Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim
We establish an a priori error analysis for the lowest-order Raviart–Thomas finite element discretization of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conforming approaches, which naturally imply upper energy bounds, the proposed mixed discretization provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.
{"title":"Mixed finite elements for the Gross–Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound","authors":"Dietmar Gallistl, Moritz Hauck, Yizhou Liang, Daniel Peterseim","doi":"10.1093/imanum/drae048","DOIUrl":"https://doi.org/10.1093/imanum/drae048","url":null,"abstract":"We establish an a priori error analysis for the lowest-order Raviart–Thomas finite element discretization of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conforming approaches, which naturally imply upper energy bounds, the proposed mixed discretization provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"8 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142042431","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}
Valentin Carlier, Martin Campos Pinto, Francesco Fambri
In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.
在这篇文章中，我们针对纳维-斯托克斯方程提出了两种有限元方案，这两种方案都是基于德拉姆序列微分算子和具有显式弱形式倾斜对称性的平流算子的重新表述。我们的第一个方案是通过符合 FEEC（有限元外部计算）的空间对这一公式进行离散化而获得的：它保留了速度、总动量和能量的无发散点约束，此外还具有压力稳健性。我们的第二种方案沿用了断裂 FEEC 方法，使用完全不连续空间和局部保角投影来定义离散微分算子。它保留了相同的不变性，直到能量耗散，以稳定数值不连续性。对于这两种方案，我们都采用了中间点时间离散化，在完全离散的水平上保留了这些不变式，并根据 CFL 条件分析了其良好拟合性。虽然我们的理论结果适用于保留 de Rham 结构的一般有限元，但我们考虑了张量乘积样条空间的一个应用。具体地说，我们进行了几个数值测试案例，以验证所产生的数值方法的高阶精度及其处理一般边界条件的能力。
{"title":"Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations","authors":"Valentin Carlier, Martin Campos Pinto, Francesco Fambri","doi":"10.1093/imanum/drae047","DOIUrl":"https://doi.org/10.1093/imanum/drae047","url":null,"abstract":"In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"95 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141991902","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}
In this paper we study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau–Lifshitz–Bloch (LLB) equation on a bounded domain in ${mathbb{R}}^{d}$ for $d=1,2$. Our main results are estimates of the rate of convergence of the Finite Element Method to the solutions of stochastic LLB. To overcome the lack of regularity of the solution in the case $d=2$, we propose a Finite Element scheme for a regularized version of the equation. We then obtain error estimates of numerical solutions and for the solution of the regularized equation as well as the rate of convergence of this solution to the solution of the stochastic LLB equation. As a consequence, the convergence in probability of the approximate solutions to the solution of the stochastic LLB equation is derived. A stronger result is obtained in the case $d=1$ due to a new regularity result for the LLB equation which allows us to avoid regularization.
{"title":"Numerical method and error estimate for stochastic Landau–Lifshitz–Bloch equation","authors":"Beniamin Goldys, Chunxi Jiao, Kim-Ngan Le","doi":"10.1093/imanum/drae046","DOIUrl":"https://doi.org/10.1093/imanum/drae046","url":null,"abstract":"In this paper we study numerical methods for solving a system of quasilinear stochastic partial differential equations known as the stochastic Landau–Lifshitz–Bloch (LLB) equation on a bounded domain in ${mathbb{R}}^{d}$ for $d=1,2$. Our main results are estimates of the rate of convergence of the Finite Element Method to the solutions of stochastic LLB. To overcome the lack of regularity of the solution in the case $d=2$, we propose a Finite Element scheme for a regularized version of the equation. We then obtain error estimates of numerical solutions and for the solution of the regularized equation as well as the rate of convergence of this solution to the solution of the stochastic LLB equation. As a consequence, the convergence in probability of the approximate solutions to the solution of the stochastic LLB equation is derived. A stronger result is obtained in the case $d=1$ due to a new regularity result for the LLB equation which allows us to avoid regularization.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"191 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141915158","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}
Simple stochastic momentum methods are widely used in machine learning optimization, but their good practical performance is at odds with an absence of theoretical guarantees of acceleration in the literature. In this work, we aim to close the gap between theory and practice by showing that stochastic heavy ball momentum retains the fast linear rate of (deterministic) heavy ball momentum on quadratic optimization problems, at least when minibatching with a sufficiently large batch size. The algorithm we study can be interpreted as an accelerated randomized Kaczmarz algorithm with minibatching and heavy ball momentum. The analysis relies on carefully decomposing the momentum transition matrix, and using new spectral norm concentration bounds for products of independent random matrices. We provide numerical illustrations demonstrating that our bounds are reasonably sharp.
