Abstract A high-performance sparse model is very important for processing high-dimensional data. Therefore, based on the quadratic approximate greed pursuit (QAGP) method, we can make full use of the information of the quadratic lower bound of its approximate function to get the relaxation quadratic approximate greed pursuit (RQAGP) method. The calculation process of the RQAGP method is to construct two inexact quadratic approximation functions by using the m -strongly convex and L -smooth characteristics of the objective function and then solve the approximation function iteratively by using the Iterative Hard Thresholding (IHT) method to get the solution of the problem. The convergence analysis is given, and the performance of the method in the sparse logistic regression model is verified on synthetic data and real data sets. The results show that the RQAGP method is effective.
{"title":"Relaxation Quadratic Approximation Greedy Pursuit Method Based on Sparse Learning","authors":"Shihai Li, Changfeng Ma","doi":"10.1515/cmam-2023-0050","DOIUrl":"https://doi.org/10.1515/cmam-2023-0050","url":null,"abstract":"Abstract A high-performance sparse model is very important for processing high-dimensional data. Therefore, based on the quadratic approximate greed pursuit (QAGP) method, we can make full use of the information of the quadratic lower bound of its approximate function to get the relaxation quadratic approximate greed pursuit (RQAGP) method. The calculation process of the RQAGP method is to construct two inexact quadratic approximation functions by using the m -strongly convex and L -smooth characteristics of the objective function and then solve the approximation function iteratively by using the Iterative Hard Thresholding (IHT) method to get the solution of the problem. The convergence analysis is given, and the performance of the method in the sparse logistic regression model is verified on synthetic data and real data sets. The results show that the RQAGP method is effective.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135549142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Recently, H(div) mathbf{H}(mathrm{div}) -conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates. In order to ensure corresponding large-scale computation, this contribution provides novel Raviart–Thomas basis functions, robust regarding the anisotropy of a given triangulation. This new set of basis functions on simplices uses a hierarchical approach, and the orientation of the basis functions is inherited from the lowest-order case. In the higher-order case, the new basis functions can be written as a combination of the lowest-order Raviart–Thomas elements and higher-order Lagrange-elements. This ensures robustness regarding the mesh anisotropy and assembling strategies as demonstrated in the numerical experiments.
{"title":"Novel Raviart–Thomas Basis Functions on Anisotropic Finite Elements","authors":"Fleurianne Bertrand","doi":"10.1515/cmam-2022-0235","DOIUrl":"https://doi.org/10.1515/cmam-2022-0235","url":null,"abstract":"Abstract Recently, <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"bold\">H</m:mi> <m:mo></m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>div</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> mathbf{H}(mathrm{div}) -conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates. In order to ensure corresponding large-scale computation, this contribution provides novel Raviart–Thomas basis functions, robust regarding the anisotropy of a given triangulation. This new set of basis functions on simplices uses a hierarchical approach, and the orientation of the basis functions is inherited from the lowest-order case. In the higher-order case, the new basis functions can be written as a combination of the lowest-order Raviart–Thomas elements and higher-order Lagrange-elements. This ensures robustness regarding the mesh anisotropy and assembling strategies as demonstrated in the numerical experiments.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135011096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.
{"title":"Space-Time Approximation of Local Strong Solutions to the 3D Stochastic Navier–Stokes Equations","authors":"Dominic Breit, Alan Dodgson","doi":"10.1515/cmam-2023-0052","DOIUrl":"https://doi.org/10.1515/cmam-2023-0052","url":null,"abstract":"Abstract We consider the 3D stochastic Navier–Stokes equation on the torus. Our main result concerns the temporal and spatio-temporal discretisation of a local strong pathwise solution. We prove optimal convergence rates for the energy error with respect to convergence in probability, that is convergence of order (up to) 1 in space and of order (up to) 1/2 in time. The result holds up to the possible blow-up of the (time-discrete) solution. Our approach is based on discrete stopping times for the (time-discrete) solution.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135747554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.
