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

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

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

the factorization method applied to the complete electrode model of impedance tomography the factorization method is a tool for recovering inclusions inside a body when the neumann to dirichlet operator which maps applied currents to measured voltages is known in practice this information is never at hand due to the discreteness and physical properties of the measurement devices the complete electrode model of impedance tomography includes these physical characteristics but leads to a finite dimensional data set called the resistivity matrix the main result of this work is an approximation link relating the resistivity matrix to the neumann to dirichlet operator in the l 2 operator norm this result allows us to extend the factorization method to the framework of real life electrode measurements using a regularized series criterion which is easy to implement in practice the truncation index of the sequence criterion which represents the stopping index of the regularization scheme can be computed solely from the measured perturbed and finite dimensional data the functionality of

hermite spectral methods with a time dependent scaling for parabolic equations in unbounded domains hermite spectral methods are investigated for linear diffusion equations and nonlinear convection diffusion equations in unbounded domains when the solution domain is unbounded the diffusion operator no longer has a compact resolvent which makes the hermite spectral methods unstable to overcome this difficulty a time dependent scaling factor is employed in the hermite expansions which yields a positive bilinear form as a consequence stability and spectral convergence can be established for this approach the present method plays a similar role in the stability of the similarity transformation technique proposed by funaro and kavian math comp 57 1991 pp 597 619 however since coordinate transformations are not required the present approach is more efficient and is easier to implement in fact with the time dependent scaling the resulting discretization system is of the same form as that associated with the classical straightforward but unstable hermite spectral method numerical experiments are carried out to support the theoretical stability and convergence results

a spectral viscosity method based on hermite functions for nonlinear conservation laws we consider the approximation by a spectral method of the solution of the cauchy problem for a scalar conservation law in one dimension posed in the whole real line we analyze a spectral viscosity method in which the orthogonal basis considered is the one of hermite functions we prove the convergence of the approximate solution to the unique entropy solution of the problem by using compensated compactness arguments

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

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

vlasov simulations using velocity scaled hermite representations the efficiency accuracy and stability of two different pseudo spectral methods using scaled hermite basis and weight functions applied to the nonlinear vlasov poisson equations in one dimension 1d 1v are explored and compared a variable velocity scaleuis introduced into the hermite basis and is shown to yield orders of magnitude reduction in errors as compared to linear kinetic theory with no increase in workload a set of fourier hermite coefficients representing a periodic gaussian distribution function are advanced through time with ano t2 accurate splitting method within this splitting scheme the advection and acceleration terms of the vlasov equation are solved separately using ano t4 accurate runge kutta method the asymmetrically weighted aw hermite basis which has been used previously by many authors conserves particles and momentum exactly and total energy too t3 however the aw hermite method doesnotconserve the square integral of the distribution and is in fact numerically unstable the symmetrically weighted sw hermite algorithm applied here to the vlasov system for the first time can either conserve particles and energy fornueven or momentum fornuodd as t 0 wherenuis the largest hermite mode number the sw hermite method conserves the square integral of the distribution and therefore remains numerically stable in addition careful velocity scaling improves the conservation properties of the sw hermite method damping and growth rates oscillation frequencies e field saturation levels and phase space evolution are seen to be qualitatively correct during simulations relative errors with respect to linear landau damping and linear bump on tail instability are shown to be less than 1 using only 64 velocity scaled hermite functions comparisons to particle in cell pic simulations show that as the number of particles increases to more than 106 the pic solutions converge to scaled sw hermite solutions that were found in only 1 10 of the run time the sw hermite method with velocity scaling is well suited to kinetic simulations of warm plasmas

