Piecewise

About: Piecewise is a research topic. Over the lifetime, 21064 publications have been published within this topic receiving 432096 citations. The topic is also known as: piecewise-defined function & hybrid function.

An aggregation-based algebraic multigrid method

Abstract: An algebraic multigrid method is presented to solve large systems of linear equations. The coarsen- ing is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the number of variables by a factor slightly less than four. The matching algorithm favors the strongest negative coupling(s), inducing a problem depen- dant coarsening. This aggregation is combined with piecewise constant (unsmoothed) prolongation, ensuring low setup cost and memory requirements. Compared with previous aggregation-based multigrid methods, the scalability is enhanced by using a so-called K-cycle multigrid scheme, providing Krylov subspace acceleration at each level. This paper is the logical continuation of (SIAM J. Sci. Comput., 30 (2008), pp. 1082-1103), where the analysis of a model anisotropic problem shows that aggregation-based two-grid methods may have optimal order convergence, and of (Numer. Lin. Alg. Appl., 15 (2008), pp. 473-487), where it is shown that K-cycle multigrid may provide optimal or near optimal convergence under mild assumptions on the two-grid scheme. Whereas in these papers only model problems with geometric aggregation were considered, here a truly algebraic method is presented and tested on a wide range of discrete second order scalar elliptic PDEs, including nonsymmetric and unstructured problems. Numerical results indicate that the proposed method may be significantly more robust as black box solver than the classical AMG method as implemented in the code AMG1R5 by K. Stuben. The parallel implementation is also discussed. Satisfactory speedups are obtained on a medium size multi-processor cluster that is typical of today com- puter market. A code implemanting the method is freely available for download both as a Fortran program and a Matlab function.

CESAM: A code for stellar evolution calculations

Abstract: The code CESAM is a consistent set of programs and routines which performs calculations of 1D quasi-static stellar evolution including diffusion and rotation. The principal new feature is the solution of the quasi-static equilibrium by collocation method based on piecewise polynomials approximations projected on their B-spline basis; that allows stable and robust calculations and the exact restitution of the solution not only at grid points. Another advantage is the monitoring by only one parameter of the accuracy and its improvement by superconvergence. An automatic mesh refinement has been designed for adjusting the localizations of grid points according to the changes of unknowns, each limit between a radiative and a convective zones being shifted to the closest grid point. For standard models, the evolution of the chemical composition is solved by stiffly stable schemes of orders up to four; for non-standard models the solution of the diffusion equation employs the Petrov-Galerkin scheme, with the mixing of chemicals in convective zones performed by strong turbulent diffusion. A precise restoration of the atmosphere is allowed for. CESAM computes evolution of stars from the pre-main sequence to the beginning of the 4 He burning cycle. In this paper a detailed description of the algorithms is presented.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms

Abstract: SUMMARY We present a new numerical approach to solve the elastic wave equation in heterogeneous media in the presence of externally given source terms with arbitrary high-order accuracy in space and time on unstructured triangular meshes. We combine a discontinuous Galerkin (DG) method with the ideas of the ADER time integration approach using Arbitrary high-order DERivatives. The time integration is performed via the so-called Cauchy-Kovalewski procedure using repeatedly the governing partial differential equation itself. In contrast to classical finite element methods we allow for discontinuities of the piecewise polynomial approximation of the solution at element interfaces. This way, we can use the well-established theory of fluxes across element interfaces based on the solution of Riemann problems as developed in the finite volume framework. In particular, we replace time derivatives in the Taylor expansion of the time integration procedure by space derivatives to obtain a numerical scheme of the same high order in space and time using only one single explicit step to evolve the solution from one time level to another. The method is specially suited for linear hyperbolic systems such as the heterogeneous elastic wave equations and allows an efficient implementation. We consider continuous sources in space and time and point sources characterized by a Delta distribution in space and some continuous source time function. Hereby, the method is able to deal with point sources at any position in the computational domain that does not necessarily need to coincide with a mesh point. Interpolation is automatically performed by evaluation of test functions at the source locations. The convergence analysis demonstrates that very high accuracy is retained even on strongly irregular meshes and by increasing the order of the ADER‐DG schemes computational time and storage space can be reduced remarkably. Applications of the proposed method to Lamb’s Problem, a problem of strong material heterogeneities and to an example of global seismic wave propagation finally confirm its accuracy, robustness and high flexibility.

Minimum contrast estimators on sieves: exponential bounds and rates of convergence

Abstract: This paper, which we dedicate to Lucien Le Cam for his seventieth birthday, has been written in the spirit of his pioneering works on the relationships between the metric structure of the parameter space and the rate of convergence of optimal estimators. It has been written in his honour as a contribution to his theory. It contains further developments of the theory of minimum contrast estimators elaborated in a previous paper. We focus on minimum contrast estimators on sieves. By a `sieve' we mean some approximating space of the set of parameters. The sieves which are commonly used in practice are D-dimensional linear spaces generated by some basis: piecewise polynomials, wavelets, Fourier, etc. It was recently pointed out that nonlinear sieves should also be considered since they provide better spatial adaptation (think of histograms built from any partition of D subintervals of [0,1] as a typical example). We introduce some metric assumptions which are closely related to the notion of finite-dimensional metric space in the sense of Le Cam. These assumptions are satisfied by the examples of practical interest and allow us to compute sharp rates of convergence for minimum contrast estimators.