We introduce a hybridizable discontinuous Galerkin (HDG) method for the convected Helmholtz equation based on the total flux formulation, in which the vector unknown represents both diffusive and convective phenomena. This HDG method is constricted with the same interpolation degree for all the unknowns and a physically informed value for the penalization parameter is computed. A detailed analysis including local and global well-posedness as well as a super-convergence result is carried out. We then provide numerical experiments to illustrate the theoretical results.
{"title":"Construction and analysis of a HDG solution for the total-flux formulation of the convected Helmholtz equation","authors":"Hélène Barucq, Nathan Rouxelin, Sébastien Tordeux","doi":"10.1090/mcom/3850","DOIUrl":"https://doi.org/10.1090/mcom/3850","url":null,"abstract":"We introduce a hybridizable discontinuous Galerkin (HDG) method for the convected Helmholtz equation based on the total flux formulation, in which the vector unknown represents both diffusive and convective phenomena. This HDG method is constricted with the same interpolation degree for all the unknowns and a physically informed value for the penalization parameter is computed. A detailed analysis including local and global well-posedness as well as a super-convergence result is carried out. We then provide numerical experiments to illustrate the theoretical results.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"204 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136265188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A singularly perturbed convection-diffusion problem posed on the unit square in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 slash 2 right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(N^{-(k+1/2)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> accurate in the energy norm induced by the bilinear form of the weak formulation, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 right-parenthesis Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mro
{"title":"Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem","authors":"Yao Cheng, Shan Jiang, Martin Stynes","doi":"10.1090/mcom/3844","DOIUrl":"https://doi.org/10.1090/mcom/3844","url":null,"abstract":"A singularly perturbed convection-diffusion problem posed on the unit square in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper R squared\"> <mml:semantics> <mml:msup> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=\"application/x-tex\">mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, whose solution has exponential boundary layers, is solved numerically using the local discontinuous Galerkin (LDG) method with tensor-product piecewise polynomials of degree at most <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on three families of layer-adapted meshes: Shishkin-type, Bakhvalov-Shishkin-type and Bakhvalov-type. On Shishkin-type meshes this method is known to be no greater than <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 slash 2 right-parenthesis Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>−<!-- − --></mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">O(N^{-(k+1/2)})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> accurate in the energy norm induced by the bilinear form of the weak formulation, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper N\"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding=\"application/x-tex\">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mesh intervals are used in each coordinate direction. (Note: all bounds in this abstract are uniform in the singular perturbation parameter and neglect logarithmic factors that will appear in our detailed analysis.) A delicate argument is used in this paper to establish <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper O left-parenthesis upper N Superscript minus left-parenthesis k plus 1 right-parenthesis Baseline right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mro","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136231610","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a number field, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finitely generated subgroup of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K Superscript times"> <mml:semantics> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>×<!-- × --></mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">K^times</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathfrak p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the order of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper G mod German p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(Gbmod mathfrak p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German p"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">mathfrak p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the order is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:
{"title":"Divisibility conditions on the order of the reductions of algebraic numbers","authors":"Pietro Sgobba","doi":"10.1090/mcom/3848","DOIUrl":"https://doi.org/10.1090/mcom/3848","url":null,"abstract":"Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a number field, and let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper G\"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=\"application/x-tex\">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a finitely generated subgroup of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K Superscript times\"> <mml:semantics> <mml:msup> <mml:mi>K</mml:mi> <mml:mo>×<!-- × --></mml:mo> </mml:msup> <mml:annotation encoding=\"application/x-tex\">K^times</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Without relying on the Generalized Riemann Hypothesis we prove an asymptotic formula for the number of primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German p\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper K\"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding=\"application/x-tex\">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that the order of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-parenthesis upper G mod German p right-parenthesis\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo lspace=\"thickmathspace\" rspace=\"thickmathspace\">mod</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">p</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">)</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">(Gbmod mathfrak p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is divisible by a fixed integer. We also provide a rational expression for the natural density of this set. Furthermore, we study the primes <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"German p\"> <mml:semantics> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"fraktur\">p</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">mathfrak p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which the order is <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k\"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding=\"application/x-tex\">k</mml:annotation> </mml:","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"235 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134922467","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Charles-Edouard Bréhier, David Cohen, Tobias Jahnke
We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie–Poisson systems, namely: stochastically perturbed Maxwell–Bloch, rigid body and sine–Euler equations.
