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optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs

JOURNAL OF NUMERICAL MATHEMATICS 2024年

作者： Bringmann, Philipp Brunner, Maximilian Praetorius, Dirk Streitberger, Julian TU Wien Inst Anal & Sci Comp Vienna Austria

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest G(u star) with a linear goal functional G and the solution u star to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

关键词： goal-oriented adaptive finite element method linear quantity of interest iterative solver nonsymmetric partial differential equations optimal convergence rates optimal complexity

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MULTIGRID SOLVERS FOR THE DE RHAM COMPLEX WITH optimal complexity IN POLYNOMIAL DEGREE\ast

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2024年第3期46卷 A1549-A1573页

作者： Brubeck, Pablo D. Farrell, Patrick E. Univ Oxford Math Inst Oxford OX2 6GG England

the construction of fast preconditioners for more complicated problems. In this work, we present multigrid solvers for high-order finite-element discretizations of these Riesz maps with the same time and space complexity as sum-factorized operator application, i.e., with optimal complexity in polynomial degree in the context of Krylov methods. The key idea of our approach is to build new finite elements for each space in the de Rham complex with orthogonality properties in both the L2and H (d)-inner products (d \in \{ grad, curl, div\} ) on the reference hexahedron. The resulting sparsity enables the fast solution of the patch problems arising in the Pavarino, Arnold--Falk--Winther, and Hiptmair space decompositions in the separable case. In the nonseparable case, the method can be applied to an auxiliary operator that is sparse by construction. With exact Cholesky factorizations of the sparse patch problems, the application complexity is optimal, but the setup costs and storage are not. We overcome this with the finer Hiptmair space decomposition and the use of incomplete Cholesky factorizations imposing the sparsity pattern arising from static condensation, which applies whether static condensation is used for the solver or not. This yields multigrid relaxations with time and space complexity that are both optimal in the polynomial degree.

关键词： computational results Key words. preconditioning de Rham complex high-order addi- tive Schwarz incomplete Cholesky optimal complexity

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Accelerated first-order methods for large-scale convex optimization: nearly optimal complexity under strong convexity

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MATHEMATICAL METHODS OF OPERATIONS RESEARCH 2019年第3期89卷 319-353页

作者： Ahookhosh, Masoud Univ Vienna Fac Math Oskar Morgenstern Pl 1 A-1090 Vienna Austria Katholieke Univ Leuven Dept Elect Engn ESAT STADIUS Kasteelpk Arenberg 10 B-3001 Leuven Belgium

We introduce four accelerated (sub)gradient algorithms (ASGA) for solving several classes of convex optimization problems. More specifically, we propose two estimation sequences majorizing the objective function and develop two iterative schemes for each of them. In both cases, the first scheme requires the smoothness parameter and a Holder constant, while the second scheme is parameter-free (except for the strong convexity parameter which we set zero if it is not available) at the price of applying a finitely terminated backtracking line search. The proposed algorithms attain the optimal complexity for smooth problems with Lipschitz continuous gradients, nonsmooth problems with bounded variation of subgradients, and weakly smooth problems with Holder continuous gradients. Further, for strongly convex problems, they are optimal for smooth problems while nearly optimal for nonsmooth and weakly smooth problems. Finally, numerical results for some applications in sparse optimization and machine learning are reported, which confirm the theoretical foundations.

关键词： Structured nonsmooth convex optimization First-order black-box oracle Estimation sequence Strong convexity optimal complexity

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Hypersequent calculi for non-normal modal and deontic logics: countermodels and optimal complexity

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JOURNAL OF LOGIC AND COMPUTATION 2021年第1期31卷 67-111页

作者： Dalmonte, Tiziano Lellmann, Bjorn Olivetti, Nicola Pimentel, Elaine Aix Marseille Univ Univ Toulon LIS CNRS F-13013 Marseille France Tech Univ Wien A-1040 Vienna Austria Univ Fed Rio Grande do Norte Dept Matemat BR-59078970 Natal RN Brazil

