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

theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions multivariate interpolation of smooth data using smooth radial basis functions is considered the behavior of the interpolants in the limit of nearly flat radial basis functions is studied both theoretically and numerically explicit criteria for different types of limits are given using the results for the limits the dependence of the error on the shape parameter of the radial basis function is investigated the mechanisms that determine the optimal shape parameter value are studied and explained through approximate expansions of the interpolation error

extensions and traces of functions of bounded variation on metric spaces abstract in the setting of a metric space equipped with a doubling measure and supporting a poincare inequality and based on results by bjorn and shanmugalingam 2007 7 we show that functions of bounded variation can be extended from any bounded uniform domain to the whole space closely related to extensions is the concept of boundary traces which have previously been studied by hakkarainen et al 2014 17 on spaces that satisfy a suitable locality condition for sets of finite perimeter we establish some basic results for the traces of functions of bounded variation our analysis of traces also produces novel pointwise results on the behavior of functions of bounded variation in their jump sets

neumann problem for p laplace equation in metric spaces using a variational approach existence boundedness and boundary regularity we employ a variational approach to study the neumann boundary value problem for the p laplacian on bounded smooth enough domains in the metric setting and show that solutions exist and are bounded the boundary data considered are borel measurable bounded functions we also study boundary continuity properties of the solutions one of the key tools utilized is the trace theorem for newton sobolev functions and another is an analog of the de giorgi inequality adapted to the neumann problem

the interior radon transform the interior radon transform arises from a limited data problem in computerized tomography when only rays travelling through a specified region of interest are measured this problem occurs due to technical restrictions of the sampling apparatus or in an endeavour to reduce the x ray dose the corresponding operator mathcal r i is investigated as a mapping between weighted l 2 spaces the main result is a singular value decomposition svd for this operator for functions of unbounded support in mathbb r 2 the proof is based on the construction of intertwining differential operators the techniques used are unified in the sense that svds for other radon transforms with rotational symmetry can easily be derived in the same way consistency conditions are also obtained for mathcal r i for functions of bounded and unbounded support many of the results generalize to higher dimensions they are stated whenever they follow directly from the two dimensional case

stable computation of multiquadric interpolants for all values of the shape parameter abstract spectrally accurate interpolation and approximation of derivatives used to be practical only on highly regular grids in very simple geometries since radial basis function rbf approxima ions permit this even for multivariate scattered data there has been much recent interest in practical algorithms to compute these approximations effectively several types of rbfs feature a free parameter e g c in the multiquadric mq case f r r 2 c 2 the limit of c increasingly flat basis functions has not received much attention because it leads to a severely ill conditioned problem we present here an algorithm which avoids this difficulty and which allows numerically stable computations of mq rbf interpolants for all parameter values we then find that the accuracy of the resulting approximations in some cases becomes orders of magnitude higher than was the case within the previously available parameter range our new method provides the first tool for the numerical exploration of mq rbf interpolants in the limit of c the method is in no way specific to mq basis functions and can without any change be applied to many other cases as well

unsymmetric and symmetric meshless schemes for the unsteady convection diffusion equation in this paper we investigate the application of unsymmetric and symmetric meshless collocation techniques with radial basis functions for solving the unsteady convection diffusion equation we employ the method of lines approach to discretize the governing operator equation the stability of both explicit and implicit time stepping schemes are analyzed numerical results are presented for 1d and 2d problems to compare the performance of the unsymmetric and symmetric collocation techniques we compare the performance of various globally supported radial basis functions such as multiquadric inverse multiquadric gaussian thin plate splines and quintics numerical studies suggest that symmetric collocation is only marginally better than the unsymmetric approach further it appears that both collocation techniques require a very dense set of collocation points in order to achieve accurate results for high pe clet numbers

the runge phenomenon and spatially variable shape parameters in rbf interpolation many studies mostly empirical have been devoted to finding an optimal shape parameter for radial basis functions rbf when exploring the underlying factors that determine what is a good such choice we are led to consider the runge phenomenon rp best known in cases of high order polynomial interpolation as a key error mechanism this observation suggests that it can be advantageous to let the shape parameter vary spatially rather than assigning a single value to it benefits typically include improvements in both accuracy and numerical conditioning still another benefit arises if one wishes to improve local accuracy by clustering nodes in selected areas this idea is routinely used when working with splines or finite element methods however local refinement with rbfs may cause rp type errors unless we use a spatially variable shape paremeter with this enhancement rbf approximations combine freedom from meshes with spectral accuracy on irregular domains and furthermore permit local node clustering to improve the resolution wherever this might be needed

