We consider a class of relaxation problems mixing slow and fast variations which can describe population dynamics models or hyperbolic systems, with varying stiffness (from non-stiff to strongly dissipative), and develop a multi-scale method by decomposing this problem into a micro-macro system where the original stiffness is broken. We show that this new problem can therefore be simulated with a uniform order of accuracy using standard explicit numerical schemes. In other words, it is possible to solve the micro-macro problem with a cost independent of the stiffness (a.k.a. uniform cost), such that the error is also uniform. This method is successfully applied to two hyperbolic systems with and without non-linearities, and is shown to circumvent the phenomenon of order reduction.
{"title":"A uniformly accurate numerical method for a class of dissipative systems","authors":"P. Chartier, M. Lemou, Léopold Trémant","doi":"10.1090/mcom/3688","DOIUrl":"https://doi.org/10.1090/mcom/3688","url":null,"abstract":"We consider a class of relaxation problems mixing slow and fast variations which can describe population dynamics models or hyperbolic systems, with varying stiffness (from non-stiff to strongly dissipative), and develop a multi-scale method by decomposing this problem into a micro-macro system where the original stiffness is broken. We show that this new problem can therefore be simulated with a uniform order of accuracy using standard explicit numerical schemes. In other words, it is possible to solve the micro-macro problem with a cost independent of the stiffness (a.k.a. uniform cost), such that the error is also uniform. This method is successfully applied to two hyperbolic systems with and without non-linearities, and is shown to circumvent the phenomenon of order reduction.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89609922","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 prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on R, for d ≥ 1. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to ±∞, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on R was initially introduced by Nichols & Kuo in 2014 to analyse the error of quasi-Monte Carlo (QMC) approximations for integrals on unbounded domains; whereas the classical Sobolev space of dominating mixed smoothness was used as the setting in a series of papers by Griebel, Kuo & Sloan on the smoothing effect of integration, in an effort to develop a rigorous theory on why QMC methods work so well for certain non-smooth integrands with kinks or jumps coming from option pricing problems. In this same setting, Griewank, Kuo, Leövey & Sloan in 2018 subsequently extended these ideas by developing a practical smoothing by preintegration technique to approximate integrals of such functions with kinks or jumps. We first prove the equivalence in one dimension (itself a non-trivial task), before following a similar, but more complicated, strategy to prove the equivalence for general dimensions. As a consequence of this equivalence, we analyse applying QMC combined with a preintegration step to approximate the fair price of an Asian option, and prove that the error of such an approximation using N points converges at a rate close to 1/N .
{"title":"Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on $mathbb{R}^d$","authors":"A. D. Gilbert, F. Kuo, I. Sloan","doi":"10.1090/mcom/3718","DOIUrl":"https://doi.org/10.1090/mcom/3718","url":null,"abstract":"We prove that a variant of the classical Sobolev space of first-order dominating mixed smoothness is equivalent (under a certain condition) to the unanchored ANOVA space on R, for d ≥ 1. Both spaces are Hilbert spaces involving weight functions, which determine the behaviour as different variables tend to ±∞, and weight parameters, which represent the influence of different subsets of variables. The unanchored ANOVA space on R was initially introduced by Nichols & Kuo in 2014 to analyse the error of quasi-Monte Carlo (QMC) approximations for integrals on unbounded domains; whereas the classical Sobolev space of dominating mixed smoothness was used as the setting in a series of papers by Griebel, Kuo & Sloan on the smoothing effect of integration, in an effort to develop a rigorous theory on why QMC methods work so well for certain non-smooth integrands with kinks or jumps coming from option pricing problems. In this same setting, Griewank, Kuo, Leövey & Sloan in 2018 subsequently extended these ideas by developing a practical smoothing by preintegration technique to approximate integrals of such functions with kinks or jumps. We first prove the equivalence in one dimension (itself a non-trivial task), before following a similar, but more complicated, strategy to prove the equivalence for general dimensions. As a consequence of this equivalence, we analyse applying QMC combined with a preintegration step to approximate the fair price of an Asian option, and prove that the error of such an approximation using N points converges at a rate close to 1/N .","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83489063","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}
For $m, d in {mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c ge 0$ and $C$ such that for all $d$ and all $m ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies [c ,dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)} le {mathbb E} D^*(P) le C, dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)}.] This discrepancy is thus smaller by a factor of $Thetabig(sqrt{frac{1+log(m/d)}{m/d}},big)$ than the one of a uniformly distributed random point set of $m^d$ points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity (2016)). It also removes the asymptotic requirement that $m$ is sufficiently large compared to $d$.
