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This is essential for approaching peak floating point performance. Display statistics.

Type Book Chapter. Authors Efendiev, Yalchin R. Date Metadata Show full item record. Publisher Springer Nature. Bank Fig. The domain left and solution right for the anisotropic equation This problem is solved by an interior point method described in [7, 1]. The state variable left, top , Lagrange multiplier right, top and optimal control bottom for equation This local solve was followed by several iterations of the DD solver. For both Plan A and Plan B, the convergence criteria for the DD iteration was Here G is the diagonal of the finite element mass matrix, introduced to account for nonuniformity of the global finite element mesh.

For the multigraph iteration on each processor, the convergence criteria was The stronger criteria was to insure that the approximation on coarse interior residuals by zero remained valid. In Tables we summarize the results of our computations.

In these tables, p is the number of processors, N is the number of vertices on the final global mesh, and DD is the number of domain decomposition iterations used in Step III. Step I is done on a single processor. For Steps II and III, average times across all processors are reported; the range of times is also included in parentheses. The increase in time with increasing p is due mostly to eigenvalue problems that are solved are part of the spectral bisection load balancing scheme. The DD algorithm in [5] is shown to converge independently of N , which was empirically verified in [6] for the version implemented here.

There is some slight, empirically logarithmic, dependence on p. Details of the multigraph solver are given in [8]. See [1] for details. In viewing the results as a whole, both paradigms scale reasonably well as a function of p; since Step III is a very costly part of the calculation, it is clearly worthwhile to try to make the convergence rate independent of p as 14 Randolph E. Bank well as N , or at least to reduce the dependence on p. This is a topic of current research interest. References 1. Bank and M. Bank, P. Jimack, S. Nadeem, and S. Bank and S.

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Bank and R. The proposed method uses compatible relaxation to select the set of coarse variables. Then, an energy minimizing coarse basis is formed using an approach aimed to minimize the trace of the coarse—level operator. Our AMG approach for solving 1 involves a stationary linear iterative smoother and a coarse-level correction. It is well known that if A is symmetric, 16 J. Brannick and L. Zikatanov then this variational form of the correction step is optimal in the energy norm. In AMG, the smoother is typically fixed and the coarse-level correction is formed to compensate for its deficiencies.

The primary task is, of course, the selection of P. Standard algebraic multigrid setup algorithms are based on properties of M -matrices e. In fact, the components and parameters associated with these approaches are often problem dependent. Developing more robust AMG solvers is currently a topic of intense research.

General approaches for selecting the set of coarse variables are presented in [12, 4]. These approaches use compatible relaxation CR to gauge the quality of as well as construct the coarse variable set, an idea first introduced by Brandt [2].

## NUMA - Publications and Talks by Neumüller

In [3], an energy-based strength-of-connection measure is developed and shown to extend the applicability of Classical AMG when coupled with adaptive AMG interpolation [7]. Recent successes in developing a more general form of interpolation include [7, 6, 17, 19]. In [7, 6], these components are computed automatically in the setup procedure using a multilevel power method iteration based on the error propagation operator of the method itself. The algorithm we propose for constructing P is motivated by the recently developed two-level theory introduced in [9] and [10].

We explore the use of this theory in developing a robust setup procedure in the setting of classical AMG. In particular, as in classical AMG, we assume that the coarse-level variables are a subset of the fine-level variables.

## Domain Decomposition Methods in Science and Engineering XX

Our coarsening algorithm constructs the coarse variable set using the CR-based algorithm introduced by Brannick and Falgout in [4]. The notion of strength of connection we use in determining the nonzero sparsity pattern of the columns of P is based on AMG Based on CR and Energy Minimization 17 a sparse approximation of the so-called two-level ideal interpolation operator. In what follows, we use several projections on the Range P. The fact that the coarse-level degrees of freedom are a subset of the fine-level degrees of freedom is reflected in the form of R.

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The matrix S corresponds to the complementary degrees of freedom, i. Zikatanov 2.

Theorem 1. Let ET G be defined as in 2. Then Assuming that the set of coarse degrees of freedom have been selected i. R is defined , the remaining task is defining a P to minimize K P. Finding such a P is of course not at all straightforward, because the dependence of K P on P given in Theorem 2 is complicated.

Also, this measure suggests that error components consisting of eigenvectors associated with small eigenvalues i. Theorem 2. Then Moreover, the asymptotic convergence factor of CR provides an upper bound for the above minimum as follows see Theorem 5.

Theorem 3. The main ideas of our algorithm, described next, are based on observations and conclusions drawn from the above results. AMG Based on CR and Energy Minimization 19 3 Compatible relaxation based coarsening In this section, we give more details on the first step of the algorithm, selecting the coarse degrees of freedom.

- Minisymposia!
- Instant Notes in Molecular Biology.
- Domain Decomposition Methods in Science and Engineering XX?

The quality of the set of coarse-level degrees of freedom, C, depends on two conflicting criteria: C1: algebraically-smooth error should be approximated well by some vector interpolated from C, and C2: C should have substantially fewer variables than on the fine level. In our adaptive AMG solver, the set of coarse variables is selected using the CR-based coarsening approach developed in [4].

The algorithm ties the selection of C to the smoother. Hence, this algorithm constructs C so that C1 is strictly enforced and C2 is satisfied as much as possible.

**gripurlaiswar.cf**

## Domain Decomposition Methods in Science and Engineering XX

The details of this algorithm are given in [4]. An advantage of this approach, over the two-pass algorithm employed in classical AMG, is the use of the asymptotic convergence factor of compatible relaxation as a measure of the quality of C and, thus, the ability to adapt C when necessary. An additional advantage of this approach is that the algorithm does not rely on the notion of strength of connections to form C, instead, only the graph of matrix A and the error generated by the CR process are used to form C.

This typically results in more aggressive coarsening than in traditional coarsening approaches, especially on coarser levels where stencils tend to grow. Additionally, this approach has been shown to work for a wide range of problems without the need for parameter tuning [4].

We conclude this section by proving the following proposition relating the spectral radii of Ef to the condition number of Af f. Proposition 1.

- Domain Decomposition Methods in Science and Engineering XXIV | Petter E. Bjørstad | Springer.
- Domain Decomposition Methods in Science and Engineering XX.
- Domain decomposition methods in science and engineering XX.