observations on the behavior of radial basis function approximations near boundaries abstract rbf approximations would appear to be very attractive for approximating spatial derivatives in numerical simulations of pdes rbfs allow arbitrarily scattered data generalize easily to several space dimensions and can be spectrally accurate however accuracy degradations near boundaries in many cases severely limit the utility of this approach with that as motivation this study aims at gaining a better understanding of the properties of rbf approximations near the ends of an interval in 1 d and towards edges in 2 d

trace theorems for functions of bounded variation in metric spaces abstract in this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1 poincare inequality and obtain l 1 estimates of the trace functions in contrast with the treatment of traces given in other papers on this subject the traces we consider do not require knowledge of the function in the exterior of the domain we also establish a maz ya type inequality for functions of bounded variation that vanish on a set of positive capacity

the approximate inverse in action ii convergence and stability the approximate inverse is a scheme for constructing stable inversion formulas for operator equations originally it is defined on l2 spaces in the present article we extend the concept of approximate inverse to more general settings which allow us to investigate the discrete version of the approximate inverse which actually underlies numerical computations indeed we show convergence if the discretization parameter tends to zero further we prove stability that is we show the regularization property finally we apply the results to the filtered backprojection algorithm in 2d tomography to obtain convergence rates

comparisons of relative bv capacities and sobolev capacity in metric spaces abstract we study relations between the variational sobolev 1 capacity and versions of variational bv capacity in a complete metric space equipped with a doubling measure and supporting a weak 1 1 poincare inequality we prove the equality of 1 modulus and the continuous 1 capacity extending the known results for 1 p to also cover the more geometric case p 1 then we give alternative definitions for variational bv capacities and obtain equivalence results between them finally we study relations between total 1 capacity and versions of bv capacity

a new method for evaluating options based on multiquadric rbf fd method abstract in this paper a new local meshless approach based on radial basis functions rbfs is presented to price the options under the black scholes model the global rbf approximations derived from the conventional global collocation method usually lead to ill conditioned matrices employing the idea of local approximants of the finite difference fd method and combining it with the radial basis function rbf method can result in a local meshless approach such as rbf fd it removes the difficulty of ill conditionness of the original method the new proposed approach is unconditionally stable as it is shown by von neumann stability analysis it is fast and produces high accurate results as shown in numerical experiments moreover we took into account the variation of shape parameter and analyzed numerically the behavior of the rbf fd method

application of the rbf meshless method to the solution of the radiative transport equation in this paper we introduce a radial basis function collocation method for computing solutions to the time dependent radiative transfer equation for these computations we use finite differences to discretize the time coordinate a discrete ordinate method to discretize the directional variable and an expansion in radial basis functions to approximate the spatial dependence of the solution the main advantages of the rbf method are that it does not require any mesh or grid achieves spectral accuracy in multi dimensions for arbitrary node layouts and it is extremely simple to implement

approximate inverse method for stable analytic continuation in a strip domain numerical analytic continuation is in general an ill posed problem and some special regularization methods are needed in this paper we apply the approximate inverse method to deal with the problem in a strip domain w z x iy c x r 0 y

a note on the extension of bv functions in metric measure spaces in this note a classical extension result for bv functions due to yu d burago and v g maz ja yu d burago v g maz ja potential theory and function theory for irregular regions seminars in math vol 3 v a steklov math inst leningrad 1969 translated from russian is generalized to the abstract setting of doubling metric measure spaces endowed with a differentiation structure called d structure

optimality of the fully discrete filtered backprojection algorithm for tomographic inversion although the filtered backprojection algorithm fba has been the standard reconstruction algorithm in 2d computerized tomography for more than 30 years its convergence behavior is not completely settled so far relying on convergence results by rieder and faridani for the semi discrete fba siam j numer anal 41 3 869 892 2003 we show optimality of the fully discrete version for reconstructing sufficiently smooth density distributions further we introduce mfba a modified version of fba and prove its optimality under weaker smoothness requirements remarkably mfba may have a larger convergence order in the angular than in the lateral variable thus allowing optimal convergence in case of angular under sampling moreover mfba can be seen as a limit of the phantom view method introduced to increase angular resolution

a finite dimensional realization of the mollifier method for compact operator equations we introduce and analyze a stable procedure for the approximation of f if where f is the least residual norm solution of the minimal norm of the ill posed equation af g with compact operator a x y between hilbert spaces and x has some smoothness assumption our method is based on a finite number of singular values of a and some finite rank operators our results are in a more general setting than the one considered by rieder and schuster 2000 and nair and lal 2003 with special reference to the mollifier method and it is also applicable under fewer smoothness assumptions on

