Pub Date : 2023-09-28DOI: 10.1007/s00211-023-01368-6
Michele Benzi, Michele Rinelli, Igor Simunec
Abstract We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix A , defined as $${{,textrm{tr},}}(f(A))$$ tr(f(A)) where $$f(x)=-xlog x$$ f(x)=-xlogx . After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.
摘要考虑一个大的、稀疏的、对称的正半定矩阵a的von Neumann熵的近似问题，定义为$${{,textrm{tr},}}(f(A))$$ tr (f (a))，其中$$f(x)=-xlog x$$ f (x) = - x log x。在建立了该矩阵函数的一些有用性质之后，我们考虑在两种近似方法中使用多项式和有理Krylov子空间算法，即随机迹估计和基于图着色的探测技术。我们开发了用于算法实现的误差界和启发式算法。对不同类型网络的密度矩阵进行了数值实验，验证了该方法的有效性。
{"title":"Computation of the von Neumann entropy of large matrices via trace estimators and rational Krylov methods","authors":"Michele Benzi, Michele Rinelli, Igor Simunec","doi":"10.1007/s00211-023-01368-6","DOIUrl":"https://doi.org/10.1007/s00211-023-01368-6","url":null,"abstract":"Abstract We consider the problem of approximating the von Neumann entropy of a large, sparse, symmetric positive semidefinite matrix A , defined as $${{,textrm{tr},}}(f(A))$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mtext>tr</mml:mtext> <mml:mspace /> </mml:mrow> <mml:mo>(</mml:mo> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> where $$f(x)=-xlog x$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>x</mml:mi> <mml:mo>log</mml:mo> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> . After establishing some useful properties of this matrix function, we consider the use of both polynomial and rational Krylov subspace algorithms within two types of approximations methods, namely, randomized trace estimators and probing techniques based on graph colorings. We develop error bounds and heuristics which are employed in the implementation of the algorithms. Numerical experiments on density matrices of different types of networks illustrate the performance of the methods.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135386246","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}
Pub Date : 2023-09-20DOI: 10.1007/s00211-023-01372-w
Daniel Kressner, Stefano Massei, Junli Zhu
Abstract This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of parallel-in-time integration leading to a class of algorithms called ParaDiag. We develop and analyze two novel approaches for the numerical solution of such equations. Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications. Both approaches avoid the use of iterative refinement needed by ParaDiag and related space-time approaches in order to attain good accuracy. In turn, our new algorithms have the potential to outperform, sometimes significantly, existing methods. This potential is demonstrated for several different types of PDEs.
{"title":"Improved ParaDiag via low-rank updates and interpolation","authors":"Daniel Kressner, Stefano Massei, Junli Zhu","doi":"10.1007/s00211-023-01372-w","DOIUrl":"https://doi.org/10.1007/s00211-023-01372-w","url":null,"abstract":"Abstract This work is concerned with linear matrix equations that arise from the space-time discretization of time-dependent linear partial differential equations (PDEs). Such matrix equations have been considered, for example, in the context of parallel-in-time integration leading to a class of algorithms called ParaDiag. We develop and analyze two novel approaches for the numerical solution of such equations. Our first approach is based on the observation that the modification of these equations performed by ParaDiag in order to solve them in parallel has low rank. Building upon previous work on low-rank updates of matrix equations, this allows us to make use of tensorized Krylov subspace methods to account for the modification. Our second approach is based on interpolating the solution of the matrix equation from the solutions of several modifications. Both approaches avoid the use of iterative refinement needed by ParaDiag and related space-time approaches in order to attain good accuracy. In turn, our new algorithms have the potential to outperform, sometimes significantly, existing methods. This potential is demonstrated for several different types of PDEs.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"59 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308154","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}
Pub Date : 2023-09-20DOI: 10.1007/s00211-023-01374-8
Konstantinos Chrysafinos, Dimitra Plaka
{"title":"Analysis and approximations of an optimal control problem for the Allen–Cahn equation","authors":"Konstantinos Chrysafinos, Dimitra Plaka","doi":"10.1007/s00211-023-01374-8","DOIUrl":"https://doi.org/10.1007/s00211-023-01374-8","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308335","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}
Pub Date : 2023-09-20DOI: 10.1007/s00211-023-01369-5
Andreas Bartel, Michael Günther, Birgit Jacob, Timo Reis
Abstract A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.
