An Introduction to Algebraic Multigrid

Algebraic multigrid (AMG) solves linear systems based on multigrid principles, but in a way that only depends on the coefficients in the underlying matrix. The author begins with a basic introduction to AMG methods, and then describes some more recent advances and theoretical development

Effect of trailing edge shape on the separated flow characteristics around an airfoil at low Reynolds number: A numerical study

Direct numerical simulations of the flow field around a NACA 0012 airfoil at Reynolds number 50 000 and angle of attack 5° with 3 different trailing edge shapes (straight, blunt, and serrated) have been performed. Both time-averaged flow characteristics and the most dominant flow structures and their frequencies are investigated using the dynamic mode decomposition method. It is shown that for the straight trailing edge airfoil, this method can capture the fundamental as well as the subharmonic of the Kelvin-Helmholtz instability that develops naturally in the separating shear layer. The fundamental frequency matches well with relevant data in the literature. The blunt trailing edge results in periodic vortex shedding, with frequency close to the subharmonic of the natural shear layer frequency. The shedding, resulting from a global instability, has an upstream effect and forces the separating shear layer. Due to forcing, the shear layer frequency locks onto the shedding frequency while the natural frequency (and its subharmonic) is suppressed. The presence of serrations in the trailing edge creates a spanwise pressure gradient, which is responsible for the development of a secondary flow pattern in the spanwise direction. This pattern affects the mean flow in the near wake. It can explain an unexpected observation, namely, that the velocity deficit downstream of a trough is smaller than the deficit after a protrusion. Furthermore, the insertion of serrations attenuates the energy of vortex shedding by de-correlating the spanwise coherence of the vortices. This results in weaker forcing of the separating shear layer, and both the subharmonics of the natural frequency and the shedding frequency appear in the spectra

Filtering Algebraic Multigrid and Adaptive Strategies

Solving linear systems arising from systems of partial differential equations, multigrid and multilevel methods have proven optimal complexity and efficiency properties. Due to shortcomings of geometric approaches, algebraic multigrid methods have been developed. One example is the filtering algebraic multigrid method introduced by C. Wagner. This paper proposes a variant of Wagner's method with substantially improved robustness properties. The method is used in an adaptive, self-correcting framework and tested numerically

An assumed partition algorithm for determining processor inter-communication

The recent advent of parallel machines with tens of thousands of processors is presenting new challenges for obtaining scalability. A particular challenge for large-scale scientific software is determining the inter-processor communications required by the computation when a global description of the data is unavailable or too costly to store. We present a type of rendezvous algorithm that determines communication partners in a scalable manner by assuming the global distribution of the data. We demonstrate the scaling properties of the algorithm on up to 32,000 processors in the context of determining communication patterns for a matrix-vector multiply in the hypre software library. Our algorithm is very general and is applicable to a variety of situations in parallel computing

Distance-two interpolation for parallel algebraic multigrid

In this paper we study the use of long distance interpolation methods with the low complexity coarsening algorithm PMIS. AMG performance and scalability is compared for classical as well as long distance interpolation methods on parallel computers. It is shown that the increased interpolation accuracy largely restores the scalability of AMG convergence factors for PMIS-coarsened grids, and in combination with complexity reducing methods, such as interpolation truncation, one obtains a class of parallel AMG methods that enjoy excellent scalability properties on large parallel computers

Adaptive Algebraic Multigrid Methods

Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems

Scaling Algebraic Multigrid Solvers: On the Road to Exascale

Algebraic Multigrid (AMG) solvers are an essential component of many large-scale scientific simulation codes. Their continued numerical scalability and efficient implementation is critical for preparing these codes for exascale. Our experiences on modern multi-core machines show that significant challenges must be addressed for AMG to perform well on such machines. We discuss our experiences and describe the techniques we have used to overcome scalability challenges for AMG on hybrid architectures in preparation for exascale

Extending the applicability of multigrid methods

Multigrid methods are ideal for solving the increasingly large-scale problems that arise in numerical simulations of physical phenomena because of their potential for computational costs and memory requirements that scale linearly with the degrees of freedom. Unfortunately, they have been historically limited by their applicability to elliptic-type problems and the need for special handling in their implementation. In this paper, we present an overview of several recent theoretical and algorithmic advances made by the TOPS multigrid partners and their collaborators in extending applicability of multigrid methods. Specific examples that are presented include quantum chromodynamics, radiation transport, and electromagnetics

Activity of recombinant dengue 2 virus NS3 protease in the presence of a truncated NS2B co-factor, small peptide substrates, and inhibitors

Recombinant forms of the dengue 2 virus NS3 protease linked to a 40-residue co-factor, corresponding to part of NS2B, have been expressed in Escherichia coli and shown to be active against para-nitroanilide substrates comprising the P6-P1 residues of four substrate cleavage sequences. The enzyme is inactive alone or after the addition of a putative 13-residue co-factor peptide but is active when fused to the 40-residue co-factor, by either a cleavable or a noncleavable glycine linker. The NS4B/NS5 cleavage site was processed most readily, with optimal processing conditions being pH 9, I = 10 mm, 1 mm CHAPS, 20% glycerol. A longer 10-residue peptide corresponding to the NS2B/NS3 cleavage site (P6-P4') was a poorer substrate than the hexapeptide (P6-P1) para-nitroanilide substrate under these conditions, suggesting that the prime side substrate residues did not contribute significantly to protease binding. We also report the first inhibitors of a co-factor-complexed, catalytically active flavivirus NS3 protease. Aprotinin was the only standard serine protease inhibitor to be active, whereas a number of peptide substrate analogues were found to be competitive inhibitors at micromolar concentrations