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Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems 基于字典的参数哈密顿系统在线自适应结构保全模型阶次缩减
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-02-05 DOI: 10.1007/s10444-023-10102-7
Robin Herkert, Patrick Buchfink, Bernard Haasdonk
{"title":"Dictionary-based online-adaptive structure-preserving model order reduction for parametric Hamiltonian systems","authors":"Robin Herkert, Patrick Buchfink, Bernard Haasdonk","doi":"10.1007/s10444-023-10102-7","DOIUrl":"https://doi.org/10.1007/s10444-023-10102-7","url":null,"abstract":"<p>Classical model order reduction (MOR) for parametric problems may become computationally inefficient due to large sizes of the required projection bases, especially for problems with slowly decaying Kolmogorov <i>n</i>-widths. Additionally, Hamiltonian structure of dynamical systems may be available and should be preserved during the reduction. In the current presentation, we address these two aspects by proposing a corresponding dictionary-based, online-adaptive MOR approach. The method requires dictionaries for the state-variable, non-linearities, and discrete empirical interpolation (DEIM) points. During the online simulation, local basis extensions/simplifications are performed in an online-efficient way, i.e., the runtime complexity of basis modifications and online simulation of the reduced models do not depend on the full state dimension. Experiments on a linear wave equation and a non-linear Sine-Gordon example demonstrate the efficiency of the approach.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139695846","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the balanced truncation error bound and sign parameters from arrowhead realizations 关于平衡截断误差约束和箭头实现的符号参数
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-01-31 DOI: 10.1007/s10444-024-10105-y
Sean Reiter, Tobias Damm, Mark Embree, Serkan Gugercin
{"title":"On the balanced truncation error bound and sign parameters from arrowhead realizations","authors":"Sean Reiter, Tobias Damm, Mark Embree, Serkan Gugercin","doi":"10.1007/s10444-024-10105-y","DOIUrl":"https://doi.org/10.1007/s10444-024-10105-y","url":null,"abstract":"<p>Balanced truncation and singular perturbation approximation for linear dynamical systems yield reduced order models that satisfy a well-known error bound involving the Hankel singular values. We show that this bound holds with equality for single-input, single-output systems, if the sign parameters corresponding to the truncated Hankel singular values are all equal. These signs are determined by a generalized state-space symmetry property of the corresponding linear model. For a special class of systems having arrowhead realizations, the signs can be determined directly from the off-diagonal entries of the corresponding arrowhead matrix. We describe how such arrowhead systems arise naturally in certain applications of network modeling and illustrate these results with a power system model that motivated this study.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139644107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Dual frames compensating for erasures—a non-canonical case 补偿擦除的双帧--非正则案例
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-01-25 DOI: 10.1007/s10444-023-10104-5
Ljiljana Arambašić, Diana Stoeva
{"title":"Dual frames compensating for erasures—a non-canonical case","authors":"Ljiljana Arambašić, Diana Stoeva","doi":"10.1007/s10444-023-10104-5","DOIUrl":"https://doi.org/10.1007/s10444-023-10104-5","url":null,"abstract":"<p>In this paper, we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set <i>E</i>. Starting from a frame <span>((x_n)_{n=1}^infty )</span> and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of <span>((x_n)_{nin E^c})</span> so that the perfect reconstruction can be obtained from the preserved frame coefficients. The work is motivated by methods using the canonical dual frame of <span>((x_n)_{n=1}^infty )</span>, which however do not extend automatically to the case when the canonical dual is replaced with another dual frame. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated. We also give several ways of computing a dual of the reduced frame, among which we are the most interested in the iterative procedure for computing this dual frame.</p>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139551114","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities 不同弹性张量和不同质量密度下弹性传输特征值问题的自适应有限元模型
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-01-17 DOI: 10.1007/s10444-023-10099-z
Shixi Wang, Hai Bi, Yidu Yang
{"title":"An adaptive FEM for the elastic transmission eigenvalue problem with different elastic tensors and different mass densities","authors":"Shixi Wang,&nbsp;Hai Bi,&nbsp;Yidu Yang","doi":"10.1007/s10444-023-10099-z","DOIUrl":"10.1007/s10444-023-10099-z","url":null,"abstract":"<div><p>The elastic transmission eigenvalue problem, arising from the inverse scattering theory, plays a critical role in the qualitative reconstruction methods for elastic media. This paper proposes and analyzes an a posteriori error estimator of the finite element method for solving the elastic transmission eigenvalue problem with different elastic tensors and different mass densities in <span>(mathbb {R}^{d}~(d=2,3))</span>. An adaptive algorithm based on the a posteriori error estimators is designed. Numerical results are provided to illustrate the efficiency of our adaptive algorithm.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139480386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A fractional osmosis model for image fusion 图像融合的分数渗透模型
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-01-08 DOI: 10.1007/s10444-023-10103-6
Mohammed Hachama, Fatiha Boutaous
{"title":"A fractional osmosis model for image fusion","authors":"Mohammed Hachama,&nbsp;Fatiha Boutaous","doi":"10.1007/s10444-023-10103-6","DOIUrl":"10.1007/s10444-023-10103-6","url":null,"abstract":"<div><p>This paper introduces a novel model for image fusion that is based on a fractional-order osmosis approach. The model incorporates a definition of osmosis energy that takes into account nonlocal pixel relationships using fractional derivatives and contrast change. The proposed model was subjected to theoretical and experimental investigation. The semigroup theory was used to demonstrate the existence and uniqueness of the evolution equation solution. Additionally, the model was validated and tested using numerical experiments and compared to local image fusion methods. The findings demonstrate that the proposed model outperforms the competitive local image fusion models.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139379370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Grassmann manifold handbook: basic geometry and computational aspects 格拉斯曼流形手册：基本几何和计算方面