{"title":"Operator splitting based dynamic iteration for linear differential-algebraic port-Hamiltonian systems","authors":"Andreas Bartel, Michael Günther, Birgit Jacob, Timo Reis","doi":"10.1007/s00211-023-01369-5","DOIUrl":"https://doi.org/10.1007/s00211-023-01369-5","url":null,"abstract":"Abstract A dynamic iteration scheme for linear differential-algebraic port-Hamiltonian systems based on Lions–Mercier-type operator splitting methods is developed. The dynamic iteration is monotone in the sense that the error is decreasing and no stability conditions are required. The developed iteration scheme is even new for linear port-Hamiltonian systems governed by ODEs. The obtained algorithm is applied to a multibody system and an electrical network.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136308339","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}
Pub Date : 2023-09-13DOI: 10.1007/s00211-023-01371-x
Hassan, Muhammad, Maday, Yvon, Wang, Yipeng
Abstract The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an $$L^2$$ L2 -type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.
{"title":"Analysis of the single reference coupled cluster method for electronic structure calculations: the full-coupled cluster equations","authors":"Hassan, Muhammad, Maday, Yvon, Wang, Yipeng","doi":"10.1007/s00211-023-01371-x","DOIUrl":"https://doi.org/10.1007/s00211-023-01371-x","url":null,"abstract":"Abstract The central problem in electronic structure theory is the computation of the eigenvalues of the electronic Hamiltonian—a semi-unbounded, self-adjoint operator acting on an $$L^2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -type Hilbert space of antisymmetric functions. Coupled cluster (CC) methods, which are based on a non-linear parameterisation of the sought-after eigenfunction and result in non-linear systems of equations, are the method of choice for high-accuracy quantum chemical simulations. The existing numerical analysis of coupled cluster methods relies on a local, strong monotonicity property of the CC function that is valid only in a perturbative regime, i.e., when the sought-after ground state CC solution is sufficiently close to zero. In this article, we introduce a new well-posedness analysis for the single reference coupled cluster method based on the invertibility of the CC derivative. Under the minimal assumption that the sought-after eigenfunction is intermediately normalisable and the associated eigenvalue is isolated and non-degenerate, we prove that the continuous (infinite-dimensional) CC equations are always locally well-posed. Under the same minimal assumptions and provided that the discretisation is fine enough, we prove that the discrete Full-CC equations are locally well-posed, and we derive residual-based error estimates with guaranteed positive constants. Preliminary numerical experiments indicate that the constants that appear in our estimates are a significant improvement over those obtained from the local monotonicity approach.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134989487","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}
Pub Date : 2023-09-11DOI: 10.1007/s00211-023-01373-9
Rob Stevenson, Johannes Storn
Abstract We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.
{"title":"Interpolation operators for parabolic problems","authors":"Rob Stevenson, Johannes Storn","doi":"10.1007/s00211-023-01373-9","DOIUrl":"https://doi.org/10.1007/s00211-023-01373-9","url":null,"abstract":"Abstract We introduce interpolation operators with approximation and stability properties suited for parabolic problems in primal and mixed formulations. We derive localized error estimates for tensor product meshes (occurring in classical time-marching schemes) as well as locally in space-time refined meshes.","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135938882","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}
Pub Date : 2023-08-30DOI: 10.1007/s00211-023-01370-y
Eric Hallman, Ilse C. F. Ipsen
{"title":"Precision-aware deterministic and probabilistic error bounds for floating point summation","authors":"Eric Hallman, Ilse C. F. Ipsen","doi":"10.1007/s00211-023-01370-y","DOIUrl":"https://doi.org/10.1007/s00211-023-01370-y","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136080147","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}
Pub Date : 2023-06-01DOI: 10.1007/s00211-023-01357-9
S. Chandler-Wilde, E. Spence
{"title":"Correction to: Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains","authors":"S. Chandler-Wilde, E. Spence","doi":"10.1007/s00211-023-01357-9","DOIUrl":"https://doi.org/10.1007/s00211-023-01357-9","url":null,"abstract":"","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":"154 1","pages":"319-321"},"PeriodicalIF":2.1,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46189064","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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