{"title":"A Convergent Entropy-Dissipating BDF2 Finite-Volume Scheme for a Population Cross-Diffusion System","authors":"Ansgar Jüngel, Martin Vetter","doi":"10.1515/cmam-2023-0009","DOIUrl":"https://doi.org/10.1515/cmam-2023-0009","url":null,"abstract":"Abstract A second-order backward differentiation formula (BDF2) finite-volume discretization for a nonlinear cross-diffusion system arising in population dynamics is studied. The numerical scheme preserves the Rao entropy structure and conserves the mass. The existence and uniqueness of discrete solutions and their large-time behavior as well as the convergence of the scheme are proved. The proofs are based on the G-stability of the BDF2 scheme, which provides an inequality for the quadratic Rao entropy and hence suitable a priori estimates. The novelty is the extension of this inequality to the system case. Some numerical experiments in one and two space dimensions underline the theoretical results.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135747555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 𝑑 approximately compared with two-scale finite element methods when Ω=(0,1)d Omega=(0,1)^{d} . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly.
摘要本文给出了一类具有对称解的偏微分方程的对称双尺度有限元方法。利用这些方法，将精细张量积网格上的有限元逼近简化为更粗网格和不变精细网格上的有限元逼近。理论和包括电子结构计算在内的数值都表明，所得到的近似仍然保持渐近最优精度。采用对称双尺度有限元方法，当Ω =(0,1) d Omega=(0,1)^{d}时，与双尺度有限元方法相比，计算成本可进一步约降低𝑑。因此，对称双尺度有限元方法大大降低了计算成本。
{"title":"Symmetrized Two-Scale Finite Element Discretizations for Partial Differential Equations with Symmetric Solutions","authors":"Pengyu Hou, Fang Liu, Aihui Zhou","doi":"10.1515/cmam-2022-0192","DOIUrl":"https://doi.org/10.1515/cmam-2022-0192","url":null,"abstract":"Abstract In this paper, some symmetrized two-scale finite element methods are proposed for a class of partial differential equations with symmetric solutions. With these methods, the finite element approximation on a fine tensor-product grid is reduced to the finite element approximations on a much coarser grid and a univariant fine grid. It is shown by both theory and numerics including electronic structure calculations that the resulting approximations still maintain an asymptotically optimal accuracy. By symmetrized two-scale finite element methods, the computational cost can be reduced further by a factor of 𝑑 approximately compared with two-scale finite element methods when <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mo>=</m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> Omega=(0,1)^{d} . Consequently, symmetrized two-scale finite element methods reduce computational cost significantly.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135747204","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain Ω on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on ∂Ω partialOmega . In this paper, easily implementable minimal residual discretizations are constructed which yield quasi-optimal approximation from the employed trial space, in which the evaluation of fractional Sobolev norms is fully avoided.
{"title":"A Convenient Inclusion of Inhomogeneous Boundary Conditions in Minimal Residual Methods","authors":"Rob Stevenson","doi":"10.1515/cmam-2023-0072","DOIUrl":"https://doi.org/10.1515/cmam-2023-0072","url":null,"abstract":"Abstract Inhomogeneous essential boundary conditions can be appended to a well-posed PDE to lead to a combined variational formulation. The domain of the corresponding operator is a Sobolev space on the domain Ω on which the PDE is posed, whereas the codomain is a Cartesian product of spaces, among them fractional Sobolev spaces of functions on <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo rspace=\"0em\">∂</m:mo> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> </m:math> partialOmega . In this paper, easily implementable minimal residual discretizations are constructed which yield quasi-optimal approximation from the employed trial space, in which the evaluation of fractional Sobolev norms is fully avoided.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135444034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.
{"title":"Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment","authors":"Pauline Achieng, F. Berntsson, V. Kozlov","doi":"10.1515/cmam-2022-0244","DOIUrl":"https://doi.org/10.1515/cmam-2022-0244","url":null,"abstract":"Abstract We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41823031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Recent Advances in Finite Element Methods","authors":"Sven Beuchler, A. Rösch","doi":"10.1515/cmam-2023-0135","DOIUrl":"https://doi.org/10.1515/cmam-2023-0135","url":null,"abstract":"","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"10 1","pages":"813 - 815"},"PeriodicalIF":1.3,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139354992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler
This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.
{"title":"Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations","authors":"Philip Freese, Dietmar Gallistl, Daniel Peterseim, Timo Sprekeler","doi":"10.1515/cmam-2023-0040","DOIUrl":"https://doi.org/10.1515/cmam-2023-0040","url":null,"abstract":"This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition. The construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization. The degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems. The rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.","PeriodicalId":48751,"journal":{"name":"Computational Methods in Applied Mathematics","volume":"113 2","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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