compensation of modelling errors due to unknown domain boundary in electrical impedance tomography electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors this instability is especially severe when absolute imaging is considered with clinical measurements accurate knowledge about the body shape is usually not available and therefore an approximate model domain has to be used in the computational model it has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect in this paper we adapt the so called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape this approach has previously been shown to be applicable to a variety of modeling errors such as coarse discretization in the numerical approximation of the forward model and domain truncation we evaluate the approach with a simulated example of thorax imaging and also with experimental data from a laboratory setting with absolute imaging considered in both cases we show that the related modeling errors can be efficiently compensated for by the approximation error approach we also show that recovery from simultaneous discretization related errors is feasible allowing the use of computationally efficient reduced order models

complete electrode model of electrical impedance tomography approximation properties and characterization of inclusions in electrical impedance tomography one tries to recover the spatial admittance distribution inside a body from boundary measurements in theoretical considerations it is usually assumed that the boundary data consists of the neumann to dirichlet map when conducting real world measurements the obtainable data is a linear finite dimensional operator mapping electrode currents onto electrode potentials in this paper it is shown that when using the complete electrode model to handle electrode measurements the corresponding current to voltage map can be seen as a discrete approximation of the traditional neumann to dirichlet operator this approximating link is utilized further in the special case of constant background conductivity with inhomogeneities it is demonstrated how inclusions with strictly higher or lower conductivities can be characterized by the limit behavior of the range of a boundary operator determined through electrode measurements when the electrodes get infinitely small and cover all

fast calculation of laurent expansions for matrix inverses previously described algorithms for calculating the laurent expansion of the inverse of a matrix valued analytic function become impractical already for singularity orders as low as around p 6 since they require over o 28p matrix multiplications and correspondingly large amounts of memory in place of using mathematically exact recursions we show that for floating point calculations a rational approximation approach can avoid this cost barrier without any significant loss in accuracy

the use of pde centres in the local rbf hermitian method for 3d convective diffusion problems in this work an extension is proposed to the local hermitian interpolation lhi method a meshless numerical method based on interpolation with small and heavily overlapping radial basis function rbf systems this extension to the lhi method uses interpolation functions which themselves satisfy the partial differential equation pde to be solved in this way a much improved reconstruction of partial derivatives can be obtained resulting in significantly improved accuracy in many cases the implementation algorithm is described and is validated via three convection diffusion reaction problems for steady and transient situations a crank nicolson implicit time stepping technique is used for the time dependent problems in the proposed approach a form of analytical upwinding is implicitly implemented by the use of the partial differential operator of the governing equation in the interpolation function which includes the desired information about the convective velocity field the implicit upwinding scheme intrinsic to the proposed numerical approach is tested by solving a one dimensional travelling front problem at peclet numbers of 500 1000 2000 5000 and infinity which corresponds to a shock front in the case of infinity in addition the accuracy of the numerical scheme is validated against a one dimensional steady state solution exhibiting strong boundary layer effects and also against a steady and a transient three dimensional convection diffusion problem on irregular datasets all the test cases are validated against the corresponding analytical solutions finally the effect of various interpolation stencil configurations is investigated and some important limitations on local data centre distribution are identified

b spline level set method for shape reconstruction in electrical impedance tomography a b spline level set bls based method is proposed for shape reconstruction in electrical impedance tomography eit we assume that the conductivity distribution to be reconstructed is piecewise constant transforming the image reconstruction problem into a shape reconstruction problem the shape interface of inclusions is implicitly represented by a level set function lsf which is modeled as a continuous parametric function expressed using b spline functions starting from modeling the conductivity distribution with the b spline based lsf we show that the shape modeling allows us to compute the solution by restricting the minimization problem to the space spanned by the b splines as a consequence the solution to the minimization problem is obtained in terms of the b spline coefficients we illustrate the behavior of this method using simulated as well as water tank data in addition robustness studies considering varying initial guesses differing numbers of control points and modeling errors caused by inhomogeneity are performed both simulation and experimental results show that the bls based approach offers clear improvements in preserving the sharp features of the inclusions in comparison to the recently published parametric level set method

local radial basis function based gridfree scheme for unsteady incompressible viscous flows a local radial basic function rbf based gridfree scheme has been developed to solve unsteady incompressible navier stokes equations in primitive variables the velocity pressure decoupling is obtained by making use of a fractional step algorithm the scheme is validated over a variety of benchmark problems and found a very good agreement with the existing results comparisons with the benchmark solutions show that the developed local rbf gridfree scheme is stable and produces accurate results on domains discretized even with non uniform distribution of nodal points

