Global Basis Functions Research Articles

A matching pursuit approach to the geophysical inverse problem of seismic traveltime tomography under the ray theory approximation

SUMMARYSeismic traveltime tomography is a geophysical imaging method to infer the 3-D interior structure of the solid Earth. Most commonly formulated as a linearized inverse problem, it maps differences between observed and expected wave traveltimes to interior regions where waves propagate faster or slower than the expected average. The Earth’s interior is typically parametrized by a single kind of localized basis function. Here we present an alternative approach that uses matching pursuits on large dictionaries of basis functions.Within the past decade the (Learning) Inverse Problem Matching Pursuits [(L)IPMPs] have been developed. They combine global and local trial functions. An approximation is built in a so-called best basis, chosen iteratively from an intentionally overcomplete set or dictionary. In each iteration, the choice for the next best basis element reduces the Tikhonov–Phillips functional. This is in contrast to classical methods that use either global or local basis functions. The LIPMPs have proven their applicability in inverse problems like the downward continuation of the gravitational potential as well as the MEG-/EEG-problem from medical imaging. Here, we remodel the Learning Regularized Functional Matching Pursuit (LRFMP), which is one of the LIPMPs, for traveltime tomography in a ray theoretical setting. In particular, we introduce the operator, some possible trial functions and the regularization. We show a numerical proof of concept for artificial traveltime delays obtained from a contrived model for velocity differences. The corresponding code is available online.

A machine learning approach for entropy due to natural convection flow of a nanofluid under the uniform inclined magnetic field

In this study, entropy generation of natural convection flow of Cu-water nanofluid under the influence of a uniform inclined magnetic field (MF) is firstly modeled by a machine learning approach using the data obtained from a numerical process. The two dimensional, time dependent dimensionless equations are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second order backward differentiation formula (BDF2) in time derivatives. In a set of combined dimensionless problem parameters, numerical simulations are performed. In the obtained data, the inputs are a set of combined dimnesionless problem parameters and the outputs are entropy indicators. Then, global and local entropy indicators are modeled by Neural Network (NN). In view of smaller mean squared error metric results, small number of layers or layer sizes in NN seems enough in smaller data while trilayer NN (TNN) is a better modeler in case of larger data.

Pseudospectral methods for continuous-time heterogeneous-agent models

We propose a pseudospectral method to solve heterogeneous-agent models in continuous time. The solution is approximated as a sum of smooth global basis functions, in our case polynomials represented by their values at Chebyshev nodes. We illustrate the method by applying it to a Krusell-Smith model. It solves the differential equations characterizing the steady-state efficiently and precisely, despite using only very few nodes. System dynamics are then automatically differentiated to simulate a linearized model. The full solution takes a third of a second and only uses standard software. A benchmark against finite differences shows that pseudospectral methods achieve far greater precision for a given number of nodes and for a given runtime. We conclude by discussing the methods' applicability, which is promising for smooth multi-dimensional models.

PARALLEL MESHLESS RADIAL BASIS FUNCTION COLLOCATION METHOD FOR NEUTRON DIFFUSION PROBLEMS

The meshless global radial basis function (RBF) collocation method is widely used to model physical phenomena in science and engineering. The method produces highly accurate solutions with an exponential convergence rate. However, due to the global approximation structure of the method, dense node distributions lead to long computation times and hinder the applicability of the technique. In order to overcome this issue, this study proposes a parallel meshless global RBF collocation algorithm. The algorithm is applied to 2-D neutron diffusion problems. The multiquadric is used as the RBF. The algorithm is developed with Mathematica and eight virtual processors are used in calculations on a multicore computer with four physical cores. The method provides accurate numerical results in a stable manner. Parallel speedup increases with the number of processors up to five and seven processors for external and fission source problems, respectively. The speedup values are limited by the constrained resource sharing of the multicore computer’s memory. On the other hand, significant time savings are achieved with parallel computation. For the four-group fission source problem, when 4316 interpolation nodes are employed, the utilization of seven processors instead of sequential computation decreases the computation time of the meshless approach by 716 s.