{"title":"On the fast convergence of minibatch heavy ball momentum","authors":"Raghu Bollapragada, Tyler Chen, Rachel Ward","doi":"10.1093/imanum/drae033","DOIUrl":"https://doi.org/10.1093/imanum/drae033","url":null,"abstract":"Simple stochastic momentum methods are widely used in machine learning optimization, but their good practical performance is at odds with an absence of theoretical guarantees of acceleration in the literature. In this work, we aim to close the gap between theory and practice by showing that stochastic heavy ball momentum retains the fast linear rate of (deterministic) heavy ball momentum on quadratic optimization problems, at least when minibatching with a sufficiently large batch size. The algorithm we study can be interpreted as an accelerated randomized Kaczmarz algorithm with minibatching and heavy ball momentum. The analysis relies on carefully decomposing the momentum transition matrix, and using new spectral norm concentration bounds for products of independent random matrices. We provide numerical illustrations demonstrating that our bounds are reasonably sharp.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"72 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141910224","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 work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.
{"title":"Long-term accuracy of numerical approximations of SPDEs with the stochastic Navier–Stokes equations as a paradigm","authors":"Nathan E Glatt-Holtz, Cecilia F Mondaini","doi":"10.1093/imanum/drae043","DOIUrl":"https://doi.org/10.1093/imanum/drae043","url":null,"abstract":"This work introduces a general framework for establishing the long time accuracy for approximations of Markovian dynamical systems on separable Banach spaces. Our results illuminate the role that a certain uniformity in Wasserstein contraction rates for the approximating dynamics bears on long time accuracy estimates. In particular, our approach yields weak consistency bounds on ${mathbb{R}}^{+}$ while providing a means to sidestepping a commonly occurring situation where certain higher order moment bounds are unavailable for the approximating dynamics. Additionally, to facilitate the analytical core of our approach, we develop a refinement of certain ‘weak Harris theorems’. This extension expands the scope of applicability of such Wasserstein contraction estimates to a variety of interesting stochastic partial differential equation examples involving weaker dissipation or stronger nonlinearity than would be covered by the existing literature. As a guiding and paradigmatic example, we apply our formalism to the stochastic 2D Navier–Stokes equations and to a semi-implicit in time and spectral Galerkin in space numerical approximation of this system. In the case of a numerical approximation, we establish quantitative estimates on the approximation of invariant measures as well as prove weak consistency on ${mathbb{R}}^{+}$. To develop these numerical analysis results, we provide a refinement of $L^{2}_{x}$ accuracy bounds in comparison to the existing literature, which are results of independent interest.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"64 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141631548","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 an analytic integrand, the error term in the Gaussian quadrature can be represented as a contour integral, where the contour is commonly taken to be an ellipse. Thus, finding its upper bound can be reduced to finding the maximum of the modulus of the kernel on the ellipse. The location of this maximum was investigated in many special cases, particularly, for the Gaussian quadrature with respect to the Chebyshev measures modified by a quadratic divisor (known as the Bernstein–Szeg̋ measures). Here, for the Gaussian quadratures with respect to the Chebyshev measures modified by a linear over linear rational factor, we examine the kernel and describe sufficient conditions for the maximum to occur on the real axis. Furthermore, an assessment of the kernel is made in each case, since in some cases the true maximum is hard to reach. Hence, we derive the error bounds for these quadrature formulas. The results are illustrated by the numerical examples. An alternative approach for estimating the error of the Gaussian quadrature with respect to the same measure can be found in [Djukić, D. L., Djukić, R. M. M., Reichel, L. & Spalević, M. M. (2023, Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. Appl. Numer. Math., ISSN 0168-9274)].
对于解析积分，高斯正交中的误差项可以表示为等值线积分，其中等值线通常被视为椭圆。因此，寻找其上限可以简化为寻找椭圆上核的模的最大值。这个最大值的位置在许多特殊情况下都得到了研究，特别是高斯正交与二次除数修正的切比雪夫度量（称为伯恩斯坦-塞格度量）的关系。在此，对于关于经线性有理因子修正的切比雪夫度量的高斯正交，我们研究了核，并描述了在实轴上出现最大值的充分条件。此外，我们还对每种情况下的核进行了评估，因为在某些情况下很难达到真正的最大值。因此，我们推导出了这些正交公式的误差范围。结果将通过数值示例加以说明。估算高斯正交对于同一度量的误差的另一种方法见 [Djukić, D. L., Djukić, R. M. M., Reichel, L. & Spalević, M. M. (2023, Weighted averaged Gaussian quadrature rules for modified Chebyshev measure.Appl.Math., ISSN 0168-9274)].
