divergence runge phenomenon for least squares polynomial approximation on an equispaced grid and mock chebyshev subset interpolation runge showed more than a century ago that polynomial interpolation of a function f x using points evenly spaced on email protected 1 1 could diverge on parts of this interval even if f x was analytic everywhere on the interval least squares fitting of a polynomial of degree n to an evenly spaced grid with p points should improve accuracy if email protected n we show through numerical experiments that such an overdetermined fit reduces but does not eliminate the runge phenomenon more precisely define b n 1 p the least squares fit will fail to converge everywhere on 1 1 as n for fixed b if f x has a singularity inside a curve d b in the complex plane the width of the region enclosed by the convergence boundary d shrinks as b diminishes i e more points for a fixed polynomial degree but shrinks to the interval 1 1 only when b 0 we also show that the runge phenomenon can be completely defeated by interpolation on a mock chebyshev grid a subset of n 1 points from an equispaced grid with o n 2 points chosen to mimic the non uniform n 1 point chebyshev lobatto grid

formulae for the derivatives of heat semigroups abstract formulae for the derivatives of solutions of diffusion equations are derived which clearly exhibit and allow estimation of the equations smoothing properties these also give formulae for the logarithmic gradient of the corresponding heat kernels extending and giving a very elementary proof of bismut s well known formula corresponding formulae are derived for solutions of heat equations for differential forms and their exterior derivatives

a comparison of numerical algorithms for fourier extension of the first second and third kinds the range of fourier methods can be significantly increased by extending a nonperiodic function f x to a periodic function f on a larger interval when f x is analytically known on the extended interval the extension is straightforward when f x is unknown outside the physical interval there is no standard recipe worse still like a radarless aircraft groping through fog the algorithm may wreck on the mountain in fog problem a function f x which is perfectly well behaved on the physical interval may very well have singularities in the extended domain in this article we compare several algorithms for successfully extending a function f x into the fog even when the analytic extension is singular the best third kind extension requires singular value decomposition with iterative refinement but achieves accuracy close to machine precision

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

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

mean field stochastic differential equations and associated pdes in this paper we consider a mean field stochastic differential equation also called the mckean vlasov equation with initial data t x 0 t rd t x 0 t rd whose coefficients depend on both the solution xt xsxst x and its law by considering square integrable random variables as initial condition for this equation we can easily show the flow property of the solution xt sxst of this new equation associating it with a process xt x p sxst x p which coincides with xt sxst when one substitutes for xx but which has the advantage to depend on only through its law p p we characterize the function v t x p e xt x p t pxt t v t x p e xtt x p pxtt under appropriate regularity conditions on the coefficients of the stochastic differential equation as the unique classical solution of a nonlocal partial differential equation of mean field type involving the first and the second order derivatives of vv with respect to its space variable and the probability law the proof bases heavily on a preliminary study of the first and second order derivatives of the solution of the mean field stochastic differential equation with respect to the probability law and a corresponding ito formula in our approach we use the notion of derivative with respect to a probability measure with finite second moment introduced by lions in cours au college de france theorie des jeu a champs moyens 2013 and we extend it in a direct way to the second order derivatives

degenerate fokker planck equations bismut formula gradient estimate and harnack inequality abstract by constructing successful couplings for degenerate diffusion processes explicit derivative formula and harnack type inequalities are presented for solutions to a class of degenerate fokker planck equations on r m r d the main results are also applied to the study of gradient estimate entropy transportation cost inequality and heat kernel inequalities

adaptive residual subsampling methods for radial basis function interpolation and collocation problems we construct a new adaptive algorithm for radial basis functions rbfs method applied to interpolation boundary value and initial boundary value problems with localized features nodes can be added and removed based on residuals evaluated at a finer point set we also adapt the shape parameters of rbfs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix the performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains

nearly holomorphic sections on compact hermitian symmetric spaces abstract let x be a kahler manifold and e be a hermitian vector bundle on x we investigate the space n x e of nearly holomorphic sections in e which generalizes the notion of nearly holomorphic functions introduced by shimura if x u k is a compact hermitian symmetric space and e is u homogeneous it turns out that n x e coincides with the space of u finite vectors in c x e and we obtain new results on the u type decomposition of the hilbert space of square integrable sections as an application we determine this decomposition for the holomorphic tangent bundle of x

quartic gaussian and inverse quartic gaussian radial basis functions the importance of a nonnegative fourier transform we catalogue the numerical properties of approximations using two novel types of radial basis functions f r the qg species is a basis of exponentials of quartic argument f x f r b f x a h j 1 na jexp a h 4 x x j 4 where the x j are the rbf centers and also the interpolation points we show that quartic gaussian rbfs fail at many discrete values of the shape parameter a we show through a detailed analysis that these singularities are directly related to zeros of q k the fourier transform of exp x 4 if we reverse the roles and take q x as the rbf all difficulties disappear because these iqg rbfs have a fourier transform which is nonnegative for all real k we explain that although the quartic gaussian exp x 4 is positive definite in the physics dynamical systems sense of being zero free and nonnegative it lacks the crucial property of being positive definition in the rbf analysis sense

a stable algorithm for flat radial basis functions on a sphere 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 the case of near flat basis functions there will typically also be a separate point at which disastrous 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 nonflat basis functions often is the more accurate of the two methods

fourier embedded domain methods extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and c a simple way to solve a partial differential equation in a non rectangular domain w is to embed the domain in a rectangle b and solve the problem more easily in the rectangle to apply a fourier spectral method on the rectangle it is necessary to extend the inhomogeneous term in the pde f x y to b in such a way that the extended function g x y is periodic and infinitely differentiable and yet is equal to f x y everywhere in the irregular domain w if the boundary of w email protected is defined as the zero isoline of a function f x y then a suitable extension is g x y f x y h email protected x y constant where h is a smoothed approximation to the step function which is defined in the main text

newton iteration with multiquadrics for the solution of nonlinear pdes abstract newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations we show how globally supported multiquadric radial basis functions can be used for this task one of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm some experiments with nash iteration are also included

polynomials and potential theory for gaussian radial basis function interpolation we explore a connection between gaussian radial basis functions and polynomials using standard tools of potential theory we find that these radial functions are susceptible to the runge phenomenon not only in the limit of increasingly flat functions but also in the finite shape parameter case we show that there exist interpolation node distributions that prevent such phenomena and allow stable approximations using polynomials also provides an explicit interpolation formula that avoids the difficulties of inverting interpolation matrices while not imposing restrictions on the shape parameter or number of points

error saturation in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation one restriction is error saturation which occurs for many types of rbfs including gaussian rbfs of the form f x a h exp email protected 2 x h 2 in the limit h 0 for fixed a the error does not converge to zero but rather to e s a previous studies have theoretically determined the saturation error for gaussian rbf on an infinite uniform interval and for the same with a single point omitted the gap enormously increases e s a we show experimentally that the saturation error on the unit interval email protected 1 1 is about 0 06exp 0 47 a 2 email protected huge compared to the o email protected a 2 exp email protected 2 email protected 2 saturation error for a grid with one point omitted we show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points the saturation error can be avoided by choosing email protected 1 the flat limit but the condition number of the interpolation matrix explodes as o exp p 2 email protected 2 the best strategy is to choose the largest a which yields an acceptably small saturation error if the user chooses an error tolerance d then a o p t i m u m d 1 2log d 0 06

the helgason fourier transform for homogeneous vector bundles over compact riemannian symmetric spaces the local theory the helgason fourier transform on noncompact riemannian symmetric spaces g k is generalized to the homogeneous vector bundles over the compact dual spaces u k the scalar theory on u k was considered by sherman the local theory for u k of arbitrary rank and the global theory for u k of rank one in this paper we extend the local theory of sherman to arbitrary homogeneous vector bundles on u k f oru k of rank one we also obtain a generalization of the cartan helgason theorem valid for any k type

