Matrix-valued radial basis functions: stability estimates and applications

Radial basis functions (RBFs) have found important applications in areas such as signal processing, medical imaging, and neural networks since the early 1980s. Several applications require that certain physical properties are satisfied by the interpolant, for example, being divergence-free in case of incompressible data. In this paper we consider a class of customized (e.g., divergence-free) RBFs that are matrix-valued and have compact support; these are matrix-valued analogues of the well-known Wendland functions. We obtain stability estimates for a wide class of interpolants based on matrix-valued RBFs, also taking into account the size of the compact support of the generating RBF. We conclude with an application based on an incompressible Navier–Stokes equation, namely the driven-cavity problem, where we use divergence-free RBFs to solve the underlying partial differential equation numerically. We discuss the impact of the size of the support of the basis function on the stability of the solution. AMS subject classification 65D05     

A density theorem for matrix-valued radial basis functions

Recently a new class of customized radial basis functions (RBFs) was introduced. We revisit this class of RBFs and derive a density result guaranteeing that any sufficiently smooth divergence-free function can be approximated arbitrarily closely by a linear combination of members of this class. This result has potential applications to numerically solving differential equations, such as fluid flows, whose solution is divergence free. AMS subject classification 41Axx, 41A30, 41A35, 41A60Svenja Lowitzsch: The results are part of the authorss dissertation written at Texas A&M University, College Station, TX 77843, USA.     

A meshfree method for the numerical solution of the RLW equation

In this paper, we present a meshfree technique for the numerical solution of the regularized long wave (RLW) equation. This approach is based on a global collocation method using the radial basis functions (RBFs). Different kinds of RBFs are used for this purpose. Accuracy of the new method is tested in terms of _L2$L_2$ and _L∞$L_{\infty }$ error norms. In case of non-availability of the exact solution, performance of the new method is compared with existing methods. Stability analysis of the method is established. Propagation of single and double solitary waves, wave undulation, and conservation properties of mass, energy and momentum of the RLW equation are discussed.     

Superconvergence of mixed finite element methods for optimal control problems

In this paper, we investigate the superconvergence property of the numerical solution of a quadratic convex optimal control problem by using rectangular mixed finite element methods. The state and co-state variables are approximated by the lowest order Raviart-Thomas mixed finite element spaces and the control variable is approximated by piecewise constant functions. Some realistic regularity assumptions are presented and applied to error estimation by using an operator interpolation technique. We derive superconvergence properties for the flux functions along the Gauss lines and for the scalar functions at the Gauss points via mixed projections. Moreover, global superconvergence results are obtained by virtue of an interpolation postprocessing technique. Thus, based on these superconvergence estimates, some asymptotic exactness a posteriori error estimators are presented for the mixed finite element methods. Finally, some numerical examples are given to demonstrate the practical side of the theoretical results about superconvergence.

     

Large step volumetric potential reduction algorithms for linear programming

We consider the construction of potential reduction algorithms using volumetric, and mixed volumetric — logarithmic, barriers. These are true large step methods, where dual updates produce constant-factor reductions in the primal-dual gap. Using a mixed volumetric — logarithmic barrier we obtain an iteration algorithm, improving on the best previously known complexity for a large step method. Our results complement those of Vaidya and Atkinson on small step volumetric, and mixed volumetric — logarithmic, barrier function algorithms. We also obtain simplified proofs of fundamental properties of the volumetric barrier, originally due to Vaidya.Research supported by a Summer Research Grant from the College of Business Administration, University of Iowa.     

A compact 9 point stencil based on integrated RBFs for the convection–diffusion equation

In this paper, a high-order compact stencil for solving the convection–diffusion equation in two dimensions is proposed. The convection and diffusion terms are both approximated by means of radial basis functions (RBFs) that are constructed over 3×3$3\times 3$ rectangular stencils. Salient features here are that (i) integration is employed to construct local RBF approximations; and (ii) through the constants of integration, values of the convection–diffusion equation at several selected nodes on the stencil are also enforced. Numerical results indicate that (i) the inclusion of the governing equation into the stencil leads to a significant improvement in accuracy; (ii) when the convection dominates, accurate solutions are obtained at a regime of the RBF width which makes the RBFs peaked; and (iii) high levels of accuracy are achieved using relatively coarse grids.     

Recent developments in error estimates for scattered-data interpolation via radial basis functions

Error estimates for scattered data interpolation by shifts of a positive definite function for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. However, apart from special cases where data is gridded, these interpolation estimates do not apply when the target functions generating the data are outside of the associated RKHS, and in fact until very recently no estimates were known in such situations. In this paper, we review these estimates in cases where the underlying space is Rⁿ and the positive definite functions are radial basis functions (RBFs). AMS subject classification 41A25, 41A05, 41A63, 42B35Research supported by grant DMS-0204449 from the National Science Foundation.     

A mesh-free method for the numerical solution of the KdV–Burgers equation

This paper formulates a simple classical radial basis functions (RBFs) collocation (Kansa) method for the numerical solution of the nonlinear dispersive and dissipative KdV–Burgers’ (KdVB) equation. The computed results show implementation of the method to nonlinear partial differential equations. This method has an edge over traditional methods such as finite-difference and finite element methods because it does not require a mesh to discretize the problem domain, and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method. Accuracy of the method is assessed in terms of error norms _L2,_L∞$L_2\text{,}{L}_{\infty }$, number of nodes in the domain of influence, parameter dependent RBFs and time step length. Numerical experiments demonstrate accuracy and robustness of the method for solving nonlinear dispersive and dissipative problems.     

A posteriori error analysis for locally conservative mixed methods

In this work we present a theoretical analysis for a residual-type error estimator for locally conservative mixed methods. This estimator was first introduced by Braess and Verfürth for the Raviart-Thomas mixed finite element method working in mesh-dependent norms. We improve and extend their results to cover any locally conservative mixed method under minimal assumptions, in particular, avoiding the saturation assumption made by Braess and Verfürth. Our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries. The main results are applied to the nonconforming finite element method and the interior penalty discontinuous Galerkin method as well as the mixed finite element method.

     

Convex Normalizations in Lift-and-Project Methods for 0–1 Programming

Branch-and-Cut algorithms for general 0–1 mixed integer programs can be successfully implemented by using Lift-and-Project (L&P) methods to generate cuts. L&P cuts are drawn from a cone of valid inequalities that is unbounded and, thus, needs to be truncated, or normalized. We consider general normalizations defined by arbitrary closed convex sets and derive dual problems for generating L&P cuts. This unified theoretical framework generalizes and covers a wide group of already known normalizations. We also give conditions for proving finite convergence of the cutting plane procedure that results from using such general L&P cuts.