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A fast solver for the Stokes equations with distributed forces in complex geometries
 J. Comput. Phys
"... We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded ..."
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Cited by 41 (10 self)
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We present a new method for the solution of the Stokes equations. The main features of our method are: (1) it can be applied to arbitrary geometries in a blackbox fashion; (2) it is second order accurate; and (3) it has optimal algorithmic complexity. Our approach, to which we refer as the Embedded Boundary Integral method, is based on Anita Mayo’s work for the Poisson’s equation: “The Fast Solution of Poisson’s and the Biharmonic Equations on Irregular Regions”, SIAM Journal on Numerical Analysis, 21 (1984), pp. 285–299. We embed the domain in a rectangular domain, for which fast solvers are available, and we impose the boundary conditions as interface (jump) conditions on the velocities and tractions. We use an indirect boundary integral formulation for the homogeneous Stokes equations to compute the jumps. The resulting equations are discretized by Nyström’s method. The rectangular domain problem is discretized by finite elements for a velocitypressure formulation with equal order interpolation bilinear elements (£¥ ¤£¥ ¤). Stabilization is used to circumvent the ¦¨§�©������� � condition for the pressure space. For the integral equations, fast matrix vector multiplications are achieved via an ���¨���� � algorithm based on a block representation of the discrete integral operator, combined with (kernel independent) singular value decomposition to sparsify lowrank blocks. The regular grid solver is a Krylov method (Conjugate Residuals) combined with an optimal twolevel Schwartzpreconditioner. For the integral equation we use GMRES. We have tested our algorithm on several numerical examples and we have observed optimal convergence rates. Key Words: Stokes equations, fast solvers, integral equations, doublelayer potential, fast multipole methods, embedded domain methods, immersed interface methods, fictitious
Spectral AMGe (ρAMGe
 SIAM J. Sci. Comput
"... Abstract. Spectral AMGe (ρAMGe), is a new algebraic multigrid method for solving discretizations that arise in Ritztype finite element methods for partial differential equations. The method assumes access to the element stiffness matrices in order to lessen certain presumptions that can limit other ..."
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Cited by 40 (13 self)
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Abstract. Spectral AMGe (ρAMGe), is a new algebraic multigrid method for solving discretizations that arise in Ritztype finite element methods for partial differential equations. The method assumes access to the element stiffness matrices in order to lessen certain presumptions that can limit other algebraic methods. ρAMGe uses the spectral decomposition of small collections of element stiffness matrices to determine local representations of algebraically “smooth” error components. This decomposition provides the basis for generating a coarse grid and for defining effective interpolation. This paper presents a theoretical foundation for ρAMGe along with numerical results that demonstrate the efficiency and robustness of the method. 1. Introduction. Computational
Multigrid with matrixdependent transfer operators for a singular perturbation problem, to appear in Computing
, 1993
"... Citation for published version (APA): Document status and date: ..."
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Cited by 38 (6 self)
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Citation for published version (APA): Document status and date:
MULTIGRID STRATEGIES FOR VISCOUS FLOW SOLVERS ON ANISOTROPIC UNSTRUCTURED MESHES
, 1998
"... Unstructured multigrid techniques for relieving the stiffness associated with highReynolds number viscous flow simulations on extremely stretched grids are investigated. One approach consists of employing a semicoarsening or directionalcoarsening technique, based on the directions of strong coup ..."
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Cited by 38 (6 self)
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Unstructured multigrid techniques for relieving the stiffness associated with highReynolds number viscous flow simulations on extremely stretched grids are investigated. One approach consists of employing a semicoarsening or directionalcoarsening technique, based on the directions of strong coupling within the mesh, in order to construct more optimal coarse grid levels. An alternate approach is developed which employs directional implicit smoothing with regular fully coarsened multigrid levels. The directional implicit smoothing is obtained by constructing implicit lines in the unstructured mesh based on the directions of strong coupling. Both approaches yield large increases in convergence rates over the traditional explicit fullcoarsening multigrid algorithm. However, maximum benefits are achieved by combining the two approaches in a coupled manner into a single algorithm. An order of magnitude increase in convergence rate over the traditional explicit fullcoarsening algorithm is demonstrated, and convergence rates for highReynolds number viscous flows which are independent of the grid aspect ratio are obtained. Further acceleration is provided by incorporating lowMachnumber preconditioning techniques, and a NewtonGMRES strategy which employs the multigrid scheme as a preconditioner. The compounding effects of these various techniques on speed of convergence is documented through several example test cases.
Multilevel Solvers For Unstructured Surface Meshes
 SIAM J. Sci. Comput
"... Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated mu ..."
