Yaming Zhang, Ning Jiang, Jiangyan Liang, Yi-Long Luo, Min Tang
We investigate the linear stability analysis of a pathway-based diffusion model (PBDM), which characterizes the dynamics of the engineered Escherichia coli populations [X. Xue, C. Xue, and M. Tang, PLoS Comput. Biol., 14 (2018), e1006178]. This stability analysis considers small perturbations of the density and chemical concentration around two nontrivial steady states, and the linearized equations are transformed into a generalized eigenvalue problem. By formal analysis, when the internal variable responds to the outside signal fast enough, the PBDM converges to an anisotropic diffusion model, for which the probability density distribution in the internal variable becomes a delta function. We introduce an asymptotic preserving (AP) scheme for the PBDM that converges to a stable limit scheme consistent with the anisotropic diffusion model. Further numerical simulations demonstrate the theoretical results of linear stability analysis, i.e., the pattern formation, and the convergence of the AP scheme.
{"title":"Pattern Formation of a Pathway-Based Diffusion Model: Linear Stability Analysis and an Asymptotic Preserving Method","authors":"Yaming Zhang, Ning Jiang, Jiangyan Liang, Yi-Long Luo, Min Tang","doi":"10.1137/22m1490958","DOIUrl":"https://doi.org/10.1137/22m1490958","url":null,"abstract":"We investigate the linear stability analysis of a pathway-based diffusion model (PBDM), which characterizes the dynamics of the engineered Escherichia coli populations [X. Xue, C. Xue, and M. Tang, PLoS Comput. Biol., 14 (2018), e1006178]. This stability analysis considers small perturbations of the density and chemical concentration around two nontrivial steady states, and the linearized equations are transformed into a generalized eigenvalue problem. By formal analysis, when the internal variable responds to the outside signal fast enough, the PBDM converges to an anisotropic diffusion model, for which the probability density distribution in the internal variable becomes a delta function. We introduce an asymptotic preserving (AP) scheme for the PBDM that converges to a stable limit scheme consistent with the anisotropic diffusion model. Further numerical simulations demonstrate the theoretical results of linear stability analysis, i.e., the pattern formation, and the convergence of the AP scheme.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135598704","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}
Yang Liu, Jian Song, Robert Burridge, Jianliang Qian
We present a butterfly-compressed representation of the Hadamard–Babich (HB) ansatz for the Green’s function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all cell centers to each other. The overall CPU time and memory requirement scale as for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency domain methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-of-the-art supercomputer at Lawrence Berkeley National Laboratory.
{"title":"A Fast Butterfly-Compressed Hadamard–Babich Integrator for High-Frequency Helmholtz Equations in Inhomogeneous Media with Arbitrary Sources","authors":"Yang Liu, Jian Song, Robert Burridge, Jianliang Qian","doi":"10.1137/21m1450422","DOIUrl":"https://doi.org/10.1137/21m1450422","url":null,"abstract":"We present a butterfly-compressed representation of the Hadamard–Babich (HB) ansatz for the Green’s function of the high-frequency Helmholtz equation in smooth inhomogeneous media. For a computational domain discretized with discretization cells, the proposed algorithm first solves and tabulates the phase and HB coefficients via eikonal and transport equations with observation points and point sources located at the Chebyshev nodes using a set of much coarser computation grids, and then butterfly compresses the resulting HB interactions from all cell centers to each other. The overall CPU time and memory requirement scale as for any bounded two-dimensional (2D) domains with arbitrary excitation sources. A direct extension of this scheme to bounded 3D domains yields an CPU complexity, which can be further reduced to quasi-linear complexities with proposed remedies. The scheme can also efficiently handle scattering problems involving inclusions in inhomogeneous media. Although the current construction of our HB integrator does not accommodate caustics, the resulting HB integrator itself can be applied to certain sources, such as concave-shaped sources, to produce caustic effects. Compared to finite-difference frequency domain methods, the proposed HB integrator is free of numerical dispersion and requires fewer discretization points per wavelength. As a result, it can solve wave propagation problems well beyond the capability of existing solvers. Remarkably, the proposed scheme can accurately model wave propagation in 2D domains with 640 wavelengths per direction and in 3D domains with 54 wavelengths per direction on a state-of-the-art supercomputer at Lawrence Berkeley National Laboratory.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136080825","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 analyze the solutions of the Schroźdinger equation with low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that the dynamics generates a nontrivial energy in the high frequencies, which does not homogenize---the high frequency component of the wave field remains random and the evolution of its energy is described by a kinetic equation. The transition from the homogenization of the low frequencies to the random limit of the high frequencies is illustrated by understanding the size of the small random fluctuations of the low frequency component.
{"title":"The Random Schroźdinger Equation","authors":"Yu Gu, L. Ryzhik","doi":"10.1137/15M1024986","DOIUrl":"https://doi.org/10.1137/15M1024986","url":null,"abstract":"We analyze the solutions of the Schroźdinger equation with low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that the dynamics generates a nontrivial energy in the high frequencies, which does not homogenize---the high frequency component of the wave field remains random and the evolution of its energy is described by a kinetic equation. The transition from the homogenization of the low frequencies to the random limit of the high frequencies is illustrated by understanding the size of the small random fluctuations of the low frequency component.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"14 1","pages":"323-363"},"PeriodicalIF":1.6,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1137/15M1024986","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64309980","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}
A phase space description of Schroźdinger dynamics is provided in terms of a quantum kinetic formalism relying on the introduction of an appropriate extension of the well-known Wigner transform, also accounting for time delocalizations. This “space-time Wigner distribution,” built up in the framework of two-time correlation functions, is shown to be governed by a non-Markovian, integro-differential equation of convolution type. Its utility in investigating long time dynamics of quantum systems is also discussed and illustrated with some examples.
