Many optimization problems are characterized by expensive objective and/or constraint function evaluations paired with a lack of derivative information. Direct search methods such as generating set search (GSS) are well understood and efficient for derivative-free optimization of unconstrained and linearly constrained problems. This paper presents a study of heuristic algorithms that address the more difficult problem of general nonlinear constraints where derivatives for objective or constraint functions are unavailable. We focus on penalty methods that use GSS to solve a sequence of linearly constrained problems, numerically comparing different penalty functions. A classical choice for penalizing constraint violations is � 2 , the squared � 2 norm, which has advantages for derivative-based optimization methods. In our numerical tests, however, we show that exact penalty functions based on the � 1, � 2 ,a nd� ∞ norms converge to good approximate solutions more quickly and thus are attractive alternatives. Unfortunately, exact penalty functions are nondifferentiable and consequently degrade the final solution accuracy, so we also consider smoothed variants. Smoothed-exact penalty functions are attractive because they retain the differentiability of the original problem. Numerically, they are a compromise between exact and � 2 , i.e., they converge to a good solution somewhat quickly without sacrificing much solution accuracy. Moreover, the smoothing is parameterized and can potentially be adjusted to balance the two considerations. Since our
{"title":"Nonlinearly Constrained Optimization Using Heuristic Penalty Methods and Asynchronous Parallel Generating Set Search","authors":"J. Griffin, T. Kolda","doi":"10.1093/AMRX/ABQ003","DOIUrl":"https://doi.org/10.1093/AMRX/ABQ003","url":null,"abstract":"Many optimization problems are characterized by expensive objective and/or constraint function evaluations paired with a lack of derivative information. Direct search methods such as generating set search (GSS) are well understood and efficient for derivative-free optimization of unconstrained and linearly constrained problems. This paper presents a study of heuristic algorithms that address the more difficult problem of general nonlinear constraints where derivatives for objective or constraint functions are unavailable. We focus on penalty methods that use GSS to solve a sequence of linearly constrained problems, numerically comparing different penalty functions. A classical choice for penalizing constraint violations is � 2 , the squared � 2 norm, which has advantages for derivative-based optimization methods. In our numerical tests, however, we show that exact penalty functions based on the � 1, � 2 ,a nd� ∞ norms converge to good approximate solutions more quickly and thus are attractive alternatives. Unfortunately, exact penalty functions are nondifferentiable and consequently degrade the final solution accuracy, so we also consider smoothed variants. Smoothed-exact penalty functions are attractive because they retain the differentiability of the original problem. Numerically, they are a compromise between exact and � 2 , i.e., they converge to a good solution somewhat quickly without sacrificing much solution accuracy. Moreover, the smoothing is parameterized and can potentially be adjusted to balance the two considerations. Since our","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"76 1","pages":"36-62"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75073138","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 revisit and complete existence and uniqueness results stated and partially established by Muller zum Hagen in 1990 for the characteristic initial value problem for quasilinear hyperbolic systems of second order with data prescribed on two intersecting smooth null hypersurfaces. The new ingredient of this investigation consists of some Moser estimates expressed in the same weighted Sobolev spaces as those used by Muller zum Hagen. These estimates, combined with energy inequalities obtained by Muller zum Hagen for the linearized Goursat problem, permit us to develop a fixed point method which leads clearly to an existence and uniqueness result for the quasilinear Goursat problem. As an application we locally solve, under finite differentiability conditions, the characteristic initial value problem for the Einstein-Yang-Mills-Higgs system using harmonic gauge for the gravitational potentials and Lorentz gauge for the Yang-Mills potentials.