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Cited by 38 (3 self)
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Parameterization of unstructured surface meshes is of fundamental importance in many applications of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner.
A MultiLevel Solution Algorithm for SteadyState Markov Chains
, 1993
"... A new iterative algorithm, the multilevel algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are w ..."
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Cited by 34 (4 self)
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A new iterative algorithm, the multilevel algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the original system to achieve accelerated convergence. It is motivated by multigrid methods, which are widely used for fast solution of partial differential equations. Initial results of numerical experiments are reported, showing signi cant reductions in computation time, often an order of magnitude or more, relative to the GaussSeidel and optimal SOR algorithms for a variety of test problems. It is shown how the wellknown iterative aggregationdisaggregation algorithm of Takahashi can be interpreted as a special case of the new method.
A comparison of eigensolvers for largescale 3d modal analysis using amgpreconditioned iterative methods
 Int. Journal for Numerical Methods in Engg
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Reducing complexity in parallel algebraic multigrid preconditioners
 SIAM J. Matrix Anal. Appl
, 2006
"... Abstract. Algebraic multigrid (AMG) is a very efficient iterative solver and preconditioner for large unstructured sparse linear systems. Traditional coarsening schemes for AMG can, however, lead to computational complexity growth as problem size increases, resulting in increased memory use and exec ..."
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Cited by 33 (8 self)
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Abstract. Algebraic multigrid (AMG) is a very efficient iterative solver and preconditioner for large unstructured sparse linear systems. Traditional coarsening schemes for AMG can, however, lead to computational complexity growth as problem size increases, resulting in increased memory use and execution time, and diminished scalability. Two new parallel AMG coarsening schemes are proposed that are based solely on enforcing a maximum independent set property, resulting in sparser coarse grids. The new coarsening techniques remedy memory and execution time complexity growth for various large threedimensional (3D) problems. If used within AMG as a preconditioner for Krylov subspace methods, the resulting iterative methods tend to converge fast. This paper discusses complexity issues that can arise in AMG, describes the new coarsening schemes, and examines the performance of the new preconditioners for various large 3D problems. Key words. parallel coarsening algorithms, algebraic multigrid, complexities, preconditioners
A new diffusionbased multilevel algorithm for computing graph partitions of very high quality
 In Proc. 22nd IPDPS
, 2008
"... Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. Highquality partitions are important for many applications, whose objective functions are often NPhard to optimize. Most stateoftheart graph p ..."
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Cited by 32 (9 self)
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Abstract. Graph partitioning requires the division of a graph's vertex set into k equally sized subsets s. t. some objective function is optimized. Highquality partitions are important for many applications, whose objective functions are often NPhard to optimize. Most stateoftheart graph partitioning libraries use a variant of the KernighanLin (KL) heuristic within a multilevel framework. While these libraries are very fast, their solutions do not always meet all user requirements. Moreover, due to its sequential nature, KL is not easy to parallelize. Its use as a load balancer in parallel numerical applications therefore requires complicated adaptations. That is why we developed previously an inherently parallel algorithm, called BubbleFOS/C (Meyerhenke et al., IPDPS'06), which optimizes partition shapes by a diffusive mechanism. However, it is too slow for practical use, despite its high solution quality. In this paper, besides proving that BubbleFOS/C converges towards a local optimum of a potential function, we develop a much faster method for the improvement of partitionings. This faster method called TruncCons is based on a different diffusive process, which is restricted to local areas of the graph and also contains a high degree of parallelism. By coupling TruncCons with BubbleFOS/C in a multilevel framework based on two different hierarchy construction methods, we obtain our new graph
Adaptive Smoothed Aggregation (αSA) Multigrid
, 2005
"... Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the part ..."
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Cited by 31 (7 self)
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Substantial effort has been focused over the last two decades on developing multilevel iterative methods capable of solving the large linear systems encountered in engineering practice. These systems often arise from discretizing partial differential equations over unstructured meshes, and the particular parameters or geometry of the physical problem being discretized may be unavailable to the solver. Algebraic multigrid (AMG) and multilevel domain decomposition methods of algebraic type have been of particular interest in this context because of their promises of optimal performance without the need for explicit knowledge of the problem geometry. These methods construct a hierarchy of coarse problems based on the linear system itself and on certain assumptions about the smooth components of the error. For smoothed aggregation (SA) multigrid methods applied to discretizations of elliptic problems, these assumptions typically consist of knowledge of the nearkernel or nearnullspace of the weak form. This paper introduces an extension of the SA method in which good convergence properties are achieved in situations where explicit knowledge of the nearkernel components is unavailable. This extension is accomplished in an adaptive process that uses the method itself to determine nearkernel components and adjusts the coarsening processes accordingly.