We present some hypersequent calculi for all systems of the classical cube and their extensions with axioms T, P and D and for every n >= 1, rule RDn+. The calculi are internal as they only employ the language of the logic, plus additional structural connectives. We show that the calculi are complete with respect to the corresponding axiomatization by a syntactic proof of cut elimination. Then, we define a terminating proof search strategy in the hypersequent calculi and show that it is optimal for coNP-complete logics. Moreover, we show that from every failed proof of a formula or hypersequent it is possible to directly extract a countermodel of it in the bi-neighbourhood semantics of polynomial size for coNP logics, and for regular logics also in the relational semantics. We finish the paper by giving a translation between hypersequent rule applications and derivations in a labelled system for the classical cube.

关键词： Non-normal modal logic deontic logic hypersequent calculus neighbourhood semantics countermodels optimal complexity

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complexity-optimal AND PARAMETER-FREE FIRST-ORDER METHODS FOR FINDING STATIONARY POINTS OF COMPOSITE OPTIMIZATION PROBLEMS

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SIAM JOURNAL ON OPTIMIZATION 2024年第3期34卷 3005-3032页

作者： Kong, Weiwei Oak Ridge Natl Lab Comp Sci & Math Div Oak Ridge TN 37830 USA

This paper develops and analyzes an accelerated proximal descent method for finding stationary points of nonconvex composite optimization problems. The objective function is of the form f + h, where h is a proper closed convex function, f is a differentiable function on the domain of h, and \nabla f is Lipschitz continuous on the domain of h. The main advantage of this method is that it is ``parameter-free"" in the sense that it does not require knowledge of the Lipschitz constant of \nabla f or of any global topological properties of f. It is shown that the proposed method can obtain an \varepsilon -approximate stationary point with iteration complexity bounds that are optimal, up to logarithmic terms over \varepsilon , in both the convex and nonconvex settings. Some discussion is also given about how the proposed method can be leveraged in other existing optimization frameworks, such as min-max smoothing and penalty frameworks for constrained programming, to create more specialized parameter-free methods. Finally, numerical experiments are presented to support the practical viability of the method.

关键词： nonconvex composite optimization first-order accelerated gradient method itera tion complexity inexact proximal point method parameter-free adaptive optimal complexity

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optimal System's complexity, An Architecture Perspective

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Procedia Computer Science 2012年 12卷 63-68页

作者： Babak Heydari Kia Dalili Stevens Institute of Technology School of Systems and Enterprises Hoboken New Jersey USA

Systems complexity is driven by more connectivity among constituents and increased environmental uncertainties. Modularity is a system's mechanism to manage complexity in the presence of environmental heterogeneities. Here, a general design decision framework for modularity and fractionation in complex systems is presented. This framework incorporates spatial and temporal heterogeneity of the environment, adaptability of the system and costs associated with modularity. It is argued that as the space-time heterogeneity of the environment increases, higher levels of modularity are needed. To determine the optimal level of complexity in a modular design, a four-level modularity pyramid is introduced. This pyramid takes into account functional and physical dimensions of modularity and allows for resource sharing to enables dynamic modularity. Moving up from each level is quantified by an M+ operation that calculates the net value of increased modularity for each sub-system.

关键词： optimal complexity Modularity Complex Networks Emergence of Modularity

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A MULTILEVEL CORRECTION TYPE OF ADAPTIVE FINITE ELEMENT METHOD FOR EIGENVALUE PROBLEMS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2018年第6期40卷 A4208-A4235页

作者： Hong, Qichen Xie, Hehu Xu, Fei Chinese Acad Sci Acad Math & Syst Sci ICMSEC LSEC Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Math Sci Beijing 100049 Peoples R China Beijing Univ Technol Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China

An adaptive finite element method for eigenvalue problems is proposed based on the multilevel correction scheme. Different from the standard adaptive finite element method which requires solving eigenvalue problems on adaptively refined triangulations, our scheme only includes solving associated boundary value problems on adaptive triangulations and small scale eigenvalue problems on a very low dimensional space. Since there is no eigenvalue problem to be solved on adaptively refined triangulations, which needs more computation and memory than solving associated boundary value problem, the efficiency of the proposed method can be improved to be similar to that of the adaptive finite element method for associated boundary value problems. The convergence and optimal complexity is theoretically verified and numerically demonstrated.

