Approximate Schur Complement Multilevel Methods for General Sparse Systems

2000 ◽  
pp. 52-58
Author(s):  
Michele Benzi ◽  
Michael DeLong
Keyword(s):  
Schur Complement ◽  
Multilevel Methods ◽  
Sparse Systems

Analysis of Robust Parameter-Free Multilevel Methods for Neumann Boundary Control Problems

2013 ◽  
Vol 13 (2) ◽  
pp. 207-235
Author(s):  
Etereldes Gonçalves ◽  
Marcus Sarkis
Keyword(s):  
Hessian Matrix ◽  
Control Variable ◽  
Neumann Boundary ◽  
Schur Complement ◽  
Multilevel Methods ◽  
Polygonal Domain ◽  
Linear Quadratic ◽  
Boundary Control Problems ◽  
Robust Parameter ◽  
The One

Abstract. We consider a linear-quadratic elliptic control problem (LQECP) where the control variable corresponds to the Neumann data on the boundary of a convex polygonal domain. The optimal control unknown is the one for which the harmonic extension approximates best a specified target in the interior of the domain. We propose multilevel preconditioners for the reduced system (the discrete Hessian system) resulting from the application of the Schur complement method to the discrete LQECP. In order to derive robust preconditioners with respect to stabilization parameters, we first show that the continuous reduced Hessian operator and the corresponding discrete Hessian matrix are associated to a linear combination of fractional negative Sobolev norms. Then we propose a preconditioner based on multilevel methods, including cases where the stabilization parameters are set equal to zero. We also present numerical experiments which agree with the theoretical results.

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

Analysis between graph-based and Power Transfer Distribution Factors (PTDF)-based model reduction methods in Electric Power Systems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Diego A. Monroy-Ortiz ◽  
Sergio A. Dorado-Rojas ◽  
Eduardo Mojica-Nava ◽  
Sergio Rivera
Keyword(s):  
Power Systems ◽  
Power System ◽  
Model Reduction ◽  
Electrical Power ◽  
Schur Complement ◽  
Power Transfer ◽  
Electrical Power System ◽  
Admittance Matrix ◽  
Test Beds ◽  
Power Transfer Distribution Factors

Abstract This article presents a comparison between two different methods to perform model reduction of an Electrical Power System (EPS). The first is the well-known Kron Reduction Method (KRM) that is used to remove the interior nodes (also known as internal, passive, or load nodes) of an EPS. This method computes the Schur complement of the primitive admittance matrix of an EPS to obtain a reduced model that preserves the information of the system as seen from to the generation nodes. Since the primitive admittance matrix is equivalent to the Laplacian of a graph that represents the interconnections between the nodes of an EPS, this procedure is also significant from the perspective of graph theory. On the other hand, the second procedure based on Power Transfer Distribution Factors (PTDF) uses approximations of DC power flows to define regions to be reduced within the system. In this study, both techniques were applied to obtain reduced-order models of two test beds: a 14-node IEEE system and the Colombian power system (1116 buses), in order to test scalability. In analyzing the reduction of the test beds, the characteristics of each method were classified and compiled in order to know its advantages depending on the type of application. Finally, it was found that the PTDF technique is more robust in terms of the definition of power transfer in congestion zones, while the KRM method may be more accurate.

scholarly journals Integrable and Chaotic Systems Associated with Fractal Groups

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 237
Author(s):  
Rostislav Grigorchuk ◽  
Supun Samarakoon
Keyword(s):  
Operator Algebras ◽  
Probabilistic Approach ◽  
Schur Complement ◽  
Automata Theory ◽  
Random Schrödinger Operators ◽  
Rational Maps ◽  
Holomorphic Dynamics ◽  
Von Neumann ◽  
Intermediate Growth ◽  
Quasi Crystals

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems.

scholarly journals Augmented Lagrangian preconditioners for the Oseen–Frank model of nematic and cholesteric liquid crystals

2021 ◽  
Author(s):  
Jingmin Xia ◽  
Patrick E. Farrell ◽  
Florian Wechsung
Keyword(s):  
Liquid Crystals ◽  
Augmented Lagrangian ◽  
Augmented Lagrangian Method ◽  
Mass Matrix ◽  
Unit Length ◽  
Mesh Size ◽  
Schur Complement ◽  
Cholesteric Liquid Crystals ◽  
Multigrid Algorithm ◽  
The Cost

AbstractWe propose a robust and efficient augmented Lagrangian-type preconditioner for solving linearizations of the Oseen–Frank model arising in nematic and cholesteric liquid crystals. By applying the augmented Lagrangian method, the Schur complement of the director block can be better approximated by the weighted mass matrix of the Lagrange multiplier, at the cost of making the augmented director block harder to solve. In order to solve the augmented director block, we develop a robust multigrid algorithm which includes an additive Schwarz relaxation that captures a pointwise version of the kernel of the semi-definite term. Furthermore, we prove that the augmented Lagrangian term improves the discrete enforcement of the unit-length constraint. Numerical experiments verify the efficiency of the algorithm and its robustness with respect to problem-related parameters (Frank constants and cholesteric pitch) and the mesh size.

Schur complement spectral bounds for large hybrid FETI-DP clusters and huge three-dimensional scalar problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Zdeněk Dostál ◽  
Tomáš Brzobohatý ◽  
Oldřich Vlach
Keyword(s):  
Condition Number ◽  
Three Dimensional ◽  
Schur Complement ◽  
Decomposition Methods ◽  
Stiffness Matrices ◽  
Spectral Bounds ◽  
Schur Complements ◽  
Linear Problems ◽  
Primal Method ◽  
Large Clusters

Abstract Bounds on the spectrum of Schur complements of subdomain stiffness matrices with respect to the interior variables are key ingredients of the convergence analysis of FETI (finite element tearing and interconnecting) based domain decomposition methods. Here we give bounds on the regular condition number of Schur complements of “floating” clusters arising from the discretization of 3D Laplacian on a cube decomposed into cube subdomains. The results show that the condition number of the cluster defined on a fixed domain decomposed into m × m × m cube subdomains connected by face and optionally edge averages increases proportionally to m. The estimates support scalability of unpreconditioned H-FETI-DP (hybrid FETI dual-primal) method. Though the research is most important for the solution of variational inequalities, the results of numerical experiments indicate that unpreconditioned H-FETI-DP with large clusters can be useful also for the solution of huge linear problems.

Rough Air-Soft Elastohydrodynamic Lubrication

2013 ◽  
Vol 420 ◽  
pp. 30-35
Author(s):  
Khanittha Wongseedakaew ◽  
Jesda Panichakorn
Keyword(s):  
Surface Roughness ◽  
Film Thickness ◽  
Modulus Of Elasticity ◽  
Contact Region ◽  
Elastohydrodynamic Lubrication ◽  
Multilevel Methods ◽  
Inlet Temperature ◽  
Film Pressure ◽  
Air Inlet ◽  
Air Film

This paper presents the effects of rough surface air-soft elastohydrodynamic lubrication (EHL) of rollers for soft material under the effect of air molecular slip. The time independent modified Reynolds equation and elasticity equation were solved numerically using finite different method, Newton-Raphson method and multigrid multilevel methods were used to obtain the film pressure profiles and film thickness in the contact region. The effects of amplitude of surface roughness, modulus of elasticity and air inlet temperature are examined. The simulation results showed surface roughness has effect on film thickness but it little effect to air film pressure. When the amplitude of surface roughness and modulus of elasticity increased, the air film thickness decreased but air film pressure increased. However, the air inlet temperature increased when the air film thickness increased.

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