Pub Date : 2023-07-26DOI: 10.1007/s10208-023-09616-9
Zheng-Chu Guo, Andreas Christmann, Lei Shi
In this paper, we study an online learning algorithm with a robust loss function (mathcal {L}_{sigma }) for regression over a reproducing kernel Hilbert space (RKHS). The loss function (mathcal {L}_{sigma }) involving a scaling parameter (sigma >0) can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression aiming to estimate the conditional mean function. For properly chosen (sigma ) and step size, we show that the last iterate of this online algorithm can achieve optimal capacity independent convergence in the mean square distance. Moreover, if additional information on the underlying function space is known, we also establish optimal capacity-dependent rates for strong convergence in RKHS. To the best of our knowledge, both of the two results are new to the existing literature of online learning.
{"title":"Optimality of Robust Online Learning","authors":"Zheng-Chu Guo, Andreas Christmann, Lei Shi","doi":"10.1007/s10208-023-09616-9","DOIUrl":"https://doi.org/10.1007/s10208-023-09616-9","url":null,"abstract":"<p>In this paper, we study an online learning algorithm with a robust loss function <span>(mathcal {L}_{sigma })</span> for regression over a reproducing kernel Hilbert space (RKHS). The loss function <span>(mathcal {L}_{sigma })</span> involving a scaling parameter <span>(sigma >0)</span> can cover a wide range of commonly used robust losses. The proposed algorithm is then a robust alternative for online least squares regression aiming to estimate the conditional mean function. For properly chosen <span>(sigma )</span> and step size, we show that the last iterate of this online algorithm can achieve optimal capacity independent convergence in the mean square distance. Moreover, if additional information on the underlying function space is known, we also establish optimal capacity-dependent rates for strong convergence in RKHS. To the best of our knowledge, both of the two results are new to the existing literature of online learning.</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"31 1","pages":""},"PeriodicalIF":3.0,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138534307","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-06-09DOI: 10.1007/s10208-023-09614-x
Robert Szczelina, Piotr Zgliczyński
Abstract We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).
{"title":"High-Order Lohner-Type Algorithm for Rigorous Computation of Poincaré Maps in Systems of Delay Differential Equations with Several Delays","authors":"Robert Szczelina, Piotr Zgliczyński","doi":"10.1007/s10208-023-09614-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09614-x","url":null,"abstract":"Abstract We present a Lohner-type algorithm for rigorous integration of systems of delay differential equations (DDEs) with multiple delays, and its application in computation of Poincaré maps, to study the dynamics of some bounded, eternal solutions. The algorithm is based on a piecewise Taylor representation of the solutions in the phase space, and it exploits the smoothing of solutions occurring in DDEs to produce enclosures of solutions of a high order. We apply the topological techniques to prove various kinds of dynamical behaviour, for example, existence of (apparently) unstable periodic orbits in Mackey–Glass equation (in the regime of parameters where chaos is numerically observed) and persistence of symbolic dynamics in a delay-perturbed chaotic ODE (the Rössler system).","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"379 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135099805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-30DOI: 10.1007/s10208-023-09613-y
Nicolas Clozeau, Marc Josien, Felix Otto, Qiang Xu
Abstract We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior $$a_{textrm{hom}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mtext>hom</mml:mtext> </mml:msub> </mml:math> of a stationary random medium. The latter is described by a coefficient field a ( x ) generated from a given ensemble $$langle cdot rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> and the corresponding linear elliptic operator $$-nabla cdot anabla $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>∇</mml:mi> <mml:mo>·</mml:mo> <mml:mi>a</mml:mi> <mml:mi>∇</mml:mi> </mml:mrow> </mml:math> . In line with the theory of homogenization, the method proceeds by computing $$d=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> correctors ( d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a ( x ) from the whole-space ensemble $$langle cdot rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> . We make this point by investigating the bias (or systematic error), i.e., the difference between $$a_{textrm{hom}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>a</mml:mi> <mml:mtext>hom</mml:mtext> </mml:msub> </mml:math> and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a ( x ), we heuristically argue that this error is generically $$O(L^{-1})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In case of a suitable periodization of $$langle cdot rangle $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> , we rigorously show that it is $$O(L^{-d})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In fact, we give a