对于两个相交的光滑零超曲面上的二阶拟线性双曲型系统的特征初值问题，我们重述并补全了Muller zum Hagen在1990年提出和部分建立的存在唯一性结果。这项研究的新成分包括一些在与Muller zum Hagen使用的相同加权Sobolev空间中表示的Moser估计。这些估计与Muller zum Hagen对线性化Goursat问题得到的能量不等式相结合，使我们能够发展出一种不动点法，从而清楚地得出拟线性Goursat问题的存在唯一性结果。作为一个应用，我们在有限可微条件下局部求解了Einstein-Yang-Mills-Higgs系统的特征初值问题，用调和规范表示引力势，用洛伦兹规范表示Yang-Mills势。
{"title":"The Characteristic Initial Value Problem for the Einstein-Yang-Mills-Higgs System in Weighted Sobolev Spaces","authors":"M. Dossa, C. Tadmon","doi":"10.1093/AMRX/ABQ014","DOIUrl":"https://doi.org/10.1093/AMRX/ABQ014","url":null,"abstract":"We revisit and complete existence and uniqueness results stated and partially established by Muller zum Hagen in 1990 for the characteristic initial value problem for quasilinear hyperbolic systems of second order with data prescribed on two intersecting smooth null hypersurfaces. The new ingredient of this investigation consists of some Moser estimates expressed in the same weighted Sobolev spaces as those used by Muller zum Hagen. These estimates, combined with energy inequalities obtained by Muller zum Hagen for the linearized Goursat problem, permit us to develop a fixed point method which leads clearly to an existence and uniqueness result for the quasilinear Goursat problem. As an application we locally solve, under finite differentiability conditions, the characteristic initial value problem for the Einstein-Yang-Mills-Higgs system using harmonic gauge for the gravitational potentials and Lorentz gauge for the Yang-Mills potentials.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"6 1","pages":"154-231"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72781091","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 present rigorous derivations of anelastic limits for compressible Euler type systems when the Mach (or Froude) number tends to zero. The first and main part is to prove local existence and uniqueness of strong solution together with uniform estimates on a time interval independent of the small parameter. The key new remark is that the systems under consideration can be written in a form where ideas from cite{MS} can be adapted. The second part of the analysis is to pass to the limit as the parameter tends to zero. In this context, the main problem is to study the averaged effect of fast acoustic waves on the slow incompressible motion. In some cases, the averaged system is completely decoupled from acoustic waves. The first example studied in this paper enters this category: it is a shallow-water type system with topography and the limiting system is the inviscid lake equation (rigid lid approximation). This is similar to the low Mach limit analysis for prepared data, following the usual terminology, where the acoustic wave disappears in a pure pressure term for the limit equation. The decoupling also occurs in infinite domains where the fast acoustic waves are rapidly dispersed at infinity and therefore have no time to interact with the slow motion (see cite{Sc,MS, Al}). In other cases, and this should be expected in general for bounded domains or periodic solutions, this phenomenon does not occur and the acoustic waves leave a nontrivial averaged term in the limit fluid equation, which cannot be incorporated in the pressure term. In this case, the limit system involves a fluid equation, coupled to a nontrivial infinite dimensional system of differential equations which models the energy exchange between the fluid and some remanent acoustic energy. This was suspected for the periodic low Mach limit problem for nonisentropic Euler equations in cite{MeSc} and proved for finite dimensional models. The second example treated in this paper, namely Euler type system with heterogeneous barotropic pressure law, is an example where this scenario is rigorously carried out. To the authors' knowledge, this is the first example in the literature where such a coupling is mathematically justified.