{"title":"Splitting integrators for stochastic Lie–Poisson systems","authors":"Charles-Edouard Bréhier, David Cohen, Tobias Jahnke","doi":"10.1090/mcom/3829","DOIUrl":"https://doi.org/10.1090/mcom/3829","url":null,"abstract":"We study stochastic Poisson integrators for a class of stochastic Poisson systems driven by Stratonovich noise. Such geometric integrators preserve Casimir functions and the Poisson map property. For this purpose, we propose explicit stochastic Poisson integrators based on a splitting strategy, and analyse their qualitative and quantitative properties: preservation of Casimir functions, existence of almost sure or moment bounds, asymptotic preserving property, and strong and weak rates of convergence. The construction of the schemes and the theoretical results are illustrated through extensive numerical experiments for three examples of stochastic Lie–Poisson systems, namely: stochastically perturbed Maxwell–Bloch, rigid body and sine–Euler equations.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136086440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For all q≥2qge 2 and for all invertible residue classes aa modulo qq, there exists a natural number that is congruent to aa modulo qq and that is the product of exactly three primes, all of which are below (1015q)5/2(10^{15}q)^{5/2}.
We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.
{"title":"Robust a posteriori estimates for the stochastic Cahn-Hilliard equation","authors":"L’ubomír Baňas, Christian Vieth","doi":"10.1090/mcom/3836","DOIUrl":"https://doi.org/10.1090/mcom/3836","url":null,"abstract":"We derive a posteriori error estimates for a fully discrete finite element approximation of the stochastic Cahn-Hilliard equation. The a posteriori bound is obtained by a splitting of the equation into a linear stochastic partial differential equation and a nonlinear random partial differential equation. The resulting estimate is robust with respect to the interfacial width parameter and is computable since it involves the discrete principal eigenvalue of a linearized (stochastic) Cahn-Hilliard operator. Furthermore, the estimate is robust with respect to topological changes as well as the intensity of the stochastic noise. We provide numerical simulations to demonstrate the practicability of the proposed adaptive algorithm.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"104 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135708611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.
本文主要研究了基于梯度的jacobi型算法求解两类具有正交约束的齐次多项式的极值问题，并建立了它们的收敛性。对于Stiefel流形约束下的第一类齐次多项式，我们将其重新表述为一个酉群上的优化问题，从而使基于梯度的Jacobi-G算法得以应用。然后，如果子问题总是可以用二次形式表示，我们建立了Jacobi-G在三种条件中的任意一种下的全局收敛性。第一个条件的收敛结果是Usevich, Li, Comon [SIAM J. Optim. 30 (2020)， pp. 2998-3028]结果的简单推广，而其他两个条件是新的。该算法及其收敛性适用于众所周知的联合近似对称张量对角化。对于Stiefel流形积约束下的第二类齐次多项式，将其重新表述为酉群积上的优化问题，并提出了一种新的基于梯度的多块雅可比型(Jacobi-MG)算法。在上述三种条件中的任意一种下，如果子问题总是可以用二次型表示，则证明了Jacobi-MG的全局收敛性。该算法及其收敛性适用于众所周知的联合近似张量对角化。作为Jacobi-G和Jacobi-MG的近端变体，我们还提出了Jacobi-GP和Jacobi-MGP算法，并证明了它们在没有任何进一步条件下的全局收敛性。数值结果表明了所提算法的有效性。
{"title":"Jacobi-type algorithms for homogeneous polynomial optimization on Stiefel manifolds with applications to tensor approximations","authors":"Zhou Sheng, Jianze Li, Qin Ni","doi":"10.1090/mcom/3834","DOIUrl":"https://doi.org/10.1090/mcom/3834","url":null,"abstract":"This paper mainly studies the gradient-based Jacobi-type algorithms to maximize two classes of homogeneous polynomials with orthogonality constraints, and establish their convergence properties. For the first class of homogeneous polynomials subject to a constraint on a Stiefel manifold, we reformulate it as an optimization problem on a unitary group, which makes it possible to apply the gradient-based Jacobi-type (Jacobi-G) algorithm. Then, if the subproblem can always be represented as a quadratic form, we establish the global convergence of Jacobi-G under any one of three conditions. The convergence result for the first condition is an easy extension of the result by Usevich, Li, and Comon [SIAM J. Optim. 30 (2020), pp. 2998–3028], while other two conditions are new ones. This algorithm and the convergence properties apply to the well-known joint approximate symmetric tensor diagonalization. For the second class of homogeneous polynomials subject to constraints on the product of Stiefel manifolds, we reformulate it as an optimization problem on the product of unitary groups, and then develop a new gradient-based multiblock Jacobi-type (Jacobi-MG) algorithm to solve it. We establish the global convergence of Jacobi-MG under any one of the above three conditions, if the subproblem can always be represented as a quadratic form. This algorithm and the convergence properties are suitable to the well-known joint approximate tensor diagonalization. As the proximal variants of Jacobi-G and Jacobi-MG, we also propose the Jacobi-GP and Jacobi-MGP algorithms, and establish their global convergence without any further condition. Some numerical results are provided indicating the efficiency of the proposed algorithms.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"122 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135956560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Benjamin Gess, Rishabh S. Gvalani, Florian Kunick, Felix Otto