摘要研究了代表性体积元(RVE)方法，它是一种近似推断平稳随机介质的有效行为$$a_{textrm{hom}}$$ a home的方法。后者由一个给定集合$$langle cdot rangle $$⟨·⟩和相应的线性椭圆算子$$-nabla cdot anabla $$ -∇·a∇生成的系数场a (x)来描述。根据均匀化理论，该方法首先计算$$d=3$$ d = 3个校正量(d表示空间维度)。为了在数值上易于处理，这种计算必须在有限域中完成:即所谓的代表性体积元素，即具有周期性边界条件的大盒子。本文的主要信息是:将集成周期化，而不是将其实现周期化。通过这一点，我们的意思是，从一个适当的周期化的集合中采样比从整个空间集合$$langle cdot rangle $$⟨·⟩中周期性地扩展实现a (x)的限制更好。我们通过研究偏差(或系统误差)来提出这一点，即$$a_{textrm{hom}}$$ a home与RVE方法的期望值之间的差异，根据其缩放w.r.t.盒子的横向大小L。在周期化a (x)的情况下，我们启发式地认为该误差一般为$$O(L^{-1})$$ O (L - 1)。在$$langle cdot rangle $$⟨·⟩的合适周期化的情况下，我们严格地表明它是$$O(L^{-d})$$ O (L - d)。事实上，我们给出了两种策略的首阶误差项的特征，并论证了即使在各向同性情况下，它也是一般非简并的。我们在高斯型的合集$$langle cdot rangle $$⟨·⟩的方便设置中进行严格的分析，它允许通过(可积的)协方差函数进行直接的周期化。这种设置还具有使Price定理和Malliavin演算可用于校正量的最优随机估计的优点。我们实际上需要控制二阶校正器来捕获前阶误差项。这是由于对格林函数应用双尺度展开时的反演对称性。作为奖励，我们提出了一种流线策略来估计格林函数的高阶双尺度展开中的误差。
{"title":"Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations","authors":"Nicolas Clozeau, Marc Josien, Felix Otto, Qiang Xu","doi":"10.1007/s10208-023-09613-y","DOIUrl":"https://doi.org/10.1007/s10208-023-09613-y","url":null,"abstract":"Abstract We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior $$a_{textrm{hom}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>a</mml:mi> <mml:mtext>hom</mml:mtext> </mml:msub> </mml:math> of a stationary random medium. The latter is described by a coefficient field a ( x ) generated from a given ensemble $$langle cdot rangle $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> and the corresponding linear elliptic operator $$-nabla cdot anabla $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>∇</mml:mi> <mml:mo>·</mml:mo> <mml:mi>a</mml:mi> <mml:mi>∇</mml:mi> </mml:mrow> </mml:math> . In line with the theory of homogenization, the method proceeds by computing $$d=3$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> correctors ( d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a ( x ) from the whole-space ensemble $$langle cdot rangle $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> . We make this point by investigating the bias (or systematic error), i.e., the difference between $$a_{textrm{hom}}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>a</mml:mi> <mml:mtext>hom</mml:mtext> </mml:msub> </mml:math> and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a ( x ), we heuristically argue that this error is generically $$O(L^{-1})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In case of a suitable periodization of $$langle cdot rangle $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⟨</mml:mo> <mml:mo>·</mml:mo> <mml:mo>⟩</mml:mo> </mml:mrow> </mml:math> , we rigorously show that it is $$O(L^{-d})$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>d</mml:mi> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . In fact, we give a ","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135692550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-05-08DOI: 10.1007/s10208-023-09612-z
Ajay Jasra, Kody J. H. Law, Neil Walton, Shangda Yang
Abstract We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of $$hbox {MSE}^{-1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>MSE</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , while single-level methods require $$hbox {MSE}^{-xi }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mtext>MSE</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>ξ</mml:mi> </mml:mrow> </mml:msup> </mml:math> for $$xi >1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where $$xi =5/4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$xi =3/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately $$xi =9/4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>9</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $$xi = 5/4 + omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , for any $$omega > 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the s
{"title":"Multi-index Sequential Monte Carlo Ratio Estimators for Bayesian Inverse problems","authors":"Ajay Jasra, Kody J. H. Law, Neil Walton, Shangda Yang","doi":"10.1007/s10208-023-09612-z","DOIUrl":"https://doi.org/10.1007/s10208-023-09612-z","url":null,"abstract":"Abstract We consider the problem of estimating expectations with respect to a target distribution with an unknown normalising constant, and where even the un-normalised target needs to be approximated at finite resolution. This setting is ubiquitous across science and engineering applications, for example in the context of Bayesian inference where a physics-based model governed by an intractable partial differential equation (PDE) appears in the likelihood. A multi-index sequential Monte Carlo (MISMC) method is used to construct ratio estimators which provably enjoy the complexity improvements of multi-index Monte Carlo (MIMC) as well as the efficiency of sequential Monte Carlo (SMC) for inference. In particular, the proposed method provably achieves the canonical complexity of $$hbox {MSE}^{-1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mtext>MSE</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> , while