, based on an extended gradient flow proposed in this paper, we propose a Kohn--Sham gr...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"599 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122361183","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}
This work presents a stochastic multiscale model reduction approach to solve PDEs with random coefficients. An ensemble-based low-rank approximation method is proposed to approximate multiscale bas...
{"title":"A Low-Rank Approximated Multiscale Method for Pdes With Random Coefficients","authors":"Na Ou, G. Lin, Lijian Jiang","doi":"10.1137/19m1288565","DOIUrl":"https://doi.org/10.1137/19m1288565","url":null,"abstract":"This work presents a stochastic multiscale model reduction approach to solve PDEs with random coefficients. An ensemble-based low-rank approximation method is proposed to approximate multiscale bas...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126003896","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}
This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. First, we derive an expression for ...
本文提出了有限温度下不相称双层异质结构久保电导率的统一建模和计算方法。首先，我们推导出…
{"title":"Modeling and Computation of Kubo Conductivity for Two-Dimensional Incommensurate Bilayers","authors":"S. Etter, Daniel Massatt, M. Luskin, C. Ortner","doi":"10.1137/19m1273499","DOIUrl":"https://doi.org/10.1137/19m1273499","url":null,"abstract":"This paper presents a unified approach to the modeling and computation of the Kubo conductivity of incommensurate bilayer heterostructures at finite temperature. First, we derive an expression for ...","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128253463","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}
We introduce an end-to-end deep learning architecture called the wide-band butterfly network (WideBNet) for approximating the inverse scattering map from wide-band scattering data. This architecture incorporates tools from computational harmonic analysis, such as the butterfly factorization, and traditional multi-scale methods, such as the Cooley-Tukey FFT algorithm, to drastically reduce the number of trainable parameters to match the inherent complexity of the problem. As a result WideBNet is efficient: it requires fewer training points than off-the-shelf architectures, and has stable training dynamics, thus it can rely on standard weight initialization strategies. The architecture automatically adapts to the dimensions of the data with only a few hyper-parameters that the user must specify. WideBNet is able to produce images that are competitive with optimization-based approaches, but at a fraction of the cost, and we also demonstrate numerically that it learns to super-resolve scatterers in the full aperture scattering setup.
{"title":"Wide-band butterfly network: stable and efficient inversion via multi-frequency neural networks","authors":"Matthew Li, L. Demanet, Leonardo Zepeda-N'unez","doi":"10.1137/20m1383276","DOIUrl":"https://doi.org/10.1137/20m1383276","url":null,"abstract":"We introduce an end-to-end deep learning architecture called the wide-band butterfly network (WideBNet) for approximating the inverse scattering map from wide-band scattering data. This architecture incorporates tools from computational harmonic analysis, such as the butterfly factorization, and traditional multi-scale methods, such as the Cooley-Tukey FFT algorithm, to drastically reduce the number of trainable parameters to match the inherent complexity of the problem. As a result WideBNet is efficient: it requires fewer training points than off-the-shelf architectures, and has stable training dynamics, thus it can rely on standard weight initialization strategies. The architecture automatically adapts to the dimensions of the data with only a few hyper-parameters that the user must specify. WideBNet is able to produce images that are competitive with optimization-based approaches, but at a fraction of the cost, and we also demonstrate numerically that it learns to super-resolve scatterers in the full aperture scattering setup.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"20 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130572968","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}
The computation of free energies is a common issue in statistical physics. A natural technique to compute such high dimensional integrals is to resort to Monte Carlo simulations. However these techniques generally suffer from a high variance in the low temperature regime, because the expectation is dominated by high values corresponding to rare system trajectories. A standard way to reduce the variance of the estimator is to modify the drift of the dynamics with a control enhancing the probability of rare event, leading to so-called importance sampling estimators. In theory, the optimal control leads to a zero-variance estimator; it is however defined implicitly and computing it is of the same difficulty as the original problem. We propose here a general strategy to build approximate optimal controls, with the first goal to reduce the variance of free energy Monte Carlo estimators. Our construction builds upon low noise asymptotics by expanding the optimal control around the instanton, which is the path describing most likely fluctuations at low temperature. This technique not only helps reducing variance, but it is also interesting as a theoretical tool since it differs from usual small temperature expansions (WKB ansatz). As a complementary consequence of our expansion, we provide a perturbative formula for computing the free energy in the small temperature regime, which refines the now standard Freidlin-Wentzell asymptotics. We compute this expansion explicitly for lower orders, and explain how our strategy can be extended to an arbitrary order of accuracy. We support our findings with illustrative numerical examples.