In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> driven by space-time white noise. The (formal) gradient flow structure of its deterministic counterpart, the so-called thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure, i.e., on the level of a discretization of energy functional and dissipative metric tensor. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, and as opposed to previous discretizations of the thin-film equation with thermal noise, the above discretization is not in need of an artificial condition at the boundary of the configuration space orthant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-brace h greater-than 0 right-brace"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence="false" stretchy="false">}</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{h>0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which, admittedly, could also be avoided by modelling a disjoining pressure). Thus, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more direct discretizations is that when writing the SDE in Itô’s form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. We perform various numerical experiments to compare the behavior and performance of our discretization to that of a particular finite difference discretization of the equation. Among other things, we study numerically the invariance and entropic repulsion of the invariant
{"title":"Thermodynamically consistent and positivity-preserving discretization of the thin-film equation with thermal noise","authors":"Benjamin Gess, Rishabh S. Gvalani, Florian Kunick, Felix Otto","doi":"10.1090/mcom/3830","DOIUrl":"https://doi.org/10.1090/mcom/3830","url":null,"abstract":"In micro-fluidics, both capillarity and thermal fluctuations play an important role. On the level of the lubrication approximation, this leads to a quasi-linear fourth-order parabolic equation for the film height <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"h\"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding=\"application/x-tex\">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> driven by space-time white noise. The (formal) gradient flow structure of its deterministic counterpart, the so-called thin-film equation, which encodes the balance between driving capillary and limiting viscous forces, provides the guidance for the thermodynamically consistent introduction of fluctuations. We follow this route on the level of a spatial discretization of the gradient flow structure, i.e., on the level of a discretization of energy functional and dissipative metric tensor. Starting from an energetically conformal finite-element (FE) discretization, we point out that the numerical mobility function introduced by Grün and Rumpf can be interpreted as a discretization of the metric tensor in the sense of a mixed FE method with lumping. While this discretization was devised in order to preserve the so-called entropy estimate, we use this to show that the resulting high-dimensional stochastic differential equation (SDE) preserves pathwise and pointwise strict positivity, at least in case of the physically relevant mobility function arising from the no-slip boundary condition. As a consequence, and as opposed to previous discretizations of the thin-film equation with thermal noise, the above discretization is not in need of an artificial condition at the boundary of the configuration space orthant <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-brace h greater-than 0 right-brace\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mi>h</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">{h>0}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (which, admittedly, could also be avoided by modelling a disjoining pressure). Thus, this discretization gives rise to a consistent invariant measure, namely a discretization of the Brownian excursion (up to the volume constraint), and thus features an entropic repulsion. The price to pay over more direct discretizations is that when writing the SDE in Itô’s form, which is the basis for the Euler-Mayurama time discretization, a correction term appears. We perform various numerical experiments to compare the behavior and performance of our discretization to that of a particular finite difference discretization of the equation. Among other things, we study numerically the invariance and entropic repulsion of the invariant ","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"68 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136088151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are quasi constant-free and apply to a large class of variational problems including the pp-Dirichlet problem, as well as degenerate minimization, obstacle and image de-noising problems. In addition, these a posteriori error estimates are based on a comparison to a given non-conforming finite element solution. For the pp-Dirichlet problem, these a posteriori error bounds are equivalent to residual type a posteriori error bounds and, hence, reliable and efficient.
