C. Piret

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Papers by C. Piret

A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces

Journal of Computational Physics, 2019

The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an em... more The closest point method (Ruuth and Merriman, J. Comput. Phys. 227(3):1943-1961, [2008]) is an embedding method developed to solve a variety of partial differential equations (PDEs) on smooth surfaces, using a closest point representation of the surface and standard Cartesian grid methods in the embedding space. Recently, a closest point method with explicit time-stepping was proposed that uses finite differences derived from radial basis functions (RBF-FD). Here, we propose a least-squares implicit formulation of the closest point method to impose the constant-along-normal extension of the solution on the surface into the embedding space. Our proposed method is particularly flexible with respect to the choice of the computational grid in the embedding space. In particular, we may compute over a computational tube that contains problematic nodes. This fact enables us to combine the proposed method with the grid based particle method (Leung and Zhao, J. Comput. Phys. 228(8):2993-3024, [2009]) to obtain a numerical method for approximating PDEs on moving surfaces. We present a number of examples to illustrate the numerical convergence properties of our proposed method. Experiments for advection-diffusion equations and Cahn-Hilliard equations that are strongly coupled to the velocity of the surface are also presented.

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Quality Surface Meshing Using Discrete Parametrizations

Proceedings of the 20th International Meshing Roundtable, 2011

We present 3 mapping/flattening techniques for triangulations of poor quality triangles. The impl... more We present 3 mapping/flattening techniques for triangulations of poor quality triangles. The implementation of those mappings as well as the boundary conditions are presented in a very comprehensive manner such that it becomes accessible to a wider community than the one of computer graphics. The resulting parameterizations are used to generate new triangulations or quadrilateral meshes for the model that are of high quality.

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A Stable Algorithm for Flat Radial Basis Functions on a Sphere

SIAM Journal on Scientific Computing, 2008

When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically ... more When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Runge-type oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in case of near-flat basis functions, there will typically also be a separate point at which disasterous ill-conditioning enters. We introduce here a new method, RBF-QR, which entirely eliminates such ill-conditioning, and we apply it in the special case when the data points are distributed over the surface of a sphere. This algorithm works even for thousands of node points, and it allows the RBF shape parameter to be optimized without the limitations imposed by stability concerns. Since interpolation in the flat RBF limit on a sphere is found to coincide with spherical harmonics interpolation, new insights are gained as to why the RBF approach (with non-flat basis functions) often is the more accurate of the two methods.

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On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere

Journal of Computational Physics, 2008

Radial basis function (RBF) approximations have been used for some time to interpolate data on a ... more Radial basis function (RBF) approximations have been used for some time to interpolate data on a sphere (as well as on many other types of domains). Their ability to solve, to spectral accuracy, convection-type PDEs over a sphere has been demonstrated only very recently. In such applications, there are two main choices that have to be made: (i) which type of radial function to use, and (ii) what value to choose for their shape parameter (denoted by e, and with flat basis functionsstretched out in the radial direction-corresponding to e ¼ 0). The recent RBF-QR algorithm has made it practical to compute stably also for small values of e. Results from solving a convective-type PDE on a sphere are compared here for many choices of radial functions over the complete range of e-values (from very large down to the limit of e ! 0). The results are analyzed with a methodology that has similarities to the customary Fourier analysis in equispaced 1-D periodic settings. In particular, we find that high accuracy can be maintained also over very long time integrations. We furthermore gain insights into why RBFs sometimes offer higher accuracy than spherical harmonics (since the latter arise as an often non-optimal special case of the former). Anticipated future application areas for RBF-based methods in spherical geometries include weather and climate modeling.

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CAD and mesh repair with Radial Basis Functions

Journal of Computational Physics, 2012

This article appeared in a journal published by Elsevier. The attached copy is furnished to the a... more This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

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Mesh and CAD Repair Based on Parametrizations with Radial Basis Functions

Proceedings of the 20th International Meshing Roundtable, 2011

The goal of this paper is to present a new repair process that includes both model/mesh repair an... more The goal of this paper is to present a new repair process that includes both model/mesh repair and mesh generation. The repair algorithm is based on an initial mesh of the CAD that may contain topological and geometrical errors. This initial mesh is then remeshed by computing a discrete parametrization with radial basis functions (RBF's). [34] showed that a discrete parametrization can be computed by solving PDE's on an initial correct triangulation using finite elements. Paradoxically, the meshless character of the RBF's makes it an attractive numerical method for solving the PDE's for the parametrization in the case where the initial mesh contains errors. In this work, we implement the Orthogonal Gradients method which was recently introduced in [32], as a technique to solve PDE's on arbitrary surfaces with RBF's. We will implement the low order version of the algorithm, which already gives great results in this context. Different examples show that the presented method is able to deal with errors such as gaps, overlaps, T-joints and simple holes and that the resulting meshes are of high quality. Moreover, the presented algorithm can be used as a hole-filling algorithm to repair meshes with undesirable holes. The overall procedure is implemented in the open-source mesh generator Gmsh [18].