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

optimal error estimates of the legendre petrov galerkin method for the korteweg de vries equation in this paper the legendre petrov galerkin method for the korteweg de vries equation with nonperiodic boundary conditions is analyzed the nonlinear term is computed with the legendre spectral method and some pseudospectral methods respectively optimal error estimates in l2 norm are obtained for both semidiscrete and fully discrete schemes the method is also applicable to some 2m 1 th order differential equations

resolution and stability analysis of an inverse problem in electrical impedance tomography dependence on the input current patterns electrical impedance tomography is a procedure by which one finds the conductivity distribution inside a domain from measurements of voltages and currents at the boundary this work addresses the i

the eigenvalues of hermite and rational spectral differentiation matrices we derive expressions for the eigenvalues of spectral differentiation matrices for unbounded domains in particular we consider galerkin and collocation methods based on hermite functions as well as rational functions a fourier series combined with a cotangent mapping we show that i first derivative matrices have purely imaginary eigenvalues and second derivative matrices have real and negative eigenvalues ii for the hermite method the eigenvalues are determined by the roots of the hermite polynomials and for the rational method they are determined by the laguerre polynomials and iii the hermite method has attractive stability properties in the sense of small condition numbers and spectral radii

time dependent hermite galerkin spectral method and its applications a time dependent hermite galerkin spectral method thgsm is investigated in this paper for the nonlinear convection diffusion equations in the unbounded domains the time dependent scaling factor and translating factor are introduced in the definition of the generalized hermite functions ghf as a consequence the thgsm based on these ghf has many advantages not only in theoretical proofs but also in numerical implementations the stability and spectral convergence of our proposed method have been established in this paper the korteweg de vries burgers kdvb equation and its special cases including the heat equation and the burgers equation as the examples have been numerically solved by our method the numerical results are presented and it surpasses the existing methods in accuracy our theoretical proof of the spectral convergence has been supported by the numerical results

pseudospectral method for non isotropic heat transfer using mixed hermite legendre interpolation abstract in this paper we develop a new mixed pseudospectral method for heat transfer by using generalized hermite functions and legendre polynomials in unbounded domains fundamental to spectral methods for various unbounded domain problems we establish some basic results on the mixed generalized hermite legendre interpolation as an example a new mixed generalized hermite legendre pseudospectral scheme is provided for non isotropic heat transfer its convergence is proved numerical results demonstrate the spectral accuracy of this approach

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

error estimation of hermite spectral method for nonlinear partial differential equations hermite approximation is investigated some inverse inequalities imbedding inequalities and approximation results are obtained a hermite spectral scheme is constructed for burgers equation the stability and convergence of the proposed scheme are proved strictly the techniques used in this paper are also applicable to other nonlinear problems in unbounded domains

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

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

a generalized laguerre fourier hermite pseudospectral method for computing the dynamics of rotating bose einstein condensates a time splitting generalized laguerre fourier hermite pseudospectral method is proposed for computing the dynamics of rotating bose einstein condensates becs in two and three dimensions the new numerical method is based on the following i the use of a time splitting technique for decoupling the nonlinearity ii the adoption of polar coordinates in two dimensions and cylindrical coordinates in three dimensions such that the angular rotation term becomes constant coefficient and iii the construction of eigenfunctions for the linear operator by properly scaling the generalized laguerre fourier and hermite functions the new method enjoys the following properties i it is explicit time reversible and time transverse invariant ii it conserves the position density and is spectrally accurate in space and second order or fourth order accurate in time and iii it solves the problem in the original whole space instead of in a truncated artificial computational domain the method is also extend

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

a laplace method for under determined bayesian optimal experimental designs abstract in long et al 2013 a new method based on the laplace approximation was developed to accelerate the estimation of the post experimental expected information gains kullback leibler divergence in model parameters and predictive quantities of interest in the bayesian framework a closed form asymptotic approximation of the inner integral and the order of the corresponding dominant error term were obtained in the cases where the parameters are determined by the experiment in this work we extend that method to the general case where the model parameters cannot be determined completely by the data from the proposed experiments we carry out the laplace approximations in the directions orthogonal to the null space of the jacobian matrix of the data model with respect to the parameters so that the information gain can be reduced to an integration against the marginal density of the transformed parameters that are not determined by the experiments furthermore the expected information gain can be approximated by an integration over the prior where the integrand is a function of the posterior covariance matrix projected over the aforementioned orthogonal directions to deal with the issue of dimensionality in a complex problem we use either monte carlo sampling or sparse quadratures for the integration over the prior probability density function depending on the regularity of the integrand function we demonstrate the accuracy efficiency and robustness of the proposed method via several nonlinear under determined test cases they include the designs of the scalar parameter in a one dimensional cubic polynomial function with two unidentifiable parameters forming a linear manifold and the boundary source locations for impedance tomography in a square domain where the unknown parameter is the conductivity which is represented as a random field

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

approximation of some diffusion evolution equations in unbounded domains by hermite functions spectral and pseudospectral approximations of the heat equation are analyzed the solution is represented in a suitable basis constructed with hermite polynomials stability and convergence estimates are given and numerical tests are discussed

the use of generalized laguerre polynomials in spectral methods for nonlinear differential equations the expansion of products of generalized laguerre polynomials l n x in terms of a series of generalized laguerre polynomials is considered the expansion coefficients which are equal to triple product integrals of generalized laguerre polynomials are expressed in terms of a three index recurrence relation this is reduced to a one index relation which facilitates computation of the expansion coefficients the results are useful in the solution of nonlinear differential equations when it is desired to express products of generalized laguerre polynomials as a linear series of these functions as an application we use the results to compute a spectral solution of a nonlinear boundary value problem namely the blasius equation on a semi infinite interval by using a truncated series containing the first eight polynomials l1 2n x a solution is obtained within 4 accuracy

stable and efficient spectral methods in unbounded domains using laguerre functions stable and efficient spectral methods using laguerre functions are proposed and analyzed for model elliptic equations on regular unbounded domains it is shown that spectral galerkin approximations based on laguerre functions are stable and convergent with spectral accuracy in the usual not weighted sobolev spaces efficient accurate and well conditioned algorithms using laguerre functions are developed and implemented numerical results indicating the spectral convergence rate and effectiveness of these algorithms are presented

a legendre petrov galerkin and chebyshev collocation method for third order differential equations a legendre petrov galerkin lpg method for the third order differential equation is developed by choosing appropriate base functions the method can be implemented efficiently also this new approach enables us to derive an optimal rate of convergence in l2 norm the method is applied to some nonlinear problems such as the korteweg de vries kdv equation with the chebyshev collocation treatment for the nonlinear term it is a legendre petrov galerkin and chebyshev collocation lpg cc method numerical experiments are given to confirm the theoretical result

the numerical computation of connecting orbits in dynamical systems a rational spectral approach abstract structural changes in dynamical systems are often related to the appearance or disappearance of orbits connecting two stationary points either heteroclinic or homoclinic to compute the connecting orbits the boundary value problem which is posed on the real line is frequently replaced by one on a finite interval then the problem is solved on the finite interval using appropriate numerical methods in this work we use a rational spectral approach to compute the connecting orbits this method avoids truncating the problem to a finite interval and produces very accurate numerical solutions with a fairly small number of computational points numerical examples indicate that the method compares favorably with the existing ones

spectral methods using rational basis functions on an infinite interval by using the map y l cot t where l is a constant differential equations on the interval y can be transformed into t 0 and solved by an ordinary fourier series in this article earlier work by grosch and orszag j comput phys 25 273 1977 cain ferziger and reynolds j comput phys 56 272 1984 and boyd j comput phys 25 43 1982 57 454 1985 siam j numer anal 1987 is extended in several ways first the series of orthogonal rational functions converge on the exterior of bipolar coordinate surfaces in the complex y plane second galerkin s method will convert differential equations with polynomial or rational coefficients into banded matrix problems third with orthogonal rational functions it is possible to obtain exponential convergence even for u y that asymptote to a constant although this behavior would wreck alternatives such as hermite or sinc expansions fourth boundary conditions are usually natural rather than essential in the sense that the singularities of the differential equation will force the numerical solution to have the correct behavior at infinity even if no constraints are imposed on the basis functions fifth mapping a finite interval to an infinite one and then applying the rational chebyshev functions gives an exponentially convergent method for functions with bounded endpoint singularities these concepts are illustrated by five numerical examples