classification of finite dimensional modules of singly atypical type over the lie superalgebras sl m n we classify the finite dimensional indecomposable sl m n modules with at least a typical or singly atypical primitive weight we do this classification not only for weight modules but also for generalized weight modules we obtain that such a generalized weight module is simply a module obtained by linking subquotient modules of generalized kac modules by applying our results to sl m 1 we have in fact completely classified all finite dimensional sl m 1 modules

utilizing prior information in the estimation of volume fraction distribution for many industrial processes the bulk volume fraction of one phase such as gas or solid can be of considerable importance for this reason it is often controlled within certain limits and therefore is known at least vaguely tomographic imaging of volume fraction can provide very valuable information not only on the bulk volume fraction but also on the spatial distribution of the phase of interest prior information on the bulk volume fraction can assist in tomographic inversion even if this information is not precise in this paper we propose an electrical impedance tomography approach for imaging volume fraction distribution a vague bulk volume fraction prior is utilized in the estimation two dimensional numerical examples are used to demonstrate the proposed method copyright 2005 john wiley sons ltd

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

karhunen loeve approximation of random fields by generalized fast multipole methods kl approximation of a possibly instationary random field a x l2 dp l d subject to prescribed meanfield ea x a x dp and covariance va x x a x ea x a x ea x dp in a polyhedral domain d rd is analyzed we show how for stationary covariances va x x ga x x with ga z analytic outside of z 0 an m term approximate kl expansion am x of a x can be computed in log linear complexity the approach applies in arbitrary domains d and for nonseparable covariances ca it involves galerkin approximation of the kl eigenvalue problem by discontinuous finite elements of degree p 0 on a quasiuniform possibly unstructured mesh of width h in d plus a generalized fast multipole accelerated krylov eigensolver the approximate kl expansion am x of a x has accuracy o exp bm1 d if ga is analytic at z 0 and accuracy o m k d if ga is ck at zero it is obtained in o mn logn b operations where n o h d

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

a weak form rbf generated finite difference method abstract in this paper the idea of direct discretization via radial basis functions rbfs is applied on a local petrov galerkin test space of a partial differential equation pde this results to a weak based rbf generated finite difference rbf fd scheme that possesses some useful properties the error and stability issues are considered when the pde solution or the basis function has low smoothness the new method gives more accurate results than the already well established strong based collocation methods although the method uses a galerkin formulation it still remains meshless because not only the approximation process relies on scattered point layouts but also integrations are done over non connected independent and well shaped subdomains some applications to potential and elasticity problems on scattered data points support the theoretical analysis and show the efficiency of the proposed method

strong and weak error estimates for elliptic partial differential equations with random coefficients we consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous dirichlet boundary conditions we focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness this model is frequently used in hydrogeology we approximate this coefficient by a finite dimensional noise using a truncated karhunen loeve expansion we give estimates of the corresponding error on the solution both a strong error estimate and a weak error estimate that is an estimate of the error commited on the law of the solution we obtain a weak rate of convergence which is twice the strong one in addition we give a complete error estimate for the stochastic collocation method in this case where neither coercivity nor boundedness is stochastically uniform to conclude we apply these results of strong and weak convergence to two classical cases of covariance kernel choices the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel yielding explicit weak and strong convergence rates

generalized linearization techniques in electrical impedance tomography electrical impedance tomography aims at reconstructing the interior electrical conductivity from surface measurements of currents and voltages as the current voltage pairs depend nonlinearly on the conductivity impedance tomography leads to a nonlinear inverse problem often the forward problem is linearized with respect to the conductivity and the resulting linear inverse problem is regarded as a subproblem in an iterative algorithm or as a simple reconstruction method as such in this paper we compare this basic linearization approach to linearizations with respect to the resistivity or the logarithm of the conductivity it is numerically demonstrated that the conductivity linearization often results in compromised accuracy in both forward and inverse computations inspired by these observations we present and analyze a new linearization technique which is based on the logarithm of the neumann to dirichlet operator the method is directly applicable to discrete settings including the complete electrode model we also consider frechet derivatives of the logarithmic operators numerical examples indicate that the proposed method is an accurate way of linearizing the problem of electrical impedance tomography