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Locality properties of radial basis function expansion coefficients for equispaced interpolation

IMA Journal of Numerical Analysis, 2007

Many types of radial basis functions (RBFs) are global, in terms of having large magnitude across... more Many types of radial basis functions (RBFs) are global, in terms of having large magnitude across the entire domain. Yet, in contrast, for example, with expansions in orthogonal polynomials, RBF expansions exhibit a strong property of locality with regard to their coefficients. That is, changing a single data value mainly affects the coefficients of the RBFs which are centered in the immediate vicinity of that data location. This locality feature can be advantageous in the development of fast and well conditioned iterative RBF algorithms. With this motivation, we employ here both analytical and numerical techniques to derive the decay rates of the expansion coefficients for cardinal data, in both 1-D and 2-D. Furthermore, we explore how these rates vary in the interesting high-accuracy limit of increasingly flat RBFs.

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A Radial Basis Function based Frames Strategy for Bypassing the Runge Phenomenon

SIAM Journal on Scientific Computing

The Radial Basis Functions Method for Improved Numerical Approximations of Geological Processes in Heterogeneous Systems

Mathematical Geosciences

$Research paper thumbnail of Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials$

Numerical solution of space-time fractional PDEs using RBF-QR and Chebyshev polynomials

Applied Numerical Mathematics

Fast RBF OGr for solving PDEs on arbitrary surfaces

The orthogonal gradients method: A radial basis functions method for solving partial differential equations on arbitrary surfaces

Journal of Computational Physics, May 1, 2012

... a surface as the zero-isosurface of a 3D function with distance dependent functions has been ... more

$Research paper thumbnail of Numerical solution of the space-time fractional diffusion equation: Alternatives to finite differences$

Numerical solution of the space-time fractional diffusion equation: Alternatives to finite differences

One of the ongoing issues with fractional-order diffusion models is the design of efficient numer... more One of the ongoing issues with fractional-order diffusion models is the design of efficient numerical schemes for the space and time discretizations. Until now, most models have relied on a low-order finite difference method to discretize both the fractional-order space and time derivatives. While the finite difference method is simple and straightforward to solve integer-order differential equations, its appeal is reduced for fractional-order differential equations as it leads to systems of linear equation defined by large full matrices. Alternatives to the finite difference method exist but a unified presentation and comparison of these methods is still missing. In this paper, we compare 4 different numerical discretizations of the space-time fractional diffusion model. These consist of the finite difference, finite element, pseudo-spectral and radial basis functions methods. We suggest that non-local methods, like the pseudo-spectral and radial basis functions method, are well-suited to discretize the non-local operators like fractional-order derivatives. These methods naturally take the global behavior of the solution into account and thus do not result in an extra computational cost when moving from an integer-order to a fractional-order diffusion model.

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A Chebyshev PseudoSpectral Method to Solve the Space-Time Tempered Fractional Diffusion Equation

SIAM Journal on Scientific Computing, 2014

The tempered fractional diffusion equation is a generalization of the standard fractional diffusi... more The tempered fractional diffusion equation is a generalization of the standard fractional diffusion equation that includes the truncation effects inherent to finite-size physical domains. As such, that equation better describes anomalous transport processes occurring in realistic complex systems. To broaden the range of applicability of tempered fractional diffusion models, efficient numerical methods are needed to solve the model equation. In this work, we have developed a pseudo-spectral scheme to discretize the space-time fractional diffusion equation with exponential tempering in both space and time. The model solution is expanded in both space and time in terms of Chebyshev polynomials and the discrete equations are obtained with the Galerkin method. Numerical examples are provided to highlight the convergence rate and the flexibility of this approach. The proposed Chebyshev pseudospectral method yields an exponential rate of convergence when the solution is smooth and allows a great flexibility to simultaneously handle fractional time and space derivatives with different levels of truncation. Our results confirm that non-local numerical methods are best suited to discretize fractional differential equations as they naturally take the global behavior of the solution into account.

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Communication à un colloque (Conference Paper)

$Research paper thumbnail of Numerical solution of the space-time tempered fractional diffusion equation: Alternatives to finite differences$

Numerical solution of the space-time tempered fractional diffusion equation: Alternatives to finite differences

$Research paper thumbnail of A radial basis functions method for solving fractional diffusion equations in one and two spatial dimensions$

A radial basis functions method for solving fractional diffusion equations in one and two spatial dimensions

Communication à un colloque (Conference Paper)

Contribution à ouvrage collectif (Book Chapter)

Localization properties of RBF expansions for cardinal interpolation

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