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-01-05 DOI: 10.1007/s10444-023-10090-8
Thomas Bendokat, Ralf Zimmermann, P.-A. Absil
{"title":"A Grassmann manifold handbook: basic geometry and computational aspects","authors":"Thomas Bendokat,&nbsp;Ralf Zimmermann,&nbsp;P.-A. Absil","doi":"10.1007/s10444-023-10090-8","DOIUrl":"10.1007/s10444-023-10090-8","url":null,"abstract":"<div><p>The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-023-10090-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139109727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Variational methods for solving numerically magnetostatic systems 数值求解磁静力系统的变量方法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2024-01-04 DOI: 10.1007/s10444-023-10089-1
Patrick Ciarlet Jr., Erell Jamelot
{"title":"Variational methods for solving numerically magnetostatic systems","authors":"Patrick Ciarlet Jr.,&nbsp;Erell Jamelot","doi":"10.1007/s10444-023-10089-1","DOIUrl":"10.1007/s10444-023-10089-1","url":null,"abstract":"<div><p>In this paper, we study some techniques for solving numerically magnetostatic systems. We consider fairly general assumptions on the magnetic permeability tensor. It is elliptic, but can be nonhermitian. In particular, we revisit existing classical variational methods and propose new numerical methods. The numerical approximation is either based on the classical edge finite elements or on continuous Lagrange finite elements. For the first type of discretization, we rely on the design of a new, mixed variational formulation that is obtained with the help of <i>T</i>-coercivity. The numerical method can be related to a perturbed approach for solving mixed problems in electromagnetism. For the second type of discretization, we rely on an augmented variational formulation obtained with the help of the weighted regularization method.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139091090","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical investigation of agent-controlled pedestrian dynamics using a structure-preserving finite volume scheme 使用结构保留有限体积方案对代理控制的行人动力学进行数值研究
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2023-12-28 DOI: 10.1007/s10444-023-10098-0
Jan-Frederik Pietschmann, Ailyn Stötzner, Max Winkler
{"title":"Numerical investigation of agent-controlled pedestrian dynamics using a structure-preserving finite volume scheme","authors":"Jan-Frederik Pietschmann,&nbsp;Ailyn Stötzner,&nbsp;Max Winkler","doi":"10.1007/s10444-023-10098-0","DOIUrl":"10.1007/s10444-023-10098-0","url":null,"abstract":"<div><p>We provide a numerical realization of an optimal control problem for pedestrian motion with agents that was analyzed in Herzog et al. (<i>Appl. Math. Optim.</i> 88(3):87, 2023). The model consists of a regularized variant of Hughes’ model for pedestrian dynamics coupled to ordinary differential equations that describe the motion of agents which are able to influence the crowd via attractive forces. We devise a finite volume scheme that preserves the box constraints that are inherent in the model and discuss some of its properties. We apply our scheme to an objective functional tailored to the case of an evacuation scenario. Finally, numerical simulations for several practically relevant geometries are performed.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-023-10098-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139050758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions 具有非均质迪里希勒边界条件的不可压缩流动的三步缺陷校正稳定算法
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2023-12-27 DOI: 10.1007/s10444-023-10101-8
Bo Zheng, Yueqiang Shang
{"title":"A three-step defect-correction stabilized algorithm for incompressible flows with non-homogeneous Dirichlet boundary conditions","authors":"Bo Zheng,&nbsp;Yueqiang Shang","doi":"10.1007/s10444-023-10101-8","DOIUrl":"10.1007/s10444-023-10101-8","url":null,"abstract":"<div><p>Based on two-grid discretizations and quadratic equal-order finite elements for the velocity and pressure approximations, we develop a three-step defect-correction stabilized algorithm for the incompressible Navier-Stokes equations, where non-homogeneous Dirichlet boundary conditions are considered and high Reynolds numbers are allowed. In this developed algorithm, we first solve an artificial viscosity stabilized nonlinear problem on a coarse grid in a defect step and then correct the resulting residual by solving two stabilized and linearized problems on a fine grid in correction steps. While the fine grid correction problems have the same stiffness matrices with only different right-hand sides. We use a variational multiscale method to stabilize the system, making the algorithm has a broad range of potential applications in the simulation of high Reynolds number flows. Under the weak uniqueness condition, we give a stability analysis of the present algorithm, analyze the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we perform a series of numerical examples to demonstrate the promise of the proposed algorithm.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139041386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations 非自邻接和不定椭圆方程 CDG 有限元的二阶超收敛性
IF 1.7 3区 数学
Advances in Computational Mathematics Pub Date : 2023-12-22 DOI: 10.1007/s10444-023-10100-9
Xiu Ye, Shangyou Zhang
{"title":"Order two superconvergence of the CDG finite elements for non-self adjoint and indefinite elliptic equations","authors":"Xiu Ye,&nbsp;Shangyou Zhang","doi":"10.1007/s10444-023-10100-9","DOIUrl":"10.1007/s10444-023-10100-9","url":null,"abstract":"<div><p>A conforming discontinuous Galerkin (CDG) finite element method is designed for solving second order non-self adjoint and indefinite elliptic equations. Unlike other discontinuous Galerkin (DG) methods, the numerical trace on the edge/triangle between two elements is not the average of two discontinuous <span>(P_k)</span> functions, but a lifted <span>(P_{k+2})</span> function from four (eight in 3D) nearby <span>(P_k)</span> functions. While all existing DG methods have the optimal order of convergence, this CDG method has a superconvergence of order two above the optimal order when solving general second order elliptic equations. Due to the superconvergence, a post-process lifts a <span>(P_k)</span> CDG solution to a quasi-optimal <span>(P_{k+2})</span> solution on each element. Numerical tests in 2D and 3D are provided confirming the theory.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138840193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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