Radial Basis Function Partition Of Unity Method Research Articles

Modeling groundwater flow with random hydraulic conductivity using radial basis function partition of unity method

Simulating groundwater flows in heterogeneous aquifers is one of the most widely studied problems. The heterogeneity is modeled through random hydraulic conductivity fields log-normally distributed. In this paper, we aim to generate the realization of the log-normal hydraulic conductivity by summing up a finite number of random periodic modes with the Kraichnan algorithm. To address Neumann conditions, we propose a change of variables that results in Dirichlet boundary conditions along all boundaries of the computational domain. Then, we suggest using the radial basis function partition of unity method (RBFPUM) and RBFPUM with QR factorization (RBFPUM-QR) to analyze groundwater flow in the heterogeneous porous medium. We evaluate the performance of the proposed methods by increasing the number of random modes and the variance of the log-hydraulic conductivity fields using Gaussian and exponential correlation. In addition, we verify the effectiveness of the presented approaches in one and two dimensions using the method of manufactured solutions. Furthermore, we provide statistical inferences through Monte Carlo simulations. Numerical results demonstrate that RBFPUM-QR performs well in handling higher heterogeneity in both single realizations and Monte Carlo simulations.

A stable radial basis function partition of unity method for solving convection–diffusion equations on surfaces

The radial basis function partition of unity (RBF-PU) method is used to solve various types of partial differential equations, since it is based on scattered node sets that can be easily implemented. In this paper, a framework for the RBF-PU method is proposed to solve the convection–diffusion equations on closed surfaces. The surface gradient and Laplace–Beltrami operators are discretized based on the RBF-PU method using the polyharmonic spline radial basis functions augmented with polynomials. Since the RBF-PU method is a local meshless method, an advantage is that the differential matrix obtained by discretizing the surface differential operator is sparse. Additionally, solving convection-dominated problems or transport equations directly using RBF-PU method can produce numerical oscillations. For this reason, we add hyperviscosity term to improve the stability of the RBF-PU method for solving the convection–diffusion equations on surfaces, and numerical experiments show that such a combination has higher-order convergence rates. Finally, we solve the discontinuous source problem on the torus and the Turing system with convection term on different surfaces to demonstrate the effectiveness of the proposed method.

A direct RBF-PU method for simulating the infiltration of cytotoxic T-lymphocytes into the tumor microenvironment

Predicting the antitumor responses of cytotoxic T-lymphocytes (i.e., CD8+T cells) can be accomplished by numerical simulations of a mathematical model of tumor-immune cell interactions to control tumor growth in cancer therapy. Cytotoxic T lymphocytes are considering as the main effector cells to target tumor cells that are in a dormant avascular state through the chemotactic mechanism and also during the cytotoxic activity of immune cells. We study a dynamical model that is formulated based on the nonlinear partial differential equations describing the interactions between tumor-infiltrating cytotoxic lymphocytes, tumor cells, TICL-tumor cell complexes, inactivated tumor-infiltrating cytotoxic lymphocytes, dead tumor cells, and chemokine. Here, a new meshless method, i.e., the direct radial basis function partition of unity (RBF-PU) technique for the spatial discretization and a semi-implicit backward differentiation formula of order 1 for the temporal discretization are employed. The proposed meshless technique is a generalized form of the original RBF-PU method such that the computational cost of computing the derivatives could be reduced in comparison with its standard form. Besides, the method depends only on the location of nodes located on the domain of the problem for the approximation, where both types of radial functions, i.e., a conditionally positive definite function, i.e., the polyharmonic splines and a positive definite function, namely Matérn function have been used The obtained fully discrete difference scheme based on the new spatial approximation is solved using an iterative method, i.e., the generalized minimal residual method with a zero-fill incomplete lower–upper preconditioner. Finally, we present some simulation results, in which we conclude that the tumor’s growth entirely depends on CTLs activation, and this behavior can be predicted by simulating the proposed mathematical model, which may have a significant role in improving cancer immunotherapy. Besides, the developed numerical method is more accurate than the classical finite element method for solving the studied model as shown in one example.

A locally stabilized radial basis function partition of unity technique for the sine–Gordon system in nonlinear optics

This paper develops a localized radial basis function partition of unity method (RBF-PUM) based on a stable algorithm for finding the solution of the sine–Gordon system. This system is one useful description for the propagation of the femtosecond laser optical pulse in a systems of two-level atoms. The proposed strategy approximates the unknown solution through two main steps. First, the time discretization of the problem is accomplished by a difference formulation with second-order accuracy. Second, the space discretization is obtained using the local RBF-PUM. This method authorizes us to tackle the high computational time related to global collocation techniques. However, this scheme has the disadvantage of instability when the shape parameter ɛ approaches to small value. In order to deal with this issue, we adopt RBF-QR scheme that provides the higher accuracy and stable computations for small values ɛ. Two examples are presented to show the high accuracy of the method and to compare with other techniques in the literature.