single-level methods require $$hbox {MSE}^{-xi }$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mtext>MSE</mml:mtext> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>ξ</mml:mi> </mml:mrow> </mml:msup> </mml:math> for $$xi >1$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . This is illustrated on examples of Bayesian inverse problems with an elliptic PDE forward model in 1 and 2 spatial dimensions, where $$xi =5/4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and $$xi =3/2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , respectively. It is also illustrated on more challenging log-Gaussian process models, where single-level complexity is approximately $$xi =9/4$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>9</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and multilevel Monte Carlo (or MIMC with an inappropriate index set) gives $$xi = 5/4 + omega $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ξ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> , for any $$omega > 0$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , whereas our method is again canonical. We also provide novel theoretical verification of the product-form convergence results which MIMC requires for Gaussian processes built in spaces of mixed regularity defined in the s","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135846075","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-04-18DOI: 10.1007/s10208-023-09615-w
Liam Yemm
Abstract We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in $$L^2$$ L2 and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.
{"title":"A New Approach to Handle Curved Meshes in the Hybrid High-Order Method","authors":"Liam Yemm","doi":"10.1007/s10208-023-09615-w","DOIUrl":"https://doi.org/10.1007/s10208-023-09615-w","url":null,"abstract":"Abstract We present here a novel approach to handling curved meshes in polytopal methods within the framework of hybrid high-order methods. The hybrid high-order method is a modern numerical scheme for the approximation of elliptic PDEs. An extension to curved meshes allows for the strong enforcement of boundary conditions on curved domains and for the capture of curved geometries that appear internally in the domain e.g. discontinuities in a diffusion coefficient. The method makes use of non-polynomial functions on the curved faces and does not require any mappings between reference elements/faces. Such an approach does not require the faces to be polynomial and has a strict upper bound on the number of degrees of freedom on a curved face for a given polynomial degree. Moreover, this approach of enriching the space of unknowns on the curved faces with non-polynomial functions should extend naturally to other polytopal methods. We show the method to be stable and consistent on curved meshes and derive optimal error estimates in $$L^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and energy norms. We present numerical examples of the method on a domain with curved boundary and for a diffusion problem such that the diffusion tensor is discontinuous along a curved arc.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"97 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135932461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-02-01DOI: 10.1007/s10208-022-09602-7
M. Helmer, Vidit Nanda
{"title":"Correction to: Conormal Spaces and Whitney Stratifications","authors":"M. Helmer, Vidit Nanda","doi":"10.1007/s10208-022-09602-7","DOIUrl":"https://doi.org/10.1007/s10208-022-09602-7","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"1 1","pages":"1-8"},"PeriodicalIF":3.0,"publicationDate":"2023-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Optimal Polynomial Meshes Exist on any Multivariate Convex Domain","authors":"Feng Dai, Andriy Prymak","doi":"10.1007/s10208-023-09606-x","DOIUrl":"https://doi.org/10.1007/s10208-023-09606-x","url":null,"abstract":"We show that optimal polynomial meshes exist for every convex body in $${mathbb {R}}^d$$ , confirming a conjecture by A. Kroó.","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"248 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136297270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-16DOI: 10.1007/s10208-022-09600-9
R. Carles, C. Su
{"title":"Scattering and Uniform in Time Error Estimates for Splitting Method in NLS","authors":"R. Carles, C. Su","doi":"10.1007/s10208-022-09600-9","DOIUrl":"https://doi.org/10.1007/s10208-022-09600-9","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47665434","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-09DOI: 10.1007/s10208-022-09595-3
Jonathan W. Siegel, Jinchao Xu
{"title":"Sharp Bounds on the Approximation Rates, Metric Entropy, and n-Widths of Shallow Neural Networks","authors":"Jonathan W. Siegel, Jinchao Xu","doi":"10.1007/s10208-022-09595-3","DOIUrl":"https://doi.org/10.1007/s10208-022-09595-3","url":null,"abstract":"","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":" ","pages":""},"PeriodicalIF":3.0,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44959269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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