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

finite element implementation of maxwell s equations for image reconstruction in electrical impedance tomography traditionally image reconstruction in electrical impedance tomography eit has been based on laplace s equation however at high frequencies the coupling between electric and magnetic fields requires solution of the full maxwell equations in this paper a formulation is presented in terms of the maxwell equations expressed in scalar and vector potentials the approach leads to boundary conditions that naturally align with the quantities measured by eit instrumentation a two dimensional implementation for image reconstruction from eit data is realized the effect of frequency on the field distribution is illustrated using the high frequency model and is compared with laplace solutions numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation the results show that scalar vector potential reconstruction produces images which are essentially indistinguishable from a laplace algorithm for frequencies below 1 mhz but superior at frequencies reaching 10 mhz

development and validation of a 3d rbf spectral model for coastal wave simulation abstract with the objective of simulating wave propagation in the nearshore zone for engineering scale applications a two dimensional 2dv model based on the euler zakharov equations 73 54 is extended to three dimensions 3d to maintain the flexibility of the approach with the goal of applying the model to irregularly shaped domains the horizontal plane is discretized with scattered nodes the horizontal derivatives are then estimated using the radial basis function finite difference rbf fd method while a spectral approach is used in the vertical dimension a sensitivity analysis examined the robustness of the rbf fd approach as a function of rbf parameters when estimating the derivatives of a representative function for a targeted stencil size between 20 and 30 nodes piecewise smooth ps polyharmonic spline phs functions are recommended avoiding the use of infinitely smooth is rbfs which are less appropriate for the desired applications because of their dependence on a shape parameter comparisons of simulation results to observations from two wave basin experiments show that nonlinear effects induced by complex bottom bathymetries are reproduced well by the model with the recommended rbf approach validating the use of this method for 3d simulations of wave propagation

n 2 super teichmuller theory abstract based on earlier work of the latter two named authors on the higher super teichmuller space with n 1 a component of the flat o s p 1 2 connections on a punctured surface here we extend to the case n 2 of flat o s p 2 2 connections indeed we construct here coordinates on the higher super teichmuller space of a surface f with at least one puncture associated to the supergroup o s p 2 2 which in particular specializes to give another treatment for n 1 which is simpler than the earlier work the minkowski space in the current case where the corresponding super fuchsian groups act is replaced by the superspace r 2 2 4 and the familiar lambda lengths are extended by odd invariants of triples of special isotropic vectors in r 2 2 4 as well as extra bosonic parameters which we call ratios defining a flat r connection on f as in the pure bosonic or n 1 cases we derive the analogue of ptolemy transformations for all these new variables

sparse tensor discretization of elliptic spdes we propose and analyze sparse deterministic stochastic tensor galerkin finite element methods sparse sgfems for the numerical solution of elliptic partial differential equations pdes with random coefficients in a physical domain d subset mathbb r d in tensor product sgfems the variational solution to the boundary value problem is approximated in tensor product finite element spaces v gamma otimes v d where v gamma and v d denote suitable finite dimensional subspaces of the stochastic and deterministic function spaces respectively these approaches lead to sgfem algorithms of complexity o n gamma n d where n gamma dim v gamma and n d dim v d in this work we use hierarchic sequences v gamma 1 subset v gamma 2 subset ldots and v d 1 subset v d 2 subset ldots of finite dimensional spaces to approximate the law of the random solution the hierarchies of approximation spaces allow us to define sparse tensor product spaces v gamma ell hat otimes v d ell ell 1 2 dots yielding algorithms of o n gamma log n d n d log n gamma work and memory we estimate the convergence rate of sgfem for algebraic decay of the input random field karhunen loeve coefficients we give an algorithm for an input adapted a priori selection of deterministic and stochastic discretization spaces the convergence rate in terms of the total number of degrees of freedom of the proposed method is superior to monte carlo approximations numerical examples illustrate the theoretical results and demonstrate superiority of the sparse tensor product discretization proposed here versus the full tensor product approach

stable computations with flat radial basis functions using vector valued rational approximations one commonly finds in applications of smooth radial basis functions rbfs that scaling the kernels so they are flat leads to smaller discretization errors however the direct numerical approach for computing with flat rbfs rbf direct is severely ill conditioned we present an algorithm for bypassing this ill conditioning that is based on a new method for rational approximation ra of vector valued analytic functions with the property that all components of the vector share the same singularities this new algorithm rbf ra is more accurate robust and easier to implement than the contour pade method which is similarly based on vector valued rational approximation in contrast to the stable rbf qr and rbf ga algorithms which are based on finding a better conditioned base in the same rbf space the new algorithm can be used with any type of smooth radial kernel and it is also applicable to a wider range of tasks including calculating hermite type implicit rbf fd stencils we present a series of numerical experiments demonstrating the effectiveness of this new method for computing rbf interpolants in the flat regime we also demonstrate the flexibility of the method by using it to compute implicit rbf fd formulas in the flat regime and then using these for solving poisson s equation in a 3 d spherical shell