{"title":"The error bounds of Gaussian quadratures for one rational modification of Chebyshev measures","authors":"Rada M Mutavdžić Djukić","doi":"10.1093/imanum/drae039","DOIUrl":"https://doi.org/10.1093/imanum/drae039","url":null,"abstract":"For an analytic integrand, the error term in the Gaussian quadrature can be represented as a contour integral, where the contour is commonly taken to be an ellipse. Thus, finding its upper bound can be reduced to finding the maximum of the modulus of the kernel on the ellipse. The location of this maximum was investigated in many special cases, particularly, for the Gaussian quadrature with respect to the Chebyshev measures modified by a quadratic divisor (known as the Bernstein–Szeg̋ measures). Here, for the Gaussian quadratures with respect to the Chebyshev measures modified by a linear over linear rational factor, we examine the kernel and describe sufficient conditions for the maximum to occur on the real axis. Furthermore, an assessment of the kernel is made in each case, since in some cases the true maximum is hard to reach. Hence, we derive the error bounds for these quadrature formulas. The results are illustrated by the numerical examples. An alternative approach for estimating the error of the Gaussian quadrature with respect to the same measure can be found in [Djukić, D. L., Djukić, R. M. M., Reichel, L. & Spalević, M. M. (2023, Weighted averaged Gaussian quadrature rules for modified Chebyshev measure. Appl. Numer. Math., ISSN 0168-9274)].","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"51 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141566247","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 have developed a rational interpolation method for analytic functions with branch point singularities, which utilizes several exponentially clustered poles proposed by Trefethen and his collaborators (2021, Exponential node clustering at singularities for rational approximation, quadrature, and PDEs. Numer. Math., 147, 227–254). The key to the feasibility of this interpolation method is that the interpolation nodes approximately satisfy the distribution of the equilibrium potential. These nodes make the convergence rate of the rational interpolation consistent with the theoretical rates, and steadily approach machine accuracy. The technique can be used, not only for the interval $[0,1]$, but can also be extended to include corner regions and the case of multiple singularities.
我们为具有支点奇异性的解析函数开发了一种有理插值方法，该方法利用了 Trefethen 及其合作者提出的几个指数簇极点（2021，Exponential node clustering at singularities for rational approximation, quadrature, and PDEs.Numer.Numer.Math.，147，227-254）。这种插值方法可行性的关键在于插值节点近似满足平衡势的分布。这些节点使得有理插值的收敛速率与理论速率一致，并稳步接近机器精度。该技术不仅可用于区间 $[0,1]$，还可扩展至角区域和多奇点情况。
{"title":"Barycentric rational interpolation of exponentially clustered poles","authors":"Kelong Zhao, Shuhuang Xiang","doi":"10.1093/imanum/drae040","DOIUrl":"https://doi.org/10.1093/imanum/drae040","url":null,"abstract":"We have developed a rational interpolation method for analytic functions with branch point singularities, which utilizes several exponentially clustered poles proposed by Trefethen and his collaborators (2021, Exponential node clustering at singularities for rational approximation, quadrature, and PDEs. Numer. Math., 147, 227–254). The key to the feasibility of this interpolation method is that the interpolation nodes approximately satisfy the distribution of the equilibrium potential. These nodes make the convergence rate of the rational interpolation consistent with the theoretical rates, and steadily approach machine accuracy. The technique can be used, not only for the interval $[0,1]$, but can also be extended to include corner regions and the case of multiple singularities.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553448","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}
Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem. In order to fill in this gap, we focus on first-order perturbation theory of the trust-region subproblem. The main contributions of this paper are three-fold. First, suppose that the TRS is in easy case, we give a sufficient condition under which the perturbed TRS is still in easy case. Secondly, with the help of the structure of the TRS and the classical eigenproblem perturbation theory, we perform first-order perturbation analysis on the Lagrange multiplier and the solution of the TRS, and define their condition numbers. Thirdly, we point out that the solution and the Lagrange multiplier could be well-conditioned even if TRS is in nearly hard case. The established results are computable, and are helpful to evaluate ill-conditioning of the large-scale TRS problem beforehand. Numerical experiments show the sharpness of the established bounds and the effectiveness of the proposed strategies.