Modeling of Average Nusselt Number by Machine Learning and Interpolation Techniques

Abstract In this study, an important heat transfer, fluid flow parameter, and average Nusselt number Nu¯ are statistically modeled by using the data obtained from a numerical process. The two-dimensional, time-dependent dimensionless equations of natural convection (NC) flow either in the absence or in the presence of a uniform inclined magnetic field (MF) are numerically solved by using global radial basis function (RBF) method in spatial derivatives and the second-order backward differentiation formula (BDF2) in time derivatives. Numerical simulations are performed in a set of combined dimensionless problem parameters. A dataset with inputs Rayleigh number Ra, Prandtl number Pr, and output Nu¯ in the absence of MF and a dataset with inputs Ra, Pr, Hartmann number Ha, inclination angle γ, and output Nu¯ in the presence of inclined uniform MF are saved. The obtained data are separated into train and test sets. Then, Nu¯ is first modeled by Neural Networks (NN). Second, interpolation is also examined. In terms of mean squared error (MSE) metric, NN outputs give the best goodness of fit results compared to curve fitting on test data. On the other side, it is shown that interpolation is also an alternative for modeling. This modeling issue enables one to get the desired result without making heavy numerical calculations many times.

Energy-Stable Global Radial Basis Function Methods on Summation-By-Parts Form

Radial basis function methods are powerful tools in numerical analysis and have demonstrated good properties in many different simulations. However, for time-dependent partial differential equations, only a few stability results are known. In particular, if boundary conditions are included, stability issues frequently occur. The question we address in this paper is how provable stability for RBF methods can be obtained. We develop a stability theory for global radial basis function methods using the general framework of summation-by-parts operators often used in the Finite Difference and Finite Element communities. Although we address their practical construction, we restrict the discussion to basic numerical simulations and focus on providing a proof of concept.

An immersed selective discontinuous Galerkin method in particle-in-cell simulation with adaptive Cartesian mesh and polynomial preserving recovery

In this paper, a selective discontinuous Galerkin (SDG) method and a polynomial preserving recovery (PPR) method are developed and integrated with the immersed-finite-element particle-in-cell (IFE-PIC) method, in order to carry out the plasma-material interaction simulation on adaptive Cartesian meshes (ACM). To significantly save the computational cost in practice, the PIC simulation often needs to use Cartesian meshes for the whole domain and the ACM for some focused regions in the plasma-material interaction problems. Therefore, we utilize the SDG method with immersed finite elements for the field solver of the IFE-PIC method, as the key technique to allow the local Cartesian mesh refinement which will generate hanging nodes. To minimize the degree of freedom of the SDG on the ACM and reduce the computational cost, a new selective technique of the discontinuous global basis functions is proposed for the SDG. Various types of nodes in the ACM are discussed for the implementation of this selective technique, including the hanging nodes. Meanwhile, the gathering and scattering steps of the PIC simulation also need to be appropriately performed on the ACM. The PPR method is a generic gradient recovery technique to be incorporated into the IFE-PIC method for more accurate force deposition on the ACM. In addition, two other algorithmic structures of the traditional IFE-PIC method have been modified to accommodate the ACM in the simulation, including a new mapping array structure for the particle positioning and a new charge deposition with some special particle positions in the ACM. As a result, the integrated immersed-selective-discontinuous-Galerkin particle-in-cell (ISDG-PIC) method can efficiently perform the plasma-material interactions simulation on the Cartesian meshes with local mesh refinement independent of the interface. Several numerical experiments are performed to demonstrate the convergence, efficiency, and applicability of the proposed ISDG-PIC method.

Numerical homogenization by an adaptive Fourier spectral method on non-uniform grids using optimal transport

This article focuses on the computation of the solutions to periodic cell problems that arise in a number of homogenization schemes. Formulated in terms of a generic constitutive relation, encompassing the linear and non-linear cases, a model problem is intended to be discretized on a Fourier-basis and solved using a Fast Fourier Transform (FFT)-based iterative scheme. Such a spectral method, which makes use of a uniform grid and global basis functions, has inherent limitations for correctly capturing any localized features of the solution, such as singularities or discontinuities. An adaptive method is proposed here to overcome these shortcomings. It relies on the introduction of a non-homogeneous bijective mapping between the original physical domain and a computational domain, where the Fourier spectral method will be used to solve a transformed problem. This makes it possible both to discretize the original equations on a non-uniform grid, concentrating grid points where necessary to increase computational accuracy, and to maintain the use of standard FFT schemes. To compute the mapping, an approach based on optimal transport is proposed and a numerical scheme based on FFT tools is described. A number of adaptive strategies are then discussed, either based on a priori material-based considerations, or a posteriori using a preliminary solution. The properties of the transported problem in the computational domain are described and the applicability, up to minor modifications, of a number of standard iterative schemes is discussed. A set of numerical examples is included to assess the performances of the method, both in the case of linear conductivity and for a strongly non-linear elastoplastic problem, evaluating its results both qualitatively with full-field comparisons and quantitatively with error measurements relatively to the number of iterations, the discretization and the original material contrast. Overall, these results show that, for a given discretization and compared with a computation on a regular grid, a systematic gain in accuracy is obtained using the adaptive method on a non-uniform grid.