takiff superalgebras and conformal field theory a class of non semisimple extensions of lie superalgebras is studied they are obtained by adjoining to the superalgebra its adjoint representation as an abelian ideal when the superalgebra is of affine kac moody type a generalization of sugawara s construction is shown to give rise to a copy of the virasoro algebra and so presumably to a conformal field theory evidence for this is detailed for the extension of the affinization of the superalgebra its highest weight irreducible modules are classified using spectral flow the irreducible supercharacters are computed and a continuum version of the verlinde formula is verified to give non negative integer structure coefficients interpreting these coefficients as those of the grothendieck ring of fusion partial results on the true fusion ring and its indecomposable structures are deduced

application of stochastic galerkin fem to the complete electrode model of electrical impedance tomography abstract the aim of electrical impedance tomography is to determine the internal conductivity distribution of some physical body from boundary measurements of current and voltage the most accurate forward model for impedance tomography is the complete electrode model which consists of the conductivity equation coupled with boundary conditions that take into account the electrode shapes and the contact resistances at the corresponding interfaces if the reconstruction task of impedance tomography is recast as a bayesian inference problem it is essential to be able to solve the complete electrode model forward problem with the conductivity and the contact resistances treated as a random field and random variables respectively in this work we apply a stochastic galerkin finite element method to the ensuing elliptic stochastic boundary value problem and compare the results with monte carlo simulations

on choosing a radial basis function and a shape parameter when solving a convective pde on a sphere radial basis function rbf approximations have been used for some time to interpolate data on a sphere as well as on many other types of domains their ability to solve to spectral accuracy convection type pdes over a sphere has been demonstrated only very recently in such applications there are two main choices that have to be made i which type of radial function to use and ii what value to choose for their shape parameter denoted by e and with flat basis functions stretched out in the radial direction corresponding to e 0 the recent rbf qr algorithm has made it practical to compute stably also for small values of e results from solving a convective type pde on a sphere are compared here for many choices of radial functions over the complete range of e values from very large down to the limit of e 0 the results are analyzed with a methodology that has similarities to the customary fourier analysis in equispaced 1 d periodic settings in particular we find that high accuracy can be maintained also over very long time integrations we furthermore gain insights into why rbfs sometimes offer higher accuracy than spherical harmonics since the latter arise as an often non optimal special case of the former anticipated future application areas for rbf based methods in spherical geometries include weather and climate modeling

radial basis function partition of unity methods for pricing vanilla basket options meshfree methods based on radial basis function rbf approximation are becoming widely used for solving pde problems they are flexible with respect to the problem geometry and highly accurate a disadvantage of these methods is that the linear system to be solved becomes dense for globally supported rbfs a remedy is to introduce localisation techniques such as partition of unity rbf partition of unity methods rbf pum allow for a significant sparsification of the linear system and lower the computational effort in this work we apply a global rbf method as well as rbf pum to problems in option pricing we consider one and two dimensional vanilla options in order to price american options we employ a penalty approach a penalty term suitable for american multi asset call options has been designed rbf pum is shown to be competitive compared with a finite difference method and a global rbf method it is as accurate as the global rbf method but significantly faster the results for rbf pum look promising for extension to higher dimensional problems

conformal algebras of two dimensional disordered systems we discuss the structure of two dimensional conformal field theories at a central charge c 0 describing critical disordered systems polymers and percolation we construct a novel extension of the c 0 virasoro algebra characterized by a number b measuring the effective number of massless degrees of freedom and by a logarithmic partner of the stress tensor it is argued to be present at a generic random critical point lacking super kac moody or other higher symmetries and is a tool to describe and classify such theories interestingly this algebra is not only consistent with but indeed naturally accommodates in general an underlying global supersymmetry polymers and percolation realize this algebra unexpectedly we find that the c 0 kac table of the degenerate fields contains two distinct theories with b 5 6 and b 5 8 which we conjecture to correspond to percolation and polymers respectively a given kac table field can be degenerate only in one of them remarkably we also find this algebra and thereby an ensuing hidden supersymmetry realized at general replica averaged critical points for which we derive an explicit formula for b

improving the forward solver for the complete electrode model in eit using algebraic multigrid image reconstruction in electrical impedance tomography is an ill posed nonlinear inverse problem linearization techniques are widely used and require the repeated solution of a linear forward problem to account correctly for the presence of electrodes and contact impedances the so called complete electrode model is applied implementing a standard finite element method for this particular forward problem yields a linear system that is symmetric and positive definite and solvable via the conjugate gradient method however preconditioners are essential for efficient convergence preconditioners based on incomplete factorization methods are commonly used but their performance depends on user tuned parameters to avoid this deficiency we apply black box algebraic multigrid using standard commercial and freely available software the suggested solution scheme dramatically reduces the time cost of solving the forward problem numerical results are presented using an anatomically detailed model of the human head