关键词： eigenvalue problem multilevel correction adaptive finite element method convergence optimal complexity

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A Universal Accelerated Primal-Dual Method for Convex Optimization Problems

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2024年第1期201卷 280-312页

作者： Luo, Hao Chongqing Normal Univ Natl Ctr Appl Math Chongqing Chongqing 401331 Peoples R China Peking Univ Chongqing Res Inst Big Data Chongqing 401121 Peoples R China

This work presents a universal accelerated primal-dual method for affinely constrained convex optimization problems. It can handle both Lipschitz and Holder gradients but does not need to know the smoothness level of the objective function. In line search part, it uses dynamically decreasing parameters and produces approximate Lipschitz constant with moderate magnitude. In addition, based on a suitable discrete Lyapunov function and tight decay estimates of some differential/difference inequalities, a universal optimal mixed-type convergence rate is established. Some numerical tests are provided to confirm the efficiency of the proposed method.

关键词： Convex optimization Primal-dual method Mixed-type estimate optimal complexity Bregman divergence Lyapunov function

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Time-Domain Method Having a Naturally Diagonal Mass Matrix Independent of Element Shape for General Electromagnetic Analysis-2-D Formulations

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION 2017年第3期65卷 1202-1214页

作者： Yan, Jin Jiao, Dan Purdue Univ Sch Elect & Comp Engn W Lafayette IN 47907 USA

In this paper, we present a new time-domain method that has a naturally diagonal mass matrix and thereby a strict linear computational complexity per time step, regardless of whether the discretization is a structured grid or an unstructured mesh. This property is obtained independent of the element shape used for discretization. No interpolations, projections, and mass lumping are required. The accuracy and stability of the proposed method are both theoretically guaranteed. In addition, no dual mesh is needed and the tangential continuity of the fields is satisfied across the element interface. The flexible framework of the proposed method also allows for a straightforward extension to higher order accuracy in both electric and magnetic fields. Numerical experiments have been conducted on a variety of unstructured triangular-element meshes. Correlations with analytical solutions and the time-domain finite-element method have validated the accuracy and generality of the proposed new time-domain method.

关键词： Diagonal mass matrix electromagnetic analysis finite-difference time-domain (FDTD) methods linear complexity optimal complexity time-domain finite-element methods (TDFEMs) time-domain methods unstructured mesh

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A novel adaptive finite element method for the ground state solution of Bose-Einstein condensates

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APPLIED MATHEMATICS AND COMPUTATION 2020年 385卷 125404-125404页

作者： Xu, Fei Huang, Qiumei Wang, Min Ma, Hongkun Beijing Univ Technol Fac Sci Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Sun Yat Sen Univ Sch Business Guangzhou 510275 Guangdong Peoples R China Zhuhai Huafa Investment Holdings Co Ltd Zhuhai 519031 Guangdong Peoples R China

In this study we propose a novel adaptive finite element method for the ground state solution of Bose-Einstein Condensates (BEC). Different from the classical adaptive scheme applied to BEC which needs to solve a nonlinear eigenvalue model directly on each adaptive finite element space, our scheme requires to solve a linear elliptic boundary value problem on current adaptive space and a nonlinear eigenvalue model on a quite low-dimensional correction space. Further, the linear elliptic boundary value problem is solved by adaptive multigrid iterations. Since there is no nonlinear eigenvalue model to be solved directly on the adaptive spaces, the solving efficiency can be improved evidently. In addition, the convergence analysis of the proposed adaptive algorithm are derived numerically and theoretically. (C) 2020 Elsevier Inc. All rights reserved.

关键词： Bose-Einstein condensates Adaptive multigrid method Multilevel correction method Convergence optimal complexity

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