Higher-continuity s-version of finite element method with B-spline functions

This paper proposes a strategy to solve the problems of the conventional s-version of finite element method (SFEM) fundamentally. Because SFEM can reasonably model an analytical domain by superimposing meshes with different spatial resolutions, it has intrinsic advantages of local high accuracy, low computation time, and simple meshing procedure. However, it has disadvantages such as accuracy of numerical integration and matrix singularity. Although several additional techniques have been proposed to mitigate these limitations, they are computationally expensive or ad-hoc, and detract from the method's strengths. To solve these issues, we propose a novel strategy called B-spline based SFEM. To improve the accuracy of numerical integration, we employed cubic B-spline basis functions with C2-continuity across element boundaries as the global basis functions. To avoid matrix singularity, we applied different basis functions to different meshes. Specifically, we employed the Lagrange basis functions as local basis functions. The numerical results indicate that using the proposed method, numerical integration can be calculated with sufficient accuracy without any additional techniques used in conventional SFEM. Furthermore, the proposed method avoids matrix singularity and is superior to conventional methods in terms of convergence for solving linear equations. Therefore, the proposed method has the potential to reduce computation time while maintaining a comparable accuracy to conventional SFEM.

The dynamic analysis of axisymmetric bodies with damping effects using the modified radial integration boundary elements method (MRIBEM)

This paper investigates the elastodynamic analysis of axisymmetric bodies under axisymmetric dynamic loads using the modified radial integration boundary elements method (MRIBEM). Purposely, the governing integral equations have been developed in cylindrical coordinates by employing the three-dimensional (3D) static fundamental solution. The axisymmetric bodies can be modeled as a two-dimensional (2D) problem, which is much simpler than 3D analysis and only requires a one-dimensional boundary mesh. In the proposed formulation, the effects of inertial and damping forces appear as 2D domain integrals. The domain variables are approximated using global radial basis functions (RBFs), and the modified radial integration method (MRIM) is employed to deal with the domain integral as the main idea of the present research. Various sample problems are provided in order to illustrate the capability of the presented method for analyzing axisymmetric bodies with concave geometry and damping effect under different types of transient loads. The Newmark and Houbolt methods were applied for time marching, and their efficiency was examined. The results of the present study were also validated by the axisymmetric FEM formulation and other available references.

Meshless structure-preserving GRBF collocation methods for stochastic Maxwell equations with multiplicative noise

In this paper, motivated by the principle that numerical methods should preserve the intrinsic properties of the original system as much as possible, we propose two novel classes of structure-preserving methods for the stochastic Maxwell equations with multiplicative noise. More precisely, due to the advantages of high-order accuracy and the simplicity to deal with high-dimensional problems, the meshless global radial basis function (GRBF) collocation method is firstly utilized to discretize the stochastic Maxwell equations in space, the resulting semi-discretization preserves energy and symplectic structure. Then we apply Padé approximation, the splitting technique and the Runge–Kutta method to propose two kinds of efficient fully-discrete methods that are proved to be symplectic, multi-symplectic and energy-preserving simultaneously. Numerical experiments are given to indicate the validity of the proposed methods.