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

tensor triangular geometry for classical lie superalgebras abstract tensor triangular geometry as introduced by balmer 3 is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category in this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the balmer spectrum for the general linear lie superalgebra g g 0 g 1 we construct a zariski space from a detecting subalgebra of g and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional g modules which are semisimple over g 0

on the hp adaptive solution of complete electrode model forward problems of electrical impedance tomography electrical impedance tomography is a noninvasive imaging technique based on measurements of currents and voltages on the boundary of the object of interest the most accurate forward model for impedance tomography is the complete electrode model that takes into account the electrode shapes and the contact impedances at the corresponding interfaces many practical reconstruction algorithms of electrical impedance tomography require repetitive computation of accurate forward solutions for the complete model in this work we introduce an hp adaptive finite element method for solving the complete electrode forward problem and demonstrate its functionality via two dimensional numerical studies

electrical impedance tomography problem with inaccurately known boundary and contact impedances in electrical impedance tomography eit electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body the resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data all the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode skin contact impedances are known a priori however in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances and therefore approximate model domain and contact impedances have to be used in the image reconstruction however it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images leading to loss of diagnostically relevant information in a recent paper kolehmainen 2006 we showed how in the 2 d case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of teichmuller mappings in this paper the theory and reconstruction method are extended to include the estimation of unknown contact impedances the method is implemented numerically and tested with experimental eit data the results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method

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

sobolev type approximation rates for divergence free and curl free rbf interpolants recently error estimates have been made available for divergence free radial basis function rbf interpolants however these results are only valid for functions within the associated reproducing kernel hilbert space rkhs of the matrix valued rbf functions within the associated rkhs also known as the native space of the rbf can be characterized as vector fields having a specific smoothness making the native space quite small in this paper we develop sobolev type error estimates when the target function is less smooth than functions in the native space

dyson maleev representation of nonlinear sigma models for nonlinear sigma models in the unitary symmetry class the nonlinear target space can be parameterized with cubic polynomials this choice of coordinates has been known previously as the dyson maleev parameterization for spin systems and we show that it can be applied to a wide range of sigma models the practical use of this parameterization includes simplification of diagrammatic calculations in perturbative methods and of algebraic manipulations in non perturbative approaches we illustrate the use and specific issues of the dyson maleev parameterization with three examples the keldysh sigma model for time dependent random hamiltonians the supersymmetric sigma model for random matrices and the supersymmetric transfer matrix technique for quasi one dimensional disordered wires we demonstrate that nonlinear sigma models of unitary like symmetry classes c and b d also admit the dyson maleev parameterization

in vivo impedance imaging with total variation regularization we show that electrical impedance tomography eit image reconstruction algorithms with regularization based on the total variation tv functional are suitable for in vivo imaging of physiological data this reconstruction approach helps to preserve discontinuities in reconstructed profiles such as step changes in electrical properties at interorgan boundaries which are typically smoothed by traditional reconstruction algorithms the use of the tv functional for regularization leads to the minimization of a nondifferentiable objective function in the inverse formulation this cannot be efficiently solved with traditional optimization techniques such as the newton method we explore two implementations methods for regularization with the tv functional the lagged diffusivity method and the primal dual interior point method pd ipm first we clarify the implementation details of these algorithms for eit reconstruction next we analyze the performance of these algorithms on noisy simulated data finally we show reconstructed eit images of in vivo data for ventilation and gastric emptying studies in comparison to traditional quadratic regularization tv regulariza tion shows improved ability to reconstruct sharp contrasts

stable computations with gaussian radial basis functions radial basis function rbf approximation is an extremely powerful tool for representing smooth functions in nontrivial geometries since the method is mesh free and can be spectrally accurate a perceived practical obstacle is that the interpolation matrix becomes increasingly ill conditioned as the rbf shape parameter becomes small corresponding to flat rbfs two stable approaches that overcome this problem exist the contour pade method and the rbf qr method however the former is limited to small node sets and the latter has until now been formulated only for the surface of the sphere this paper focuses on an rbf qr formulation for node sets in one two and three dimensions the algorithm is stable for arbitrarily small shape parameters it can be used for thousands of node points in two dimensions and still more in three dimensions a sample matlab code for the two dimensional case is provided

a least squares radial basis function partition of unity method for solving pdes recently collocation based radial basis function rbf partition of unity methods pums for solving partial differential equations have been formulated and investigated numerically and theoretically when combined with stable evaluation methods such as the rbf qr method high order convergence rates can be achieved and sustained under refinement however some numerical issues remain the method is sensitive to the node layout and condition numbers increase with the refinement level here we propose a modified formulation based on least squares approximation we show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling we derive theoretical error estimates both for the collocation and least squares rbf pums numerical experiments are performed for the poisson equation in two and three space dimensions for regular and irregular geometries the convergence experiments confirm the theoretical estimates and the least squares formulation is shown to