{"title":"Anelastic Limits for Euler Type Systems","authors":"D. Bresch, G. Métivier","doi":"10.1093/AMRX/ABQ012","DOIUrl":"https://doi.org/10.1093/AMRX/ABQ012","url":null,"abstract":"In this paper, we present rigorous derivations of anelastic limits for compressible Euler type systems when the Mach (or Froude) number tends to zero. The first and main part is to prove local existence and uniqueness of strong solution together with uniform estimates on a time interval independent of the small parameter. The key new remark is that the systems under consideration can be written in a form where ideas from cite{MS} can be adapted. The second part of the analysis is to pass to the limit as the parameter tends to zero. In this context, the main problem is to study the averaged effect of fast acoustic waves on the slow incompressible motion. In some cases, the averaged system is completely decoupled from acoustic waves. The first example studied in this paper enters this category: it is a shallow-water type system with topography and the limiting system is the inviscid lake equation (rigid lid approximation). This is similar to the low Mach limit analysis for prepared data, following the usual terminology, where the acoustic wave disappears in a pure pressure term for the limit equation. The decoupling also occurs in infinite domains where the fast acoustic waves are rapidly dispersed at infinity and therefore have no time to interact with the slow motion (see cite{Sc,MS, Al}). In other cases, and this should be expected in general for bounded domains or periodic solutions, this phenomenon does not occur and the acoustic waves leave a nontrivial averaged term in the limit fluid equation, which cannot be incorporated in the pressure term. In this case, the limit system involves a fluid equation, coupled to a nontrivial infinite dimensional system of differential equations which models the energy exchange between the fluid and some remanent acoustic energy. This was suspected for the periodic low Mach limit problem for nonisentropic Euler equations in cite{MeSc} and proved for finite dimensional models. The second example treated in this paper, namely Euler type system with heterogeneous barotropic pressure law, is an example where this scenario is rigorously carried out. To the authors' knowledge, this is the first example in the literature where such a coupling is mathematically justified.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"1 1","pages":"119-141"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89744389","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}
D. Work, Sebastien Blandin, Olli-Pekka Tossavainen, B. Piccoli, A. Bayen
This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data
{"title":"A traffic model for velocity data assimilation","authors":"D. Work, Sebastien Blandin, Olli-Pekka Tossavainen, B. Piccoli, A. Bayen","doi":"10.1093/AMRX/ABQ002","DOIUrl":"https://doi.org/10.1093/AMRX/ABQ002","url":null,"abstract":"This article is motivated by the practical problem of highway traffic estimation using velocity measurements from GPS enabled mobile devices such as cell phones. In order to simplify the estimation procedure, a velocity model for highway traffic is constructed, which results in a dynamical system in which the observation operator is linear. This article presents a new scalar hyperbolic partial differential equation (PDE) model for traffic velocity evolution on highways, based on the seminal Lighthill-Whitham-Richards (LWR) PDE for density. Equivalence of the solution of the new velocity PDE and the solution of the LWR PDE is shown for quadratic flux functions. Because this equivalence does not hold for general flux functions, a discretized model of velocity evolution based on the Godunov scheme applied to the LWR PDE is proposed. Using an explicit instantiation of the weak boundary conditions of the PDE, the discrete velocity evolution model is generalized to a network, thus making the model applicable to arbitrary highway networks. The resulting velocity model is a nonlinear and nondifferentiable discrete time dynamical system with a linear observation operator, for which a Monte Carlo based ensemble Kalman filtering data","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"29 1","pages":"1-35"},"PeriodicalIF":0.0,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89874448","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}
A. Baraviera, C. F. Lardizabal, A. Lopes, M. T. Cunha
We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the denitions and results we describe here are natural generalizations of the classical concepts of Thermo- dynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is well-known that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism. We consider the operatoracting on the space of density matr ices MN over anite N-dimensional complex Hilbert space �( �) := k X i=1 tr(WiW � i ) ViV � i tr(ViV � i ) ,
我们考虑了作用于密度矩阵的平稳系统的熵和压力的新概念，它们推广了遍历理论中常用的概念。我们的部分工作是证明为什么我们在这里描述的结论和结果是热力学形式主义经典概念的自然推广(在R. Bowen, Y. Sinai和D. Ruelle的意义上)。众所周知，在我们考虑量子形式的情况下，密度算子的概念应该取代测度的概念。我们考虑在一个n维复希尔伯特空间()上作用于密度矩阵空间MN的算子:= k X i=1 tr(WiW i) ViV i tr(ViV i)，
{"title":"A Thermodynamic Formalism for Density Matrices in Quantum Information","authors":"A. Baraviera, C. F. Lardizabal, A. Lopes, M. T. Cunha","doi":"10.1093/amrx/abq008","DOIUrl":"https://doi.org/10.1093/amrx/abq008","url":null,"abstract":"We consider new concepts of entropy and pressure for stationary systems acting on density matrices which generalize the usual ones in Ergodic Theory. Part of our work is to justify why the denitions and results we describe here are natural generalizations of the classical concepts of Thermo- dynamic Formalism (in the sense of R. Bowen, Y. Sinai and D. Ruelle). It is well-known that the concept of density operator should replace the concept of measure for the cases in which we consider a quantum formalism. We consider the operatoracting on the space of density matr ices MN over anite N-dimensional complex Hilbert space �( �) := k X i=1 tr(WiW � i ) ViV � i tr(ViV � i ) ,","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"201 1","pages":"63-118"},"PeriodicalIF":0.0,"publicationDate":"2009-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79161938","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 study the minimum energy configurations of N particles in of charge −1 (“electrons”) in the potential of M particles of charges Zα > 0 (“atomic nuclei”). In a suitable large-N limit, we determine the asymptotic electron distribution explicitly, showing in particular that the number of electrons surrounding each nucleus is asymptotic to the nuclear charge (“screening”). The proof proceeds by establishing, via Gamma-convergence, a coarse-grained variational principle for the limit distribution, which can be solved explicitly.
