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We derive stability estimates for three commonly used radial basis function (RBF) methods to solve hyperbolic time-dependent PDEs: the RBF generated finite difference (RBF-FD) method, the RBF partition of unity method (RBF-PUM) and Kansa's (global) RBF method. We give the estimates in the discrete l(2)-norm 2-norm intrinsic to each of the three methods. The results show that Kansa's method and RBF-PUM can be l(2)-stable 2-stable in time under a sufficiently large oversampling of the discretized system of equations. The RBF-FD method in addition requires stabilization of the spurious jump terms due to the discontinuous RBF-FD cardinal basis functions. Numerical experiments show an agreement with our theoretical observations.

A methodology is presented for the numerical solution of nonlinear elliptic systems in unbounded domains, consisting of three elements. First, the problem is posed on a finite domain by means of a proper nonlinear change of variables. The compressed domain is then discretised, regardless of its final shape, via the radial basis function partition of unity method. Finally, the system of nonlinear algebraic collocation equations is solved with the trust-region algorithm, taking advantage of analytically derived Jacobians. We validate the methodology on a benchmark of computational fluid mechanics: the steady viscous flow past a circular cylinder. The resulting flow characteristics compare very well with the literature. Then, we stress-test the methodology on less smooth obstacles-rounded and sharp square cylinders. As expected, in the latter scenario the solution is polluted by spurious oscillations, owing to the presence of boundary singularities.

This work proposes a polynomial-augmented radial basis function (RBF) collocation method with polyharmonic to solve the advection-diffusion-reaction (ADR) equations associated with the percolation process for espresso extraction. Numerical methods for solving these equations are useful for many applications where we have chemical reactions and transport in a porous medium. Polynomial augmentation in RBF collocation is useful when it is also necessary to approximate the derivatives, to overcome the stagnation error problem. Moreover, the polyharmonic RBF avoids the hassle of determining the shape parameter. The proposed meshless method allows for discretising the ADR equations and obtaining a numerical solution used to evaluate the efficiency of the extraction process in espresso coffee; this method can be easily generalised to higher dimensions or more complex domains. The numerical results have been compared to measurements carried out in the laboratory.

Clusters employ workload schedulers such as the Sturm Workload Manager to allocate computing jobs onto nodes. These schedulers usually aim at a good trade-off between increasing resource utilization and user satisfaction (decreasing job waiting time). However, these schedulers are typically unaware of jobs sharing large input files, which may happen in data intensive scenarios. The same input files may end up being loaded several times, leading to a waste of resources. We study how to design a data-aware job scheduler that is able to keep large input files on the computing nodes, without impacting other memory needs, and can benefit from previously-loaded tiles to decrease data transfers in order to reduce the waiting times ofjobs. We present three schedulers capable of distributing the load between the computing nodes as well as re-using input files already loaded in the memory of some node as much as possible. We perform simulations with single node jobs using traces of real HPC-cluster usage, to compare them to classical job schedulers. The results show that keeping data in local memory between successive jobs and using data -locality information to schedule jobs improves performance compared to a widely -used scheduler (FCFS, with and without backfilling): a reduction in job waiting time (a 7.5% improvement in stretch), and a decrease in the amount of data transfers (7%).

Infinitely smooth radial basis functions (RBFs) have a shape parameter that controls their shapes. When using these RBFs (e.g., the Gaussian RBF) for interpolation problems, we have ill-conditioning when the shape parameter is very small, while in some cases small shape parameters lead to high accuracy. In this study, we are going to reduce the effect of the ill-conditioning of the infinitely smooth RBFs. We propose a new basis augmenting the infinitely smooth RBFs at different locations with radial polynomials of different even powers. Numerical experiments show that the new basis is stable for all values of the shape parameter.

This paper studies the influence of scaling on the behavior of radial basis function interpolation. It focuses on certain central aspects, but does not try to be exhaustive. The most important questions are: How does the error of a kernel-based interpolant vary with the scale of the kernel chosen? How does the standard error bound vary? And since fixed functions may be in spaces that allow scalings, like global Sobolev spaces, is there a scale of the space that matches the function best? The last question is answered in the affirmative for Sobolev spaces, but the required scale may be hard to estimate. Scalability of functions turns out to be restricted for spaces generated by analytic kernels, unless the functions are band-limited. In contrast to other papers, polynomials and polyharmonics are included as flat limits when checking scales experimentally, with an independent computation. The numerical results show that the hunt for near-flat scales is questionable, if users include the flat limit cases right from the start. When there are not enough data to evaluate errors directly, the scale of the standard error bound can be varied, up to replacing the norm of the unknown function by the norm of the interpolant. This follows the behavior of the actual error qualitatively well, but is only of limited value for estimating error-optimal scales. For kernels and functions with unlimited smoothness, the given interpolation data are proven to be insufficient for determining useful scales.

In a recent paper by Tominec, Larsson and Heryudono a convergence proof for an oversampled version of the RBF-FD method, using polyharmonic spline basis functions augmented with polynomials, was derived. In this paper, we take a closer look at the individual estimates involved in this proof. We investigate how large the bounds are and how they depend on the node layout, the stencil size, and the polynomial degree. We find that a moderate amount of oversampling is sufficient for the method to be stable when Halton nodes are used for the stencil approximations, while a random node layout may require a very high oversampling factor. From a practical perspective, this indicates the importance of having a locally quasi uniform node layout for the method to be stable and give reliable results. We see an overall growth of the error constant with the polynomial degree and with the stencil size.

The thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite difference (RBF-FD) framework. Then the boundary conditions are reformulated as a Robin boundary condition with smooth coefficients. The same framework is also used to approximate the boundary curve of the diaphragm cross section based on data obtained from a slice of a computed tomography (CT) scan. To solve the PDE we employ the unfitted least-squares RBF-FD method. This makes it easier to handle the geometry of the diaphragm, which is thin and non-convex. We show numerically that our solution converges with high-order towards a finite element solution evaluated on a fine grid. Through this simplified numerical model we also gain an insight into the challenges associated with the diaphragm geometry and the boundary conditions before approaching a more complex three-dimensional model. 

Localized collocation methods based on radial basis functions (RBFs) for elliptic problems appear to be nonrobust in the presence of Neumann boundary conditions. In this paper, we overcome this issue by formulating the RBF-generated finite difference method in a discrete least squares setting instead. This allows us to prove high-order convergence under node refinement and to numerically verify that the least squares formulation is more accurate and robust than the collocation formulation. The implementation effort for the modified algorithm is comparable to that for the collocation method.