reconstruction of permittivity profile from boundary capacitance data we present numerical methods of reconstructing permittivity profile of material flow from data collected in electrical capacitance tomography ect based on a formula for the sensitivity maps from a partial differential equation model we propose a unified framework of iterative algorithms using updated sensitivity maps to properly incorporate the non linear nature of the problem and a regularization mechanism to address the ill posedness for two and three phase flows the total variation tv regularization method is shown to be effective in reconstructing the permittivity profiles accurately in a few iterations without any a priori knowledge of the unknown permittivity values

mixed tensors of the general linear supergroup abstract we describe the image of the canonical tensor functor from deligne s interpolating category re p g l m n to r e p g l m n attached to the standard representation this implies explicit tensor product decompositions between any two projective modules and any two kostant modules of g l m n covering the decomposition between any two irreducible g l m 1 representations we also obtain character and dimension formulas for m n we classify the mixed tensors with non vanishing superdimension for m n we characterize the maximally atypical mixed tensors and show some applications regarding tensor products

on the role of polynomials in rbf fd approximations radial basis function generated finite difference rbf fd approximations generalize classical grid based finite differences fd from lattice based to scattered node layouts this greatly increases the geometric flexibility of the discretizations and makes it easier to carry out local refinement in critical areas many different types of radial functions have been considered in this rbf fd context in this study we find that i polyharmonic splines phs in conjunction with supplementary polynomials provide a very simple way to defeat stagnation also known as saturation error and ii give particularly good accuracy for the tasks of interpolation and derivative approximations without the hassle of determining a shape parameter in follow up studies we will focus on how to best use these hybrid rbf polynomial bases for fd approximations in the contexts of solving elliptic and hyperbolic type pdes

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

rbf loi augmenting radial basis functions rbfs with least orthogonal interpolation loi for solving pdes on surfaces abstract we present a new method for the solution of pdes on manifolds m r d of co dimension one using stable scale free radial basis function rbf interpolation our method involves augmenting polyharmonic spline phs rbfs with polynomials to generate rbf finite difference rbf fd formulas these polynomial basis elements are obtained using the recently developed least orthogonal interpolation technique loi on each rbf fd stencil to obtain local restrictions of polynomials in r 3 to stencils on m the resulting rbf loi method uses cartesian coordinates does not require any intrinsic coordinate systems or projections of points onto tangent planes and our tests illustrate robustness to stagnation errors we show that our method produces high orders of convergence for pdes on the sphere and torus and present some applications to reaction diffusion pdes motivated by biology

a weighted minimum gradient problem with complete electrode model boundary conditions for conductivity imaging we consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes the required interior data can be obtained by means of mri measurements on the boundary we only require knowledge of the electrodes their impedances and the corresponding average input currents from the mathematical point of view this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the complete electrode model cem of somersalo cheney and isaacson siam j appl math 52 1992 pp 1023 1040 we show that this variational problem has nonunique solutions the surprising discovery is that the physical data is still sufficient to determine the geometry of the connected components of the level sets of the minimizers we thus obtain an interesting phase retrieval result knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude we characterize locally the nonuniqueness in the variational problem in two and three dimensions we also show that additional measurements of the voltage potential along a curve joining the electrodes yield unique determination of the conductivity the proofs involve a maximum principle and a new regularity up to the boundary result for the cem boundary conditions a nonlinear algorithm is proposed and implemented to illustrate the theoretical results

superspace conformal field theory conformal sigma models and wess zumino witten wzw models on coset superspaces provide important examples of logarithmic conformal field theories they possess many applications to problems in string and condensed matter theory we review recent results and developments including the general construction of wzw models on type i supergroups the classification of conformal sigma models and their embedding into string theory

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

galerkin finite element approximations of stochastic elliptic partial differential equations we describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous dirichlet boundary conditions here the aim of the com putations is to approximate statistical moments of the solution and in particular we give a priori error estimates for the computation of the expected value of the solution the first method gener ates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard galerkin finite element variational formulation the monte carlo method then uses these approximations to compute corresponding sample averages the sec ond method is based on a finite dimensional approximation of the stochastic coefficients turning the original stochastic problem into a deterministic parametric elliptic problem a galerkin finite element method of either the h o rp version then approximates the corresponding deterministic solution yielding approximations of the desired statistics we present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy this comparison suggests intuitive conditions for an optimal selection of the numerical approximation

indecomposable representations of special linear lie superalgebras abstract the main result of this paper is a classification of finite dimensional representations of the lie superalgebras sl m 1 of supertraceless endomorphisms of the vector superspace of dimension m 1 the classification extends to the so called singly atypical blocks of the lie superalgebras sl m n but there exist wild representation type blocks as soon as bothmandnare 2

super integrable hierarchies associated with spl 2 1 and darboux transformations based on the lie superalgebra spl 2 1 and the tah scheme we generate 1 1 dimensional and 2 1 dimensional super soliton hierarchies associated with spl 2 1 and by using the super trace identity on lie superalgebras we construct the super hamiltonian structures of these new super soliton hierarchies at last we also obtain the darboux transformations of the 1 1 dimensional super soliton hierarchy