我们将基于凸对偶关系的凸最小化问题的一般后验误差估计的系统方法与最近导出的广义Marini公式相结合。后验误差估计是准常数自由的，并适用于一大类变分问题，包括p -Dirichlet问题，以及退化最小化，障碍和图像去噪问题。此外，这些后验误差估计是基于与给定非一致性有限元解的比较。对于p p -Dirichlet问题，这些后验误差界等价于残差型后验误差界，因此是可靠和有效的。
{"title":"Explicit and efficient error estimation for convex minimization problems","authors":"Sören Bartels, Alex Kaltenbach","doi":"10.1090/mcom/3821","DOIUrl":"https://doi.org/10.1090/mcom/3821","url":null,"abstract":"We combine a systematic approach for deriving general a posteriori error estimates for convex minimization problems based on convex duality relations with a recently derived generalized Marini formula. The a posteriori error estimates are quasi constant-free and apply to a large class of variational problems including the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dirichlet problem, as well as degenerate minimization, obstacle and image de-noising problems. In addition, these a posteriori error estimates are based on a comparison to a given non-conforming finite element solution. For the <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"p\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\"application/x-tex\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-Dirichlet problem, these a posteriori error bounds are equivalent to residual type a posteriori error bounds and, hence, reliable and efficient.","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"204 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136174585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shuhuang Xiang, Desong Kong, Guidong Liu, Li-Lian Wang
As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue x minus a EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|x-a|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue a EndAbsoluteValue greater-than 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">|a|>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="95"> <mml:semantics> <mml:mn>95</mml:mn> <mml:annotation encoding="application/x-tex">95%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> range of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket negative 1 comma 1 right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[-1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> except for a small neighbourhood near the singular point <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x equals a period"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">x=a.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In this paper, we rigorously show that the Jacobi expansion for a more general class of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi"> <mml:semantics> <mml:mi mathvariant="normal">Φ<!-- Φ --></mml:mi> <mml:annotation encoding="application/x-tex">Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions also enjoys such a local convergence
作为多项式插值和正交的一个神话，Trefethen[数学]。Today (Southend-on-Sea) 47 (2011)， pp. 184-188]揭示了|x−a| |x-a|的切比雪夫插值(与| a| ＆gt;1 |a|＆gt;1)在克伦肖-柯蒂斯点上的误差比最佳多项式近似(在最大范数中)在大约95 95的误差要小得多% range of [ − 1 , 1 ] [-1,1] except for a small neighbourhood near the singular point x = a . x=a. In this paper, we rigorously show that the Jacobi expansion for a more general class of Φ Phi -functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired log n log n -factor in the pointwise error estimate for the Legendre expansion recently stated in Babus̆ka and Hakula [Comput. Methods Appl. Mech Engrg. 345 (2019), pp. 748–773] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.
{"title":"Pointwise error estimates and local superconvergence of Jacobi expansions","authors":"Shuhuang Xiang, Desong Kong, Guidong Liu, Li-Lian Wang","doi":"10.1090/mcom/3835","DOIUrl":"https://doi.org/10.1090/mcom/3835","url":null,"abstract":"As one myth of polynomial interpolation and quadrature, Trefethen [Math. Today (Southend-on-Sea) 47 (2011), pp. 184–188] revealed that the Chebyshev interpolation of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue x minus a EndAbsoluteValue\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mi>a</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|x-a|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (with <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue a EndAbsoluteValue greater-than 1\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>a</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|a|>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"95\"> <mml:semantics> <mml:mn>95</mml:mn> <mml:annotation encoding=\"application/x-tex\">95%</mml:annotation> </mml:semantics> </mml:math> </inline-formula> range of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"left-bracket negative 1 comma 1 right-bracket\"> <mml:semantics> <mml:mrow> <mml:mo stretchy=\"false\">[</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=\"false\">]</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">[-1,1]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> except for a small neighbourhood near the singular point <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"x equals a period\"> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">x=a.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> In this paper, we rigorously show that the Jacobi expansion for a more general class of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"normal upper Phi\"> <mml:semantics> <mml:mi mathvariant=\"normal\">Φ<!-- Φ --></mml:mi> <mml:annotation encoding=\"application/x-tex\">Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-functions also enjoys such a local convergence ","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136338734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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