Elisabeth Larsson

A numerical study of some radial basis function based solution methods for elliptic PDEs

Abstract During the last decade, three main variations have been proposed for solving elliptic PDEs by means of collocation with radial basis functions (RBFs). In this study, we have implemented them for infinitely smooth RBFs, and then compared them across the full range of values for the shape parameter of the RBFs. This was made possible by a recently discovered numerical procedure that bypasses the ill conditioning, which has previously limited the range that could be used for this parameter. We find that the best values for it often fall outside the range that was previously available. We have also looked at piecewise smooth versus infinitely smooth RBFs, and found that for PDE applications with smooth solutions, the infinitely smooth RBFs are preferable, mainly because they lead to higher accuracy. In a comparison of RBF-based methods against two standard techniques (a second-order finite difference method and a pseudospectral method), the former gave a much superior accuracy.

Stable Computations with Gaussian Radial Basis Functions

Radial basis function (RBF) approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh-free and can be spectrally accurate. A perceived practical obstacle is that the interpolation matrix becomes increasingly ill-conditioned as the RBF shape parameter becomes small, corresponding to flat RBFs. Two stable approaches that overcome this problem exist: the Contour-Pade method and the RBF-QR method. However, the former is limited to small node sets, and the latter has until now been formulated only for the surface of the sphere. This paper focuses on an RBF-QR formulation for node sets in one, two, and three dimensions. The algorithm is stable for arbitrarily small shape parameters. It can be used for thousands of node points in two dimensions and still more in three dimensions. A sample MATLAB code for the two-dimensional case is provided.

Postoperative analgesia and sedation following pediatric cardiac surgery using a constant infusion of ketamine

Constant rate infusions of ketamine supplemented with intermittent doses of midazolam were given postoperatively to 10 children in order to provide analgesia and sedation during mechanical ventilation after cardiac surgery as well as during weaning from the ventilator and during spontaneous breathing. The aims of the study were to determine the pharmacokinetics of ketamine and evaluate the suitability of ketamine as an analgesic and sedative in postoperative pediatric cardiac patients. The children were between one week and 30 months old. Five children were given 1 mg/kg/h of ketamine and five children had 2 mg/kg/h. Blood was sampled during infusion and up to 24 hours after infusion for plasma concentrations of ketamine and the main plasma metabolite, norketamine, which were determined by gas chromatography and were compared to the degree of sedation. The children were arousable when ketamine concentrations were below 1.0 to 1.5 micrograms/mL. Plasma ketamine concentrations at steady state were within a narrow range for each infusion regimen and the calculated pharmacokinetic parameters were similar. Mean plasma clearance of ketamine was 0.94 +/- 0.22 L/kg/h. The elimination half-life was 3.1 +/- 1.6 hours, but in some children late samples indicated an even longer elimination half-life. Norketamine did not reach a steady state, but at the end of the infusion, the mean plasma concentration was higher than that of ketamine. The elimination half-life of norketamine was estimated to be 6.0 +/- 1.8 hours. Both ketamine infusion regimens were supplemented with midazolam and provided similarly acceptable analgesia and sedation during mechanical ventilation and during and after weaning from the ventilator.(ABSTRACT TRUNCATED AT 250 WORDS)

A new class of oscillatory radial basis functions

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only nonoscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics).

Iterative Solution of the Helmholtz Equation by a Second-Order Method

The numerical solution of the Helmholtz equation subject to nonlocal radiation boundary conditions is studied. The specific problem is the propagation of hydroacoustic waves in a two-dimensional curvilinear duct. The problem is discretized with a second-order accurate finite difference method, resulting in a linear system of equations. To solve the system of equations, a preconditioned Krylov subspace method is employed. We construct a preconditioner that is based on fast transforms and yields a direct fast Helmholtz solver for rectangular domains. Numerical experiments for curved ducts demonstrate that the rate of convergence is high. The fast transform preconditioner is compared with a symmetric successive over-relaxation (SSOR) preconditioner, and also with band Gaussian elimination. For the preconditioned iterative methods, the gains in storage requirement are large compared with band Gaussian elimination. Regarding the arithmetic complexity, the fast transform preconditioner yields a significant gain, whereas the SSOR preconditioner performs worse than band Gaussian elimination.

Stable Computation of Differentiation Matrices and Scattered Node Stencils Based on Gaussian Radial Basis Functions

Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions

Radial basis function partition of unity methods for pricing vanilla basket options

Meshfree methods based on radial basis function (RBF) approximation are becoming widely used for solving PDE problems. They are flexible with respect to the problem geometry and highly accurate. A?disadvantage of these methods is that the linear system to be solved becomes dense for globally supported RBFs. A remedy is to introduce localisation techniques such as partition of unity. RBF partition of unity methods (RBF-PUM) allow for a significant sparsification of the linear system and lower the computational effort. In this work we apply a global RBF method as well as RBF-PUM to problems in option pricing. We consider one- and two-dimensional vanilla options. In order to price American options we employ a penalty approach. A penalty term, suitable for American multi-asset call options, has been designed. RBF-PUM is shown to be competitive compared with a finite difference method and a global RBF method. It is as accurate as the global RBF method, but significantly faster. The results for RBF-PUM look promising for extension to higher-dimensional problems.

BENCHOP – The BENCHmarking project in option pricing

The aim of the BENCHOP project is to provide the finance community with a common suite of benchmark problems for option pricing. We provide a detailed description of the six benchmark problems together with methods to compute reference solutions. We have implemented fifteen different numerical methods for these problems, and compare their relative performance. All implementations are available on line and can be used for future development and comparisons.

Resource-Aware Task Scheduling

Dependency-aware task-based parallel programming models have proven to be successful for developing efficient application software for multicore-based computer architectures. The programming model is amenable to programmers, thereby supporting productivity, whereas hardware performance is achieved through a runtime system that dynamically schedules tasks onto cores in such a way that all dependencies are respected. However, even if the scheduling is completely successful with respect to load balancing, the scaling with the number of cores may be suboptimal due to resource contention. Here we consider the problem of scheduling tasks not only with respect to their interdependencies but also with respect to their usage of resources, such as memory and bandwidth. At the software level, this is achieved by user annotations of the task resource consumption. In the runtime system, the annotations are translated into scheduling constraints. Experimental results for different hardware, demonstrating performance gains both for model examples and real applications, are presented. Furthermore, we provide a set of tools to detect resource sensitivity and predict the performance improvements that can be achieved by resource-aware scheduling. These tools are solely based on parallel execution traces and require no instrumentation or modification of the application code.

Preconditioning for Radial Basis Function Partition of Unity Methods

Meshfree radial basis function (RBF) methods are of interest for solving partial differential equations due to attractive convergence properties, flexibility with respect to geometry, and ease of implementation. For global RBF methods, the computational cost grows rapidly with dimension and problem size, so localised approaches, such as partition of unity or stencil based RBF methods, are currently being developed. An RBF partition of unity method (RBF–PUM) approximates functions through a combination of local RBF approximations. The linear systems that arise are locally unstructured, but with a global structure due to the partitioning of the domain. Due to the sparsity of the matrices, for large scale problems, iterative solution methods are needed both for computational reasons and to reduce memory requirements. In this paper we implement and test different algebraic preconditioning strategies based on the structure of the matrix in combination with incomplete factorisations. We compare their performance for different orderings and problem settings and find that a no-fill incomplete factorisation of the central band of the original discretisation matrix provides a robust and efficient preconditioner.