{"title":"Minimum Energy Configurations of Classical Charges: Large N Asymptotics","authors":"Stéphane Capet, G. Friesecke","doi":"10.1093/amrx/abp002","DOIUrl":"https://doi.org/10.1093/amrx/abp002","url":null,"abstract":"We study the minimum energy configurations of N particles in of charge −1 (“electrons”) in the potential of M particles of charges Zα > 0 (“atomic nuclei”). In a suitable large-N limit, we determine the asymptotic electron distribution explicitly, showing in particular that the number of electrons surrounding each nucleus is asymptotic to the nuclear charge (“screening”). The proof proceeds by establishing, via Gamma-convergence, a coarse-grained variational principle for the limit distribution, which can be solved explicitly.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"4 1","pages":"47-73"},"PeriodicalIF":0.0,"publicationDate":"2009-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85745612","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}
Pub Date : 2009-06-01Epub Date: 2010-02-11DOI: 10.1093/amrx/abp008
Tamara Broderick, Kong Fatt Wong-Lin, Philip Holmes
In free response choice tasks, decision making is often modeled as a first-passage problem for a stochastic differential equation. In particular, drift-diffusion processes with constant or time-varying drift rates and noise can reproduce behavioral data (accuracy and response-time distributions) and neuronal firing rates. However, no exact solutions are known for the first-passage problem with time-varying data. Recognizing the importance of simple closed-form expressions for modeling and inference, we show that an interrogation or cued-response protocol, appropriately interpreted, can yield approximate first-passage (response time) distributions for a specific class of time-varying processes used to model evidence accumulation. We test these against exact expressions for the constant drift case and compare them with data from a class of sigmoidal functions. We find that both the direct interrogation approximation and an error-minimizing interrogation approximation can capture a variety of distribution shapes and mode numbers but that the direct approximation, in particular, is systematically biased away from the correct free response distribution.
{"title":"Closed-Form Approximations of First-Passage Distributions for a Stochastic Decision-Making Model.","authors":"Tamara Broderick, Kong Fatt Wong-Lin, Philip Holmes","doi":"10.1093/amrx/abp008","DOIUrl":"10.1093/amrx/abp008","url":null,"abstract":"<p><p>In free response choice tasks, decision making is often modeled as a first-passage problem for a stochastic differential equation. In particular, drift-diffusion processes with constant or time-varying drift rates and noise can reproduce behavioral data (accuracy and response-time distributions) and neuronal firing rates. However, no exact solutions are known for the first-passage problem with time-varying data. Recognizing the importance of simple closed-form expressions for modeling and inference, we show that an interrogation or cued-response protocol, appropriately interpreted, can yield approximate first-passage (response time) distributions for a specific class of time-varying processes used to model evidence accumulation. We test these against exact expressions for the constant drift case and compare them with data from a class of sigmoidal functions. We find that both the direct interrogation approximation and an error-minimizing interrogation approximation can capture a variety of distribution shapes and mode numbers but that the direct approximation, in particular, is systematically biased away from the correct free response distribution.</p>","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"2009 2","pages":"123-141"},"PeriodicalIF":0.0,"publicationDate":"2009-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3480186/pdf/nihms-377937.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"31009860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove resolvent estimates for semiclassical operators such as − h 2 Δ + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h − M in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.