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

representation theory of sl 2 1 abstract in this note we present a complete analysis of finite dimensional representations of the lie superalgebra click to view the mathml source this includes in particular the decomposition of all tensor products into their indecomposable building blocks our derivation makes use of a close relation with the representation theory of click to view the mathml source for which analogous results are described and derived keywords lie superalgebras representation theory

a stabilized radial basis finite difference rbf fd method with hybrid kernels abstract recent developments have made it possible to overcome grid based limitations of finite difference fd methods by adopting the kernel based meshless framework using radial basis functions rbfs such an approach provides a meshless implementation and is referred to as the radial basis generated finite difference rbf fd method in this paper we propose a stabilized rbf fd approach with a hybrid kernel generated through a hybridization of the gaussian and cubic rbf this hybrid kernel was found to improve the condition of the system matrix consequently the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard rbf fd methods coupled with present stable algorithms unlike other rbf fd approaches the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity and the size of the stencils as an application we solve the frequency domain acoustic wave equation in a 2d half space in order to suppress spurious reflections from truncated computational boundaries absorbing boundary conditions have been effectively implemented

on moduli of rings and quadrilaterals algorithms and experiments moduli of rings and quadrilaterals are frequently applied in geometric function theory see e g the handbook by kuhnau handbook of complex analysis geometric function theory vols 1 and 2 north holland amsterdam 2005 yet their exact values are known only in a few special cases previously the class of planar domains with polygonal boundary has been studied by many authors from the point of view of numerical computation we present here a new hp fem algorithm for the computation of moduli of rings and quadrilaterals and compare its accuracy and performance with previously known methods such as the schwarz christoffel toolbox of driscoll and trefethen we also demonstrate that the hp fem algorithm applies to the case of nonpolygonal boundary and report results with concrete error bounds

local radial basis function based differential quadrature method and its application to solve two dimensional incompressible navier stokes equations local radial basis function based differential quadrature method is presented in detail in this paper the method is a natural mesh free approach like the conventional differential quadrature dq method it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots which may be distributed randomly however different from the conventional dq method the weighting coefficients in present method are determined by taking the radial basis functions rbfs instead of high order polynomials as the test functions the method works in a similar fashion as conventional finite difference schemes but with truly mesh free property in this paper we mainly concentrate on the multiquadric rbfs since they have exponential convergence the effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied and how the value of optimal c varies with the number of local support knots is also numerically demonstrated the proposed method is validated by its application to the simulation of natural convection in a square cavity excellent numerical results are obtained on an irregular knot distribution

stable evaluation of gaussian radial basis function interpolants we provide a new way to compute and evaluate gaussian radial basis function interpolants in a stable way with a special focus on small values of the shape parameter i e for flat kernels this work is motivated by the fundamental ideas proposed earlier by bengt fornberg and his coworkers however following mercer s theorem an l 2 mathbb r d rho orthonormal expansion of the gaussian kernel allows us to come up with an algorithm that is simpler than the one proposed by fornberg larsson and flyer and that is applicable in arbitrary space dimensions d in addition to obtaining an accurate approximation of the radial basis function interpolant using many terms in the series expansion of the kernel we also propose and investigate a highly accurate least squares approximation based on early truncation of the kernel expansion

hyperviscosity based stabilization for radial basis function finite difference rbf fd discretizations of advection diffusion equations abstract we present a novel hyperviscosity formulation for stabilizing rbf fd discretizations of the advection diffusion equation the amount of hyperviscosity is determined quasi analytically for commonly used explicit implicit and implicit explicit imex time integrators by using a simple 1d semi discrete von neumann analysis the analysis is applied to an analytical model of spurious growth in rbf fd solutions that uses auxiliary differential operators mimicking the undesirable properties of rbf fd differentiation matrices the resulting hyperviscosity formulation is a generalization of existing ones in the literature but is free of any tuning parameters and can be computed efficiently to further improve robustness we introduce a simple new scaling law for polynomial augmented rbf fd that relates the degree of polyharmonic spline phs rbfs to the degree of the appended polynomial when used in a novel ghost node formulation in conjunction with the recently developed overlapped rbf fd method the resulting method is robust and free of stagnation errors we validate the high order convergence rates of our method on 2d and 3d test cases over a wide range of peclet numbers 1 1000 we then use our method to solve a 3d coupled problem motivated by models of platelet aggregation and coagulation again demonstrating high order convergence rates