对于$m, d in {mathbb N}$，在$[0,1)^d$中有$N = m^d$个点的抖动采样点集$P$是通过将单位立方体$[0,1)^d$划分为$m^d$个大小相等的轴向立方体，然后在每个立方体中独立且均匀地随机放置一个点来构建的。我们表明，存在常数$c ge 0$和$C$，使得对于所有$d$和所有$m ge d$，抖动采样点集的预期非归一化星形差异满足[c ,dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)} le {mathbb E} D^*(P) le C, dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)}.]，因此，该差异比均匀分布的随机点集$m^d$点的差异小$Thetabig(sqrt{frac{1+log(m/d)}{m/d}},big)$倍。该结果改善了Pausinger和Steinerberger (Journal of Complexity(2016))给出的抖动采样差异的上界和下界。它还消除了$m$与$d$相比足够大的渐近要求。
{"title":"A Sharp Discrepancy Bound for Jittered Sampling","authors":"Benjamin Doerr","doi":"10.1090/mcom/3727","DOIUrl":"https://doi.org/10.1090/mcom/3727","url":null,"abstract":"For $m, d in {mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c ge 0$ and $C$ such that for all $d$ and all $m ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies [c ,dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)} le {mathbb E} D^*(P) le C, dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)}.] \u0000This discrepancy is thus smaller by a factor of $Thetabig(sqrt{frac{1+log(m/d)}{m/d}},big)$ than the one of a uniformly distributed random point set of $m^d$ points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity (2016)). It also removes the asymptotic requirement that $m$ is sufficiently large compared to $d$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86535235","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}
{"title":"Block FETI-DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity","authors":"O. Widlund, S. Zampini, S. Scacchi, L. Pavarino","doi":"10.1090/MCOM/3614","DOIUrl":"https://doi.org/10.1090/MCOM/3614","url":null,"abstract":",","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90335903","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}
By a closer inspection of the Friedman-Jorgenson-Kramer algorithm related to the prime geodesic theorem on cofinite Fuchsian groups of the first kind, we refine the constants therein. The newly obtained effective upper bound for Huber’s constant is in the modular surface case approximately 74000 74000 -times smaller than the previously claimed one. The degree of reduction in the case of an upper bound for Faltings’s delta function ranges from 10 8 10^{8} to 10 16 10^{16} .
{"title":"Effective bounds for Huber's constant and Faltings's delta function","authors":"Muharem Avdispahić","doi":"10.1090/MCOM/3631","DOIUrl":"https://doi.org/10.1090/MCOM/3631","url":null,"abstract":"By a closer inspection of the Friedman-Jorgenson-Kramer algorithm related to the prime geodesic theorem on cofinite Fuchsian groups of the first kind, we refine the constants therein. The newly obtained effective upper bound for Huber’s constant is in the modular surface case approximately \u0000\u0000 \u0000 74000\u0000 74000\u0000 \u0000\u0000-times smaller than the previously claimed one. The degree of reduction in the case of an upper bound for Faltings’s delta function ranges from \u0000\u0000 \u0000 \u0000 10\u0000 \u0000 8\u0000 \u0000 \u0000 10^{8}\u0000 \u0000\u0000 to \u0000\u0000 \u0000 \u0000 10\u0000 \u0000 16\u0000 \u0000 \u0000 10^{16}\u0000 \u0000\u0000.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78491525","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 considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity $kappa(x)$, in what becomes a nonlinear hyperbolic equation with nonlocal terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from $kappa$ to the overposed data used to recover it and from this basis develop and analyse Newton-type schemes for its effective recovery.