six strategies for defeating the runge phenomenon in gaussian radial basis functions on a finite interval radial basis function rbf interpolation is a meshless strategy with great promise for adaptive approximation because it is meshless there is no canonical grid to act as a starting point for function adaptive grid modification uniform grids are therefore common with rbfs like polynomial interpolation which is the limit of rbf interpolation when the rbf functions are very wide the flat limit rbf interpolants are vulnerable to the runge phenomenon whether the grid is uniform or irregular the n point interpolant may diverge as n on the interval spanned by the interpolation points even for a function f x that is analytic everywhere on and near this interval in this work we discuss six strategies for defeating the runge phenomenon specializing to a uniform grid on a finite interval and one particular species of rbfs in one space dimension f x exp a 2 h 2 x 2 where h is the grid spacing three strategies fail but three are successful one good strategy is to vary a with the number of interpolation points n as n 1 4 unfortunately this gives both a subgeometric rate of convergence for the approximation error falling as exp qn for some constant q and a matrix condition number that grows as exp pn for some p 0 in order to explain why fixed a independent of n is not a convergent strategy we digress to discuss error saturation and show experimentally that the saturation error on the finite interval is about 0 6exp 0 47 a 2 f if a user chosen error tolerance d is acceptable then the optimum choice is a o p t i m u m d 1 2log d 0 06 the second good strategy is rbf extension which uses rbf functions with centers on a interval slightly larger than the target interval to approximate f x a third strategy is to split the interval into two boundary layers and a large middle interval and apply separate rbf approximations on each the slow decrease of a strategy is much cheaper but rbf extension is much more resistant to ill conditioning and therefore can achieve much lower errors the three interval method is the least accurate but is robust and inexpensive

bismut formula for lions derivative of distribution dependent sdes and applications abstract by using malliavin calculus bismut type formulas are established for the lions derivative of p t f e f x t where t 0 f is a bounded measurable function and x t solves a distribution dependent sde with initial distribution as applications explicit estimates are derived for the lions derivative and the total variational distance between distributions of solutions with different initial data both degenerate and non degenerate situations are considered due to the lack of the semigroup property and the invalidity of the formula p t f p t f x d x essential difficulties are overcome in the study

smoothing properties of mckean vlasov sdes in this article we develop integration by parts formulae on wiener space for solutions of sdes with general mckean vlasov interaction and uniformly elliptic coefficients these integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of lions they allows us to prove the existence of a classical solution to a related pde with irregular terminal condition we also develop bounds for the derivatives of the density of the solutions of mckean vlasov sdes

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

an analytic approximation to the cardinal functions of gaussian radial basis functions on an infinite lattice gaussian radial basis functions rbfs have been very useful in computer graphics and for numerical solutions of partial differential equations where these rbfs are defined on a grid with uniform spacing h as translates of the master function f x a h exp a 2 h 2 x 2 where a is a user choosable constant unfortunately computing the coefficients of f x jh a h requires solving a linear system with a dense matrix it would be much more efficient to rearrange the basis functions into the equivalent lagrangian or cardinal basis because the interpolation matrix in the new basis is the identity matrix the cardinal basis c j x a h is defined by the set of linear combinations of the gaussians such that c j kh 1 when k j and c j kh 0 for all integers k j we show that the cardinal functions for the uniform grid are c j x h a c x h j a where c x a a 2 p sin px sinh a 2x the relative error is only about 4exp email protected 2 a 2 as demonstrated by the explicit second order approximation it has long been known that the error in a series of gaussian rbfs does not converge to zero for fixed a as h 0 but only to an error saturation proportional to exp email protected 2 a 2 because the error in our approximation to the master cardinal function c x a is the square of the error saturation there is no penalty for using our new approximations to obtain matrix free interpolating rbf approximations to an arbitrary function f x the master cardinal function on a uniform grid in d dimensions is just the direct product of the one dimensional cardinal functions thus in two dimensions c x y a a 4 p 2 sin px sin py sinh a 2x sinh a 2y we show that the matrix free interpolation can be extended to non uniform grids by a smooth change of coordinates

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem