Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAnisotropic, interpolatory subdivision and multigrid
267
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this paper, we present a family of multivariate grid transfer operators appropriate for anisotropic multigrid methods. Our grid transfer operators are derived from a new family of anisotropic interpolatory subdivision schemes. We study the minimality, polynomial reproduction and convergence properties of these interpolatory schemes and link their properties to the convergence and optimality of the corresponding multigrid methods. We compare the performance of our interpolarory grid transfer operators with the ones derived from a family of corresponding approximating subdivision schemes.
rate research
Read More
The convergence rate of a multigrid method depends on the properties of the smoother and the so-called grid transfer operator. In this paper we define and analyze new grid transfer operators with a generic cutting size which are applicable for high order problems. We enlarge the class of available geometric grid transfer operators by relating the symbol analysis of the coarse grid correction with the approximation properties of univariate subdivision schemes. We show that the polynomial generation property and stability of a subdivision scheme are crucial for convergence and optimality of the corresponding multigrid method. We construct a new class of grid transfer operators from primal binary and ternary pseudo-spline symbols. Our numerical results illustrate the behavior of the new grid transfer operators.
A new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts.
A standard construction in approximation theory is mesh refinement. For a simplicial or polyhedral mesh D in R^k, we study the subdivision D obtained by subdividing a maximal cell of D. We give sufficient conditions for the module of splines on D to split as the direct sum of splines on D and splines on the subdivided cell. As a consequence, we obtain dimension formulas and explicit bases for several commonly used subdivisions and their multivariate generalizations.
Positive interpolatory cubature formulas (CFs) are constructed for quite general integration domains and weight functions. These CFs are exact for general vector spaces of continuous real-valued functions that contain constants. At the same time, the number of data points -- all of which lie inside the domain of integration -- and cubature weights -- all positive -- is less or equal to the dimension of that vector space. The existence of such CFs has been ensured by Tchakaloff in 1957. Yet, to the best of the authors knowledge, this work is the first to provide a procedure to successfully construct them.
The parallel full approximation scheme in space and time (PFASST) introduced by Emmett and Minion in 2012 is an iterative strategy for the temporal parallelization of ODEs and discretized PDEs. As the name suggests, PFASST is similar in spirit to a space-time FAS multigrid method performed over multiple time-steps in parallel. However, since the original focus of PFASST has been on the performance of the method in terms of time parallelism, the solution of any spatial system arising from the use of implicit or semi-implicit temporal methods within PFASST have simply been assumed to be solved to some desired accuracy completely at each sub-step and each iteration by some unspecified procedure. It hence is natural to investigate how iterative solvers in the spatial dimensions can be interwoven with the PFASST iterations and whether this strategy leads to a more efficient overall approach. This paper presents an initial investigation on the relative performance of different strategies for coupling PFASST iterations with multigrid methods for the implicit treatment of diffusion terms in PDEs. In particular, we compare full accuracy multigrid solves at each sub-step with a small fixed number of multigrid V-cycles. This reduces the cost of each PFASST iteration at the possible expense of a corresponding increase in the number of PFASST iterations needed for convergence. Parallel efficiency of the resulting methods is explored through numerical examples.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