Local-basis difference potentials method for elliptic PDEs in complex geometry

We develop efficient and high-order accurate finite difference methods for elliptic partial differential equations in complex geometry in the difference potentials framework. The main novelty of the developed schemes is the use of local basis functions defined at near-boundary grid points. The use of local basis functions allows unified numerical treatment of (i) explicitly and implicitly defined geometry; (ii) geometry of more complicated shapes, such as those with corners, multi-connected domain, etc; and (iii) different types of boundary conditions. This geometrically flexible approach is complementary to the classical difference potentials method using global basis functions, especially in the case where a large number of global basis functions are needed to resolve the boundary, or where the optimal global basis functions are difficult to obtain. Fast Poisson solvers based on FFT are employed for standard, centered finite difference stencils regardless of the designed order of accuracy. Proofs of convergence of difference potentials in maximum norm are outlined both theoretically and numerically.

A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives

This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered.

Voxel‐Based Full‐Field Eigenstrain Reconstruction of Residual Stresses

Inverse eigenstrain (inherent strain) analysis methods are shown to be effective for the reconstruction of residual stresses in plane eigenstrain problems (continuously processed bodies) while conversely residual stress reconstruction in discontinuously processed bodies is extremely challenging and necessitates the use of complex regularizing assumptions. Herein, a new generic inverse eigenstrain method suitable for the reconstruction of residual stresses along with residual elastic strains and displacements in discontinuously processed bodies is introduced. The proposed method uses the superposition of eigenstrain radial basis functions together with a set of limited experimental data for model‐free (unconstrained) determination of unknown eigenstrain fields. This approach eliminates the limitations introduced by global basis functions such as polynomials. The novel point of this method is the ability to account for all six components of strain in an isotropic body without using regularizing assumptions. By lifting complex guiding formulation, the fidelity of full‐field eigenstrain reconstruction becomes directly related to the quality of experimental data and proper discretisation of the model domain. The FEniCS implementation has been validated using the experimental data of pointwise high‐energy synchrotron X‐ray diffraction measurements from a bent titanium alloy bar. A hybrid high throughput computing approach is also introduced for effective parallel computing.

Towards stability results for global radial basis function based quadrature formulas

Quadrature formulas (QFs) based on radial basis functions (RBFs) have become an essential tool for multivariate numerical integration of scattered data. Although numerous works have been published on RBF-QFs, their stability theory can still be considered as underdeveloped. Here, we strive to pave the way towards a more mature stability theory for global and function-independent RBF-QFs. In particular, we prove stability of these for compactly supported RBFs under certain conditions on the shape parameter and the data points. As an alternative to changing the shape parameter, we demonstrate how the least-squares approach can be used to construct stable RBF-QFs by allowing the number of data points used for numerical integration to be larger than the number of centers used to generate the RBF approximation space. Moreover, it is shown that asymptotic stability of many global RBF-QFs is independent of polynomial terms, which are often included in RBF approximations. While our findings provide some novel conditions for stability of global RBF-QFs, the present work also demonstrates that there are still many gaps to fill in future investigations.

Numerical Inverse Laplace Transform Methods for Advection-Diffusion Problems

Partial differential equations arising in engineering and other sciences describe nature adequately in terms of symmetry properties. This article develops a numerical method based on the Laplace transform and the numerical inverse Laplace transform for numerical modeling of diffusion problems. This method transforms the time-dependent problem to a corresponding time-independent inhomogeneous problem by employing the Laplace transform. Then a local radial basis functions method is employed to solve the transformed problem in the Laplace domain. The main feature of the local radial basis functions method is the collocation on overlapping sub-domains of influence instead of on the whole domain, which reduces the size of the collocation matrix; hence, the problem of ill-conditioning in global radial basis functions is resolved. The Laplace transform is used in comparison with a finite difference technique to deal with the time derivative and avoid the effect of the time step on numerical stability and accuracy. However, using the Laplace transform sometimes leads to a solution in the Laplace domain that cannot be converted back into the real domain by analytic methods. Therefore, in such a case, the Laplace transform is inverted numerically. In this investigation, two inversion techniques are utilized; (i) the contour integration method, and (ii) the Stehfest method. Three test problems are used to evaluate the proposed numerical method. The numerical results demonstrate that the proposed method is computationally efficient and highly accurate.