{"title":"Approximate Optimal Controls via Instanton Expansion for Low Temperature Free Energy Computation","authors":"Gr'egoire Ferr'e, T. Grafke","doi":"10.1137/20m1385809","DOIUrl":"https://doi.org/10.1137/20m1385809","url":null,"abstract":"The computation of free energies is a common issue in statistical physics. A natural technique to compute such high dimensional integrals is to resort to Monte Carlo simulations. However these techniques generally suffer from a high variance in the low temperature regime, because the expectation is dominated by high values corresponding to rare system trajectories. A standard way to reduce the variance of the estimator is to modify the drift of the dynamics with a control enhancing the probability of rare event, leading to so-called importance sampling estimators. In theory, the optimal control leads to a zero-variance estimator; it is however defined implicitly and computing it is of the same difficulty as the original problem. We propose here a general strategy to build approximate optimal controls, with the first goal to reduce the variance of free energy Monte Carlo estimators. Our construction builds upon low noise asymptotics by expanding the optimal control around the instanton, which is the path describing most likely fluctuations at low temperature. This technique not only helps reducing variance, but it is also interesting as a theoretical tool since it differs from usual small temperature expansions (WKB ansatz). As a complementary consequence of our expansion, we provide a perturbative formula for computing the free energy in the small temperature regime, which refines the now standard Freidlin-Wentzell asymptotics. We compute this expansion explicitly for lower orders, and explain how our strategy can be extended to an arbitrary order of accuracy. We support our findings with illustrative numerical examples.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130596781","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}
We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than the size of the search domain. The small target assumption allows us to use matched asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in powers of the non-dimensionalized target size $epsilon$. One major advantage of working directly with fluxes is that one can generate statistical quantities such as splitting probabilities and conditional first passage time moments without having to solve a separate boundary value problem in each case. However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate Green's function singularities that arise in the limit $srightarrow 0$, where $s$ is the Laplace variable. We achieve this by considering a triple expansion in $epsilon$, $s$ and $Lambdasim epsilon /s$. This allows us to perform partial summations over infinite power series in $Lambda$, which leads to multiplicative factors of the form $Lambda^n/(1+Lambda)^n $. Since $Lambda^n/(1+Lambda)^n rightarrow 1$ as $srightarrow 0$, the singularities in $s$ are eliminated. We then show how corresponding asymptotic expansions of the splitting probabilities and conditional MFPTs can be derived in the small-$s$ limit. Finally, we illustrate the theory by considering a pair of targets in a spherical search domain, for which the Green's functions can be calculated explicitly.
{"title":"Asymptotic Analysis of Target Fluxes in the Three-Dimensional Narrow Capture Problem","authors":"P. Bressloff","doi":"10.1137/20M1380326","DOIUrl":"https://doi.org/10.1137/20M1380326","url":null,"abstract":"We develop an asymptotic analysis of target fluxes in the three-dimensional (3D) narrow capture problem. The latter concerns a diffusive search process in which the targets are much smaller than the size of the search domain. The small target assumption allows us to use matched asymptotic expansions and Green's functions to solve the diffusion equation in Laplace space. In particular, we derive an asymptotic expansion of the Laplace transformed flux into each target in powers of the non-dimensionalized target size $epsilon$. One major advantage of working directly with fluxes is that one can generate statistical quantities such as splitting probabilities and conditional first passage time moments without having to solve a separate boundary value problem in each case. However, in order to derive asymptotic expansions of these quantities, it is necessary to eliminate Green's function singularities that arise in the limit $srightarrow 0$, where $s$ is the Laplace variable. We achieve this by considering a triple expansion in $epsilon$, $s$ and $Lambdasim epsilon /s$. This allows us to perform partial summations over infinite power series in $Lambda$, which leads to multiplicative factors of the form $Lambda^n/(1+Lambda)^n $. Since $Lambda^n/(1+Lambda)^n rightarrow 1$ as $srightarrow 0$, the singularities in $s$ are eliminated. We then show how corresponding asymptotic expansions of the splitting probabilities and conditional MFPTs can be derived in the small-$s$ limit. Finally, we illustrate the theory by considering a pair of targets in a spherical search domain, for which the Green's functions can be calculated explicitly.","PeriodicalId":313703,"journal":{"name":"Multiscale Model. Simul.","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130840785","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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