incomplete data problems in x ray computerized tomography in the present paper we study truncated projections for the fanbeam geometry in computerized tomography first we derive consistency conditions for the divergent beam transform then we study a singular value decomposition for the case where only the interior rays in the fan are provided as for example in region of interest tomography we show that the high angular frequency components of the searched for densities are well determined and we present reconstructions from real data where the missing information is approximated based on the singular value decomposition

interpolation in the limit of increasingly flat radial basis functions abstract many types of radial basis functions such as multiquadrics contain a free parameter in the limit where the basis functions become increasingly flat the linear system to solve becomes highly ill conditioned and the expansion coefficients diverge nevertheless we find in this study that limiting interpolants often exist and take the form of polynomials in the 1 d case we prove that with simple conditions on the basis function the interpolants converge to the lagrange interpolating polynomial hence differentiation of this limit is equivalent to the standard finite difference method we also summarize some preliminary observations regarding the limit in 2 d

the approximate inverse in action with an application to computerized tomography the approximate inverse is a scheme to obtain stable numerical inversion formulae for linear operator equations of the first kind yet in some applications the computation of a crucial ingredient the reconstruction kernel is time consuming and instable it may even happen that the kernel does not exist for a particular semidiscrete system to cure this dilemma we propose and analyze a technique that is based on a singular value decomposition of the underlying operator the results are applied to the reconstruction problem in 2d computerized tomography where they enable the design of reconstruction filters and lead to a novel error analysis of the filtered backprojection algorithm

incomplete data problems in x ray computerized tomography i singular value decomposition of the limited angle transform etude des problemes intrinseques relatifs aux donnees incompletes dans la tomographie aux rayons x assistee par ordinateur decomposition de la transformation de radon

a mollification framework for improperly posed problems various authors have explored the role of mollification in the solution of improperly posed problems such as numerical differentiation and the numerical solution of first kind integral equations for example murio 13 exploiting earlier suggestions of manselli and miller 11 as well as vasin 16 has shown how mollified numerical differentiators can be constructed for the differentiation of observational data but the underlying regularity requires the existence of the second derivative of the function being differentiated on the other hand hao et al 4 proposed and examined a mollification procedure where the regularity reduced to only requiring the existence of the fractional derivative being evaluated independently anderssen and de hoog 2 examined the use of multi point moving average finite difference formulas for the numerical differentiation of observational data and show how the length of such formulas must be related to the size of the discretization of the data before convergence and stability bounded amplification error can be guaranteed as the size of the discretization is decreased this represents an explicit exemplification of the situation which occurs in mollification where some appropriate measure of the extent of the mollification must be related in some suitable way to the discretization of the data before convergence and stability results can be derived however to date such investigations have not explicitly exploited the natural translation and dilation properties of fractional differentiation operators the purpose of this paper is to show how using the classical semi group theory of hille and phillips 8 a quite general theory can be constructed for the mollification of operator equations

equivalent definitions of bv space and of total variation on metric measure spaces abstract in this paper we introduce a new definition of bv based on measure upper gradients and prove the equivalence of this definition and the coincidence of the corresponding notions of total variation with the definitions based on relaxation of l 1 norm of the slope of lipschitz functions or upper gradients as in the previous work by the first author with gigli and savare in the sobolev case the proof requires neither local compactness nor doubling and poincare

a galerkin least squares approach for photoacoustic tomography the development of fast and accurate image reconstruction algorithms is a central aspect of computed tomography in this paper we address this issue for photoacoustic computed tomography in circular geometry we investigate the galerkin least squares method for that purpose for approximating the function to be recovered we use subspaces of translation invariant spaces generated by a single function this includes many systems that have previously been employed in photoacoustic tomography such as generalized kaiser bessel basis functions or the natural pixel basis by exploiting an isometry property of the forward problem we are able to efficiently set up the galerkin equation for a wide class of generating functions and devise efficient algorithms for its solution we establish a convergence analysis and present numerical simulations that demonstrate the efficiency and accuracy of the derived algorithm