The RBF-PU Method for Solving 2D Nonlocal Diffusion and Peridynamic Equations

The radial basis function partition of unity (RBF-PU) method  was applied to obtain the numerical solution of 2D nonlocal diffusion and peridynamic problems. The main idea is to partition the original domain into several patches, use the RBF approximation on each local domain, and then give weighting to obtain the global approximation of the unknown function. The radial basis function method based on the strong form of the equation has many advantages, such as avoiding an additional layer of integral calculation, no need to deal with intersections of neighborhoods with the mesh, and easiness of implementation. The numerical results show that, this method can solve nonlocal diffusion equations and peridynamic equations accurately and efficiently.

A stable RBF partition of unity local method for elliptic interface problems in two dimensions

The interface problems are faced with multiple connected domains and consequently, their solutions or derivatives might be discontinuous. This paper proposes the use of collocation based radial basis function partition of unity method (RBF-PUM) for solving two-dimensional elliptic interface problems. The RBF-PUM is a local method that allows overcoming the high computational cost associated with the global RBF methods. In the RBF-PUM, the domain is split into overlapping patches forming a covering of it. However, this method suffers from instability when the RBF shape parameter ε tends to zero. To overcome this issue, we use the RBF-QR algorithm which offers stable computations for all values of ε and provides higher accuracy. To obtain the appropriate solution in the vicinity of the interface, the domain decomposition technique is used. In this technique, the approximation in each subdomain is built separately, and proper jump conditions are then imposed across the interface. We illustrate how to apply the proposed method to Sturm-Liouville, Sturm-Liouville eigenvalue and elastostatic interface problems. The proposed method in dealing with arbitrary interfaces within different domain sizes is validated. We present some numerical examples in which the results are compared with exact solutions and those provided by other numerical methods.

The RBF partition of unity method for solving the Klein-Gordon equation

In this paper, a localized radial basis function (RBF) method is applied to obtain a global approximation of the solution of two dimensional Klein-Gordon equation on a given bounded domain. We use the RBF partition of unity (RBF-PU) method which is based on partitioning the original domain to several patches and using the RBF approximation on each local domain. Low computational cost and well conditioned final linear system are the main advantages of this method comparing with the original (global) RBF techniques. Numerical experiments show that the given problem could be solved successfully with a reasonable accuracy.

A Stable Radial Basis Function Partition of Unity Method with d-Rectangular Patches for Modelling Water Flow in Porous Media

Richards’ equation is used to describe water flow in porous media. It is strongly nonlinear, which makes the approximation of its solution a challenging task. In this study, we aim to linearize Richards’ equation using the exponential Gardner models of hydraulic conductivity and moisture content variables. Then, we propose a new formulation of radial basis function partition of unity method (RBFPUM) based on d-rectangular patches as local subdomains in order to approximate the solution of the linearized Richards’ equation. In particular, the multivariate weights are expressed as a product of one dimensional Wendland compactly supported radial basis functions. With small values of the shape parameter contained in the Gaussian radial basis function, safe computations of the solution will be achieved by adopting RBF-QR algorithm (Forenberg et al. in SIAM J Sci Comput 33(2):869–892, 2011). Hence, an efficient numerical solution can be obtained not only in terms of safe computations but also in terms of accuracy, as we will see in the numerical examples. In order to deal with layered soils, another method is proposed for the 2D and the 3D cases. It is based on the domain decomposition principle coupled with RBFPUM by using d-rectangular patches and the RBF-QR algorithm. The matching conditions, i.e. the continuity of mass fluxes and pressure head, are imposed at the interface between zones via the Steklov–Poincare equation. The efficiency of the proposed method will be examined via some examples.

Error indicators and refinement strategies for solving Poisson problems through a RBF partition of unity collocation scheme

In this article adaptive refinement algorithms are presented to solve Poisson problems by a radial basis function partition of unity (RBF-PU) collocation scheme. Since in this context the problem of constructing an adaptive discretization method to be really effective is still open, we propose some error indicators and refinement strategies, so that each of these two essential ingredients takes advantage of the potentiality of the other one. More precisely, the refinement techniques coupled with a local error indicator is an ad-hoc strategy for the RBF-PU method. The resulting scheme turns out to be flexible and the use of efficient searching procedures enables us a fast detection of the regions that adaptively need the addition/removal of points. Several numerical experiments and applications support our study by illustrating the performance of our adaptive algorithms.

A greedy algorithm for partition of unity collocation method in pricing American options

A greedy algorithm in combination with radial basis functions partition of unity collocation (GRBF‐PUC) scheme is used as a locally meshless method for American option pricing. The radial basis function partition of unity method (RBF‐PUM) is a localization technique. Because of having interpolation matrices with large condition numbers, global approximants and some local ones suffer from instability. To overcome this, a greedy algorithm is added to RBF‐PUM. The greedy algorithm furnishes a subset of best nodes among the points X. Such nodes are then used as points of trial in a locally supported RBF approximant for each partition. Using of greedy selected points leads to decreasing the condition number of interpolation matrices and reducing the burdensome in pricing American options.