{"title":"On an inverse problem of nonlinear imaging with fractional damping","authors":"B. Kaltenbacher, W. Rundell","doi":"10.1090/mcom/3683","DOIUrl":"https://doi.org/10.1090/mcom/3683","url":null,"abstract":"This paper considers the attenuated Westervelt equation in pressure formulation. The attenuation is by various models proposed in the literature and characterised by the inclusion of non-local operators that give power law damping as opposed to the exponential of classical models. The goal is the inverse problem of recovering a spatially dependent coefficient in the equation, the parameter of nonlinearity $kappa(x)$, in what becomes a nonlinear hyperbolic equation with nonlocal terms. The overposed measured data is a time trace taken on a subset of the domain or its boundary. We shall show injectivity of the linearised map from $kappa$ to the overposed data used to recover it and from this basis develop and analyse Newton-type schemes for its effective recovery.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78886898","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 investigate a spectral Galerkin method for the two-dimensional fractional diffusion-reaction equations on a disk. We first prove regularity estimates of solutions in the weighted Sobolev space. Then we obtain optimal convergence orders of a spectral Galerkin method for the fractional diffusion-reaction equations in the L 2 L^2 and energy norm. We present numerical results to verify the theoretical analysis.
{"title":"Sharp error estimates of a spectral Galerkin method for a diffusion-reaction equation with integral fractional Laplacian on a disk","authors":"Zhaopeng Hao, Hui-yuan Li, Zhimin Zhang, Zhongqiang Zhang","doi":"10.1090/MCOM/3645","DOIUrl":"https://doi.org/10.1090/MCOM/3645","url":null,"abstract":"We investigate a spectral Galerkin method for the two-dimensional fractional diffusion-reaction equations on a disk. We first prove regularity estimates of solutions in the weighted Sobolev space. Then we obtain optimal convergence orders of a spectral Galerkin method for the fractional diffusion-reaction equations in the \u0000\u0000 \u0000 \u0000 L\u0000 2\u0000 \u0000 L^2\u0000 \u0000\u0000 and energy norm. We present numerical results to verify the theoretical analysis.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89905975","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}
{"title":"Quasi-Monte Carlo Bayesian estimation under Besov priors in elliptic inverse problems","authors":"L. Herrmann, Magdalena Keller, C. Schwab","doi":"10.1090/mcom/3615","DOIUrl":"https://doi.org/10.1090/mcom/3615","url":null,"abstract":"","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89760430","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}
Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(Delta, x_0)$ such that for all $x geq x_0$ there exists at least one prime in the interval $(x(1 - Delta^{-1}), x]$.
{"title":"Explicit interval estimates for prime numbers","authors":"Michaela Cully-Hugill, Ethan S. Lee","doi":"10.1090/mcom/3719","DOIUrl":"https://doi.org/10.1090/mcom/3719","url":null,"abstract":"Using a smoothing function and recent knowledge on the zeros of the Riemann zeta-function, we compute pairs of $(Delta, x_0)$ such that for all $x geq x_0$ there exists at least one prime in the interval $(x(1 - Delta^{-1}), x]$.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80956421","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 design adaptive finite differences discretizations, which are degenerate elliptic and second order consistent, of linear and quasi-linear partial differential operators featuring both a first order term and an anisotropic second order term. Our approach requires the domain to be discretized on a Cartesian grid, and takes advantage of techniques from the field of low-dimensional lattice geometry. We prove that the stencil of our numerical scheme is optimally compact, in dimension two, and that our approach is quasi-optimal in terms of the compatibility condition required of the first and second order operators, in dimensions two and three. Numerical experiments illustrate the efficiency of our method in several contexts.
{"title":"Second order monotone finite differences discretization of linear anisotropic differential operators","authors":"J. Bonnans, G. Bonnet, J. Mirebeau","doi":"10.1090/mcom/3671","DOIUrl":"https://doi.org/10.1090/mcom/3671","url":null,"abstract":"We design adaptive finite differences discretizations, which are degenerate elliptic and second order consistent, of linear and quasi-linear partial differential operators featuring both a first order term and an anisotropic second order term. Our approach requires the domain to be discretized on a Cartesian grid, and takes advantage of techniques from the field of low-dimensional lattice geometry. We prove that the stencil of our numerical scheme is optimally compact, in dimension two, and that our approach is quasi-optimal in terms of the compatibility condition required of the first and second order operators, in dimensions two and three. Numerical experiments illustrate the efficiency of our method in several contexts.","PeriodicalId":18301,"journal":{"name":"Math. Comput. Model.","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75793542","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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