eigenvalue stability of radial basis function discretizations for time dependent problems differentiation matrices obtained with infinitely smooth radial basis function rbf collocation methods have under many conditions eigenvalues with positive real part preventing the use of such methods for time dependent problems we explore this difficulty at theoretical and practical levels theoretically we prove that differentiation matrices for conditionally positive definite rbfs are stable for periodic domains we also show that for gaussian rbfs special node distributions can achieve stability in 1 d and tensor product nonperiodic domains as a more practical approach for bounded domains we consider differentiation matrices based on least squares rbf approximations and show that such schemes can lead to stable methods on less regular nodes by separating centers and nodes least squares techniques open the possibility of the separation of accuracy and stability characteristics

convergence analysis for a second order elliptic equation by a collocation method using scattered points a collocation method using scattered points applied to a second order elliptic differential equation is analyzed by establishing a new quadrature formula for the space of the polynomials we show that a polynomial solution possesses stability and preserves a similar convergence property occurred in the classical high order collocation method

some observations regarding interpolants in the limit of flat radial basis functions abstract radial basis functions rbfs form a primary tool for multivariate interpolation some of the most commonly used radial functions feature a shape parameter allowing them to vary from being nearly flat e small to sharply peaked e large the former limit can be particularly accurate when interpolating a smooth function based on scattered data this study discusses theoretical and computational aspects of the e 0 limit and includes the conjecture that gaussian rbf interpolants will never diverge in this limit

limit problems for interpolation by analytic radial basis functions interpolation problems for analytic radial basis functions like the gaussian and inverse multiquadrics can degenerate in two ways the radial basis functions can be scaled to become increasingly flat or the data points coalesce in the limit while the radial basis functions stay fixed both cases call for a careful regularization which if carried out explicitly yields a preconditioning technique for the degenerating linear systems behind these interpolation problems this paper deals with both cases for the increasingly flat limit we recover results by larsson and fornberg together with lee yoon and yoon concerning convergence of interpolants towards polynomials with slight modifications the same technique can also handle scenarios with coalescing data points for fixed radial basis functions the results show that the degenerating local lagrange interpolation problems converge towards certain hermite birkhoff problems this is an important prerequisite for dealing with approximation by radial basis functions adaptively using freely varying data sites

poincare inequalities uniform domains and extension properties for newton sobolev functions in metric spaces in the setting of metric measure spaces equipped with a doubling measure supporting a weak p poincare inequality with 1 p 1 we obtain a near characterization of n1 p extension domains with loc

quasiconformal maps in metric spaces with controlled geometry this paper develops the foundations of the theory of quasiconformal maps in metric spaces that satisfy certain bounds on their mass and geometry the principal message is that such a theory is both relevant and viable the first main issue is the problem of definition which we next describe quasiconformal maps are commonly understood as homeomorphisms that distort the shape of infinitesimal balls by a uniformly bounded amount this requirement makes sense in every metric space given a homeomorphism f from a metric space x to a metric space y then for x x and r 0 set

singular value decompositions and inversion methods for the exterior radon transform and a spherical transform abstract the classical radon transform r maps an integrable function in rn to its integrals over all n 1 dimensional hyperplanes and the exterior radon transform is the transform r restricted to hyperplanes that do not intersect a given disc a singular value decomposition for the exterior transform is given for spaces of square integrable functions on the exterior of the disc this decomposition in orthogonal functions explicitly produces the null space and range of the exterior transform and gives a new method for inverting the transform modulo the null space a modification of this method is given that will exactly invert functions of compact support these results generalize theorems of r m perry and the author a singular value decomposition for the radon transform that integrates over spheres in rn containing the origin is also given this follows from the singular value decomposition for r and yields the null space and a new inversion method for this transform

the parameter r2 in multiquadric interpolation the multiquadric mq method is an effective bivariate interpolant to three dimensional data xi yi zi where the xi yi are arbitrarily located in the plane the accuracy of the mq method is dependent on a user defined parameter r2 and most practitioners select r2 based upon the number of data points and the locations of the xi yi in the plane we observe that the optimal value of r2 is a strong function of the zi and that it is essentially independent of both the number and the locationsof the xi yi points contrary to some opinions we observe that the mq method can effectively interpolate to track data that is data sampled densely along tracks in the plane together with several other observations we present an algorithm that generally yields an effective value for r2

the approximate inverse in action iii 3d doppler tomography the approximate inverse is a powerful tool for solving first kind operator equations in a stable way its abstract convergence and stability theory developed in our articles siam j numer anal 37 1909 1929 2000 and math comp 72 1399 1415 2003 is applied to the reconstruction problem of 3d vector field tomography resulting in a reconstruction algorithm of filtered backprojection type for an analytically computed reconstruction filter reconstruction kernel convergence with rates as well as the regularization property are established