RBFPUM with QR Factorization for Solving Water Flow Problem in Multilayered Soil

Abstract We discuss the numerical modeling of infiltration in variably-saturated porous media. Richards’ equation is used to describe the infiltration towards the water table. It is difficult to accurately approximate its solution especially when we deal with layered soil due to its highly nonlinear fact. In this work, the nonlinearity is handled by using Gardner model. In the case of homogeneous soil, the linearized equation is solved using radial basis function partition of unity method (RBFPUM) with the introduction of QR factorization of Gaussian in order to enhance the numerical solution for small values of the so-called “shape parameter.” In the case of layered soil, domain decomposition principle is introduced. It is based on decomposing the general problem into many subproblems. The latter are solved by RBFPUM-QR and patched by using the Steklov–Poincaré equation. Infiltration towards water table in homogeneous and layered soil is considered as a numerical test.

A RBF partition of unity collocation method based on finite difference for initial–boundary value problems

Meshfree radial basis function (RBF) methods are popular tools used to numerically solve partial differential equations (PDEs). They take advantage of being flexible with respect to geometry, easy to implement in higher dimensions, and can also provide high order convergence. Since one of the main disadvantages of global RBF-based methods is generally the computational cost associated with the solution of large linear systems, in this paper we focus on a localizing RBF partition of unity method (RBF-PUM) based on a finite difference (FD) scheme. Specifically, we propose a new RBF-PUM-FD collocation method, which can successfully be applied to solve time-dependent PDEs. This approach allows to significantly decrease ill-conditioning of traditional RBF-based methods. Moreover, the RBF-PUM-FD scheme results in a sparse matrix system, reducing the computational effort but maintaining at the same time a high level of accuracy. Numerical experiments show performances of our collocation scheme on two benchmark problems, involving unsteady convection–diffusion and pseudo-parabolic equations.

Radial basis function partition of unity method for modelling water flow in porous media

Water flow in variably-saturated porous media is modelled by using the highly nonlinear parabolic Richards’ equation. The nonlinearity is due to the hydraulic conductivity and moisture content variables. The latter were estimated by using experimental models, including Gardner, Burdine, Mualem and van Genuchten models. The aim of this work is to develop a new technique based on the radial basis function partition of unity method (RBFPUM) and Gardner model in order to solve Richards’ equation in one and two dimensions. We have used Gardner model to handle the nonlinearity issue and the RBFPUM is used to approximate the solution of the linearized Richards’ equation. Our proposed algorithm is based on testing many configurations of the partitions number and selecting the optimal shape parameter for each case. Then we pick up the optimal configuration (partitions number-shape parameter) that yields the best solution in terms of error and conditioning number. By following this procedure, an optimal solution is ensured for our given problem. As numerical tests, we consider the vertical infiltration of water in soils in order to validate our proposed method.

OpenCL Based Parallel Algorithm for RBF-PUM Interpolation

We present a parallel algorithm for multivariate Radial Basis Function Partition of Unity Method (RBF-PUM) interpolation. The concurrent nature of the RBF-PUM enables designing parallel algorithms for dealing with a large number of scattered data-points in high space dimensions. To efficiently exploit this concurrency, our algorithm makes use of shared-memory parallel processors through the OpenCL standard. This efficiency is achieved by a parallel space partitioning strategy with linear computational time complexity with respect to the input and evaluation points. The speed of our approach allows for computationally more intensive construction of the interpolant. In fact, the RBF-PUM can be coupled with a cross-validation technique that searches for optimal values of the shape parameters associated with each local RBF interpolant, thus reducing the global interpolation error. The numerical experiments support our claims by illustrating the interpolation errors and the running times of our algorithm.

A meshfree approach to non-Newtonian free surface ice flow: Application to the Haut Glacier d'Arolla

Numerical models of glacier and ice sheet dynamics traditionally employ finite difference or finite element methods. Although these are highly developed and mature methods, they suffer from some drawbacks, such as inability to handle complex geometries (finite differences) or a costly assembly procedure for nonlinear problems (finite elements). Additionally, they are mesh-based, and therefore moving domains become a challenge. In this paper, we introduce a novel meshfree approach based on a radial basis function (RBF) method. The meshfree nature of RBF methods enables efficient handling of moving margins and free ice surface. RBF methods are also accurate, easy to implement, and allow for reduction the computational cost associated with the linear system assembly, since stated in strong form. To demonstrate the global RBF method we model the velocity field of ice flow in the Haut Glacier d'Arolla, which is governed by the nonlinear Stokes equations. We test the method for different basal conditions and for a free moving surface. We also compare the global RBF method with its localized counterpart—the RBF partition of unity method (RBF-PUM)—that allows for a significant gain in the computational efficiency. Both RBF methods are compared with the classical finite element method in terms of accuracy and efficiency. We find that the RBF methods are more efficient than the finite element method and well suited for ice dynamics modeling, especially the partition of unity approach.

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.