The solution of elastodynamic problems using modified radial integration boundary element method (MRIBEM)

In this paper, the modified radial integration boundary elements method (MRIBEM) is applied to analyze two-dimensional (2D) elastodynamic problems. The boundary and domain variables are approximated by Lagrangian shape functions and global radial basis functions (RBFs), respectively. As a key idea of this study, the domain integrals that result from implying the inertia force effect are computed by means of a modified radial integration method (MRIM). With the MRIBEM, we are able to deal with problems with concave shapes when the domain variables are approximated by global RBFs by defining an auxiliary point or, in case of complex domain shapes, two or more auxiliary points as the origin for radial integration. Two-time marching schemes, i.e. Newmark method and the Houbolt method, are implemented for the solution of governing differential equations along time. The performance of the proposed method is examined in the solution of four numerical examples.

High-Order Semi-Lagrangian Schemes for the Transport Equation on Icosahedron Spherical Grids

The transport process is an important part of the research of fluid dynamics, especially when it comes to tracer advection in the atmosphere or ocean dynamics. In this paper, a series of high-order semi-Lagrangian methods for the transport process on the sphere are considered. The methods are formulated entirely in three-dimensional Cartesian coordinates, thus avoiding any apparent artificial singularities associated with surface-based coordinate systems. The underlying idea of the semi-Lagrangian method is to find the value of the field/tracer at the departure point through interpolating the values of its surrounding grid points to the departure point. The implementation of the semi-Lagrangian method is divided into the following two main procedures: finding the departure point by integrating the characteristic equation backward and then interpolate on the departure point. In the first procedure, three methods are utilized to solve the characteristic equation for the locations of departure points, including the commonly used midpoint-rule method and two explicit high-order Runge–Kutta (RK) methods. In the second one, for interpolation, four new methods are presented, including (1) linear interpolation; (2) polynomial fitting based on the least square method; (3) global radial basis function stencils (RBFs), and (4) local RBFs. For the latter two interpolation methods, we find that it is crucial to select an optimal value for the shape parameter of the basis function. A Gauss hill advection case is used to compare and contrast the methods in terms of their accuracy, and conservation properties. In addition, the proposed method is applied to standard test cases, which include solid body rotation, shear deformation of twin slotted cylinders, and the evolution of a moving vortex. It demonstrates that the proposed method could simulate all test cases with reasonable accuracy and efficiency.

Determination of an Extremal in Two-Dimensional Variational Problems Based on the RBF Collocation Method

This paper introduces a direct method derived from the global radial basis function (RBF) interpolation over arbitrary collocation nodes occurring in variational problems involving functionals that depend on functions of a number of independent variables. This technique parameterizes solutions with an arbitrary RBF and transforms the two-dimensional variational problem (2DVP) into a constrained optimization problem via arbitrary collocation nodes. The advantage of this method lies in its flexibility in selecting between different RBFs for the interpolation and parameterizing a wide range of arbitrary nodal points. Arbitrary collocation points for the center of the RBFs are applied in order to reduce the constrained variation problem into one of a constrained optimization. The Lagrange multiplier technique is used to transform the optimization problem into an algebraic equation system. Three numerical examples indicate the high efficiency and accuracy of the proposed technique.

Implicit Surface Reconstruction via RBF Interpolation: A Review

Background: Implicit surface is a kind of surface modeling tool, which is widely used in point cloud reconstruction, deformation and fusion due to its advantages of good smoothness and Boolean operation. The most typical method is the surface reconstruction with Radial Basis Functions (RBF) under normal constraints. RBF has become one of the main methods of point cloud fitting because it has a strong mathematical foundation, an advantage of computation simplicity, and the ability of processing nonuniform points. Objective: Techniques and patents of implicit surface reconstruction interpolation with RBF are surveyed. Theory, algorithm, and application are discussed to provide a comprehensive summary for implicit surface reconstruction in RBF and Hermite Radial Basis Functions (HRBF) interpolation. Methods: RBF implicit surface reconstruction interpolation can be divided into RBF interpolation under the constraints of points and HRBF interpolation under the constraints of points and corresponding normals. Results: A total of 125 articles were reviewed, in which more than 30% were related to RBF in the last decade. The continuity properties and application fields of the popular global supported radial basis functions and compactly supported radial basis functions are analyzed. Different methods of RBF and HRBF implicit surface reconstruction are evaluated, and the challenges of these methods are discussed. Conclusion: In future work, implicit surface reconstruction via RBF and HRBF should be further studied in fitting accuracy, computation speed, and other fundamental problems. In addition, it is a more challenging but valuable research direction to construct a new RBF with both compact support and improved fitting accuracy.