multifrequency trans admittance scanner mathematical framework and feasibility a trans admittance scanner tas is a device for breast cancer diagnosis based on numerous experimental findings that complex conductivities of breast tumors significantly differ from those of surrounding normal tissues in tas we apply a sinusoidal voltage between a hand held electrode and a scanning probe placed on the breast skin to make current travel through the breast the scanning probe has an array of electrodes at zero voltage we measure exit currents neumann data through the electrodes that provide a map of trans admittance data over the breast surface the inverse problem of tas is to detect a suspicious abnormality underneath the breast skin from the measured neumann data previous anomaly detection methods used the difference between the measured neumann data and a reference neumann data obtained beforehand in the absence of anomaly however in practice the reference data is not available and its computation is not possible since the inhomogeneous complex conductivity of the normal breast is unknown to deal with this problem we propose a frequency difference tas fdtas in which a weighted frequency difference of the trans admittance data measured at a certain moment is used for anomaly detection this paper provides a mathematical framework and the feasibility of fdtas by showing the relationship between the anomaly information and the weighted frequency difference of the neumann data

tikhonov regularization and prior information in electrical impedance tomography the solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method the generalized tikhonov regularization methods have been popular in the solution of many inverse problems the regularization matrices that are usually used with the tikhonov method are more or less ad hoc and the implicit prior assumptions are thus in many cases inappropriate in this paper the authors propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution the approach is based on the construction of an approximating subspace for the expected impedance distributions it is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object on the other hand when the prior is incompatible with the true object the method will still give reasonable estimates

effect of domain shape modeling and measurement errors on the 2 d d bar method for eit the d bar algorithm based on nachman s 2 d global uniqueness proof for the inverse conductivity problem nachman 1996 is implemented on a chest shaped domain the scattering transform is computed on this chest shaped domain using trigonometric and adjacent current patterns and the complete electrode model for the forward problem is computed with the finite element method in order to obtain simulated voltage measurements the robustness and effectiveness of the method is demonstrated on a simulated chest with errors in input currents output voltages electrode placement and domain modeling

a stochastic collocation method for elliptic partial differential equations with random input data in this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms input data of the model the input data are assumed to depend on a finite number of random variables the method consists in a galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials gauss points in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the monte carlo approach it can be seen as a generalization of the stochastic galerkin method proposed in i babuscka r tempone and g e zouraris siam j numer anal 42 2004 pp 800 825 and allows one to treat easily a wider range of situations such as input data that depend nonlinearly on the random variables diffusivity coefficients with unbounded second moments and random variables that are correlated or even unbounded we provide a rigorous convergence analysis and demonstrate exponential convergence of the probability error with respect to the number of gauss points in each direction in the probability space under some regularity assumptions on the random input data numerical examples show the effectiveness of the method

projective electrical impedance reconstruction with two measurements this paper presents a projective reconstruction method of electrical impedance distribution using two boundary measurements of current voltage data the proposed method has a major advantage over existing electrical impedance tomography systems relatively high resolution impedance image in a near surface region underneath the voltage sensing probe that is placed on a skin surface we attach two pairs of electrodes on the border of the probe in such a way that each pair of electrodes generates electrical current flowing underneath the voltage sensing probe and two currents flow across each other a careful analysis of the measured data with respect to local perturbation of admittivity leads to the projective reconstruction method and its rationale numerical simulations show that the proposed method successfully reconstructs high resolution image of local perturbation of admittivity due to object position changes or admittivity value changes or the both in a near surface region underneath the probe

higher level appell functions modular transformations and characters we study modular transformation properties of a class of indefinite theta series involved in characters of infinite dimensional lie superalgebras the level l appell functions open image in new window satisfy open quasiperiodicity relations with additive theta function terms emerging in translating by the period generalizing the well known interpretation of theta functions as sections of line bundles the open image in new window function enters the construction of a section of a rank l 1 bundle open image in new window we evaluate modular transformations of the open image in new window functions and construct the action of an sl 2 subgroup that leaves the section of open image in new window constructed from open image in new window invariant

existence and uniqueness for electrode models for electric current computed tomography the following experiment is considered to a body of given conductivity a number of electrodes are attached through which current is sent on the same electrodes the resulting voltages are measured this experiment can be described by a number of mathematical models k s cheng et al ieee transactions on biomedical engineering 36 1989 pp 918 924 these models are discussed and their predictions compared with experiment in particular a model is exhibited that is capable of predicting the experimentally measured voltages to within 0 1 percent for this model existence and uniqueness of the associated electrical potential is proved