{"title":"Semiclassical resolvent estimates in chaotic scattering","authors":"S. Nonnenmacher, M. Zworski","doi":"10.1093/AMRX/ABP003","DOIUrl":"https://doi.org/10.1093/AMRX/ABP003","url":null,"abstract":"We prove resolvent estimates for semiclassical operators such as − h 2 Δ + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h − M in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"16 1","pages":"74-86"},"PeriodicalIF":0.0,"publicationDate":"2009-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81793346","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 consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution of an arriving customer, conditioned on the number of other customers present. A new formula is obtained for the conditional sojourn time distribution, using a discrete Green’s function. This is shown to be equivalent to some classic results of Pollaczek and Vaulot from 1946. Then various asymptotic limits are studied, including large time and/or large number of customers present, and heavy traffic, where the arrival rate is only slightly less than the service rate.
{"title":"On Sojourn Times in the M / M /1-PS Model, Conditioned on the Number of Other Users","authors":"Qiang Zhen, C. Knessl","doi":"10.1093/amrx/abq001","DOIUrl":"https://doi.org/10.1093/amrx/abq001","url":null,"abstract":"We consider the M/M/1 queue with processor sharing. We study the conditional sojourn time distribution of an arriving customer, conditioned on the number of other customers present. A new formula is obtained for the conditional sojourn time distribution, using a discrete Green’s function. This is shown to be equivalent to some classic results of Pollaczek and Vaulot from 1946. Then various asymptotic limits are studied, including large time and/or large number of customers present, and heavy traffic, where the arrival rate is only slightly less than the service rate.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"32 1","pages":"142-167"},"PeriodicalIF":0.0,"publicationDate":"2009-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73516740","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 studies the asymptotic behavior of a particle with large initial velocity and subject to a force field which is randomly time dependent and inhomogeneous in space. We analyze the diffusive limit � → 0 of the position‐velocity pair under the appropriate space‐time rescaling: (� 3 Y(s/� 2 ), � ˙ Y(s/� 2 )). Two alternative approaches are proposed. The first one is based on hydrodynamic limits and homogenization techniques for the underlying kinetic equation; the second one is based on homogenization of the random distribution of trajectories. Time randomness is embodied into an underlying Markov process. Space inhomogeneity is modeled by a periodic structure in the first approach, and by a random field in the second one. In the first case, the analysis relies on the dissipation properties of the Markov process, whereas in the second case, the mixing properties of the random field are used. We point out more analogies and differences of the two obtained results.
{"title":"Stochastic Acceleration in an Inhomogeneous Time Random Force Field","authors":"T. Goudon, Mathias Rousset","doi":"10.1093/AMRX/ABP001","DOIUrl":"https://doi.org/10.1093/AMRX/ABP001","url":null,"abstract":"This paper studies the asymptotic behavior of a particle with large initial velocity and subject to a force field which is randomly time dependent and inhomogeneous in space. We analyze the diffusive limit � → 0 of the position‐velocity pair under the appropriate space‐time rescaling: (� 3 Y(s/� 2 ), � ˙ Y(s/� 2 )). Two alternative approaches are proposed. The first one is based on hydrodynamic limits and homogenization techniques for the underlying kinetic equation; the second one is based on homogenization of the random distribution of trajectories. Time randomness is embodied into an underlying Markov process. Space inhomogeneity is modeled by a periodic structure in the first approach, and by a random field in the second one. In the first case, the analysis relies on the dissipation properties of the Markov process, whereas in the second case, the mixing properties of the random field are used. We point out more analogies and differences of the two obtained results.","PeriodicalId":89656,"journal":{"name":"Applied mathematics research express : AMRX","volume":"9 1","pages":"1-46"},"PeriodicalIF":0.0,"publicationDate":"2009-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78466351","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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