{"title":"First-Order Perturbation Theory of Trust-Region Subproblem","authors":"Bo Feng, Gang Wu","doi":"10.1093/imanum/drae042","DOIUrl":"https://doi.org/10.1093/imanum/drae042","url":null,"abstract":"Trust-region subproblem (TRS) is an important problem arising in many applications such as numerical optimization, Tikhonov regularization of ill-posed problems and constrained eigenvalue problems. In recent decades, extensive works focus on how to solve the trust-region subproblem efficiently. To the best of our knowledge, there are few results on perturbation analysis of the trust-region subproblem. In order to fill in this gap, we focus on first-order perturbation theory of the trust-region subproblem. The main contributions of this paper are three-fold. First, suppose that the TRS is in easy case, we give a sufficient condition under which the perturbed TRS is still in easy case. Secondly, with the help of the structure of the TRS and the classical eigenproblem perturbation theory, we perform first-order perturbation analysis on the Lagrange multiplier and the solution of the TRS, and define their condition numbers. Thirdly, we point out that the solution and the Lagrange multiplier could be well-conditioned even if TRS is in nearly hard case. The established results are computable, and are helpful to evaluate ill-conditioning of the large-scale TRS problem beforehand. Numerical experiments show the sharpness of the established bounds and the effectiveness of the proposed strategies.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141553449","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}
Matthias J Ehrhardt, Erlend S Riis, Torbjørn Ringholm, Carola-Bibiane Schönlieb
Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimization that provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $text{O}(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak–Łojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients—the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh–Abe discrete gradient—as well as for a randomised Itoh–Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.
{"title":"A geometric integration approach to smooth optimization: foundations of the discrete gradient method","authors":"Matthias J Ehrhardt, Erlend S Riis, Torbjørn Ringholm, Carola-Bibiane Schönlieb","doi":"10.1093/imanum/drae037","DOIUrl":"https://doi.org/10.1093/imanum/drae037","url":null,"abstract":"Discrete gradient methods are geometric integration techniques that can preserve the dissipative structure of gradient flows. Due to the monotonic decay of the function values, they are well suited for general convex and nonconvex optimization problems. Both zero- and first-order algorithms can be derived from the discrete gradient method by selecting different discrete gradients. In this paper, we present a thorough analysis of the discrete gradient method for optimization that provides a solid theoretical foundation. We show that the discrete gradient method is well-posed by proving the existence of iterates for any positive time step, as well as uniqueness in some cases, and propose an efficient method for solving the associated discrete gradient equation. Moreover, we establish an $text{O}(1/k)$ convergence rate for convex objectives and prove linear convergence if instead the Polyak–Łojasiewicz inequality is satisfied. The analysis is carried out for three discrete gradients—the Gonzalez discrete gradient, the mean value discrete gradient, and the Itoh–Abe discrete gradient—as well as for a randomised Itoh–Abe method. Our theoretical results are illustrated with a variety of numerical experiments, and we furthermore demonstrate that the methods are robust with respect to stiffness.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"13 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489284","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 novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $alpha in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960le tau _{k}/tau _{k-1}<r^{*}(alpha )$, where $tau _{k}$ is the $k$th time-step size and $r^{*}(alpha )ge 4.660$ for $alpha in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.
通过局部-非局部拆分技术构建了近似于阶数为 $alpha in (0,1)$ 的 Caputo 分导数的变步长分式 BDF2 公式的新型离散梯度结构，即将分式 BDF2 公式拆分为类似于一阶导数的两步反向微分公式（BDF2）的局部部分和类似于 Caputo 导数的 L1 型公式的非局部部分。然后在弱步长比约束$0下建立了时间分式Cahn-Hilliard模型的变步长分式BDF2隐式方案的局部离散能量耗散规律。3960le tau _{k}/tau _{k-1}<r^{*}(alpha )$，其中$tau _{k}$ 是第k$个时间步长，$r^{*}(alpha )ge 4.660$ for $alpha in (0,1)$.本结果为[SINUM, 57: 218-237, Remark 6]中的开放问题提供了一个实际答案，并大大放宽了严格的步长比限制[Math.］更有趣的是，离散能量和相应的能量耗散规律分别与经典 Cahn-Hilliard 方程的变步长 BDF2 方法的相关离散能量和能量耗散规律渐近兼容。据我们所知，这种类型的能量耗散规律是首次针对卡普托导数的变步长 L2 型公式建立的。为了证明我们提出的方法的准确性和有效性，我们提供了带有自适应步进程序的数值示例。
{"title":"Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model","authors":"Hong-lin Liao, Nan Liu, Xuan Zhao","doi":"10.1093/imanum/drae034","DOIUrl":"https://doi.org/10.1093/imanum/drae034","url":null,"abstract":"A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $alpha in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960le tau _{k}/tau _{k-1}&lt;r^{*}(alpha )$, where $tau _{k}$ is the $k$th time-step size and $r^{*}(alpha )ge 4.660$ for $alpha in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.","PeriodicalId":56295,"journal":{"name":"IMA Journal of Numerical Analysis","volume":"334 1","pages":""},"PeriodicalIF":2.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141453029","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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