{"title":"A Non-Markovian Phase Space Approach to Schroźdinger Dynamics","authors":"Joseź Luis Loźpez, J. Soler","doi":"10.1137/15M101899X","DOIUrl":"https://doi.org/10.1137/15M101899X","url":null,"abstract":"A phase space description of Schroźdinger dynamics is provided in terms of a quantum kinetic formalism relying on the introduction of an appropriate extension of the well-known Wigner transform, also accounting for time delocalizations. This “space-time Wigner distribution,” built up in the framework of two-time correlation functions, is shown to be governed by a non-Markovian, integro-differential equation of convolution type. Its utility in investigating long time dynamics of quantum systems is also discussed and illustrated with some examples.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"14 1","pages":"430-451"},"PeriodicalIF":1.6,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1137/15M101899X","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64309810","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}
Pub Date : 2012-12-11DOI: 10.2140/camcos.2014.9.47
A. Donev, A. Nonaka, Yifei Sun, T. Fai, Alejandro L. Garcia, J. Bell
We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatio-temporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplifications such as neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.
{"title":"Low Mach Number Fluctuating Hydrodynamics of Diffusively Mixing Fluids","authors":"A. Donev, A. Nonaka, Yifei Sun, T. Fai, Alejandro L. Garcia, J. Bell","doi":"10.2140/camcos.2014.9.47","DOIUrl":"https://doi.org/10.2140/camcos.2014.9.47","url":null,"abstract":"We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatio-temporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplifications such as neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2012-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2140/camcos.2014.9.47","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68005154","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}
Pub Date : 2012-01-01DOI: 10.1007/978-3-540-70529-1_70
Roeland M. H. Merks, B. Enquist
{"title":"Cell-Based Modeling","authors":"Roeland M. H. Merks, B. Enquist","doi":"10.1007/978-3-540-70529-1_70","DOIUrl":"https://doi.org/10.1007/978-3-540-70529-1_70","url":null,"abstract":"","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2012-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/978-3-540-70529-1_70","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51061334","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 introduce a new locally adaptive wavelet transform, called easy path wavelet transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of two-dimensional data.
{"title":"The Easy Path Wavelet Transform: A New Adaptive Wavelet Transform for Sparse Representation of Two-Dimensional Data","authors":"G. Plonka","doi":"10.1137/080719248","DOIUrl":"https://doi.org/10.1137/080719248","url":null,"abstract":"We introduce a new locally adaptive wavelet transform, called easy path wavelet transform (EPWT), that works along pathways through the array of function values and exploits the local correlations of the data in a simple appropriate manner. The usual discrete orthogonal and biorthogonal wavelet transform can be formulated in this approach. The EPWT can be incorporated into a multiresolution analysis structure and generates data dependent scaling spaces and wavelet spaces. Numerical results show the enormous efficiency of the EPWT for representation of two-dimensional data.","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"7 1","pages":"1474-1496"},"PeriodicalIF":1.6,"publicationDate":"2009-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1137/080719248","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64019023","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 collective behavior of bacterial populations provides an example of how cell-level decision making translates into population-level behavior and illustrates clearly the difficult multiscale mat...
{"title":"From Signal Transduction to Spatial Pattern Formation in E. coli: A Paradigm for Multiscale Modeling in Biology","authors":"R. Erban, H. Othmer","doi":"10.1137/040603565","DOIUrl":"https://doi.org/10.1137/040603565","url":null,"abstract":"The collective behavior of bacterial populations provides an example of how cell-level decision making translates into population-level behavior and illustrates clearly the difficult multiscale mat...","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"3 1","pages":"362-394"},"PeriodicalIF":1.6,"publicationDate":"2005-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1137/040603565","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63750382","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}
Pub Date : 2004-01-01DOI: 10.1007/978-3-642-18756-8_2
A. Abdulle, S. Attinger
{"title":"Homogenization method for transport of DNA particles in heterogeneous arrays","authors":"A. Abdulle, S. Attinger","doi":"10.1007/978-3-642-18756-8_2","DOIUrl":"https://doi.org/10.1007/978-3-642-18756-8_2","url":null,"abstract":"","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"198 1","pages":"23-33"},"PeriodicalIF":1.6,"publicationDate":"2004-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/978-3-642-18756-8_2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51078670","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}
Pub Date : 2003-10-02DOI: 10.1007/978-3-642-18756-8_6
G. Samaey, I. Kevrekidis, D. Roose
{"title":"Damping factors for the gap-tooth scheme","authors":"G. Samaey, I. Kevrekidis, D. Roose","doi":"10.1007/978-3-642-18756-8_6","DOIUrl":"https://doi.org/10.1007/978-3-642-18756-8_6","url":null,"abstract":"","PeriodicalId":49791,"journal":{"name":"Multiscale Modeling & Simulation","volume":"36 1","pages":"93-102"},"PeriodicalIF":1.6,"publicationDate":"2003-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"51078730","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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