laguerre galerkin method for nonlinear partial differential equations on a semi infinite interval summary a laguerre galerkin method is proposed and analyzed for the burgers equation and benjamin bona mahony bbm equation on a semi infinite interval by reformulating these equations with suitable functional transforms it is shown that the laguerre galerkin approximations are convergent on a semi infinite interval with spectral accuracy an efficient and accurate algorithm based on the laguerre galerkin approximations to the transformed equations is developed and implemented numerical results indicating the high accuracy and effectiveness of this algorithm are presented

statistical estimation of resistance conductance by electrical impedance tomography measurements this paper is built upon the assumption that in electrical impedance tomography vectors of voltages and currents are linearly dependent through a resistance matrix this linear relationship was confirmed experimentally and may be derived analytically under certain assumptions regarding electrodes isaacson 1991 given measurement data consisting of voltages and currents we treat this relationship as a linear statistical model thus our goal is not to reconstruct the image but directly estimate its electromagnetic properties reflected in the resistance and or conductance matrix using electrical impedance tomography eit measurements of voltages and currents on the periphery of the body since no inverse problem is involved the algorithm for estimation merely reduces to one matrix inversion we estimate the impedance resistance matrix using well established statistical inference techniques for linear regression models we provide a comprehensive treatment for a two dimensional homogeneous body of a circular shape by which many concepts of electrical impedance tomography such as width of electrodes the difference between voltage current and current voltage systems are illustrated our theory may be applied to various tests including eit hardware calibration and whether the medium is homogeneous these tests are illustrated on phantom agar data

generalized hermite spectral method and its applications to problems in unbounded domains in this paper we develop a spectral method based on generalized hermite functions with weight chi x equiv1 we also establish some basic results on generalized hermite orthogonal approximations which play an important role in spectral methods as examples the generalized ginzburg landau equation in a population problem and an elliptic equation with a harmonic potential are considered related spectral schemes are proposed and their convergence is proved numerical results demonstrate the spectral accuracy of this approach

hermite approximation of a hyperbolic fokker planck optimality system to control a piecewise deterministic process abstractthe hermite spectral approximation of a hyperbolic fokker planck fp optimality system arising in the control of an unbounded piecewise deterministic process pdp is discussed to control the probability density function pdf corresponding to the pdp process an optimal control based on an fp strategy is considered the resulting optimality system consists of a hyperbolic system with opposite time orientation and an integral optimality condition equation a hermite spectral discretisation is investigated to approximate solutions to the optimality system in unbounded domains it is proven that the proposed scheme satisfies the conservativity requirement of the pdfs the spectral convergence rate of the discretisation scheme is proved and validated by numerical experiments

a generalized laguerre hermite pseudospectral method for computing symmetric and central vortex states in bose einstein condensates a generalized laguerre hermite pseudospectral method is proposed for computing symmetric and central vortex states in bose einstein condensates becs in three dimensions with cylindrical symmetry the new method is based on the properly scaled generalized laguerre hermite functions and a normalized gradient flow it enjoys three important advantages i it reduces a three dimensional 3d problem with cylindrical symmetry into an effective two dimensional 2d problem ii it solves the problem in the whole space instead of in a truncated artificial computational domain and iii it is spectrally accurate extensive numerical results for computing symmetric and central vortex states in becs are presented for one dimensional 1d bec 2d bec with radial symmetry and 3d bec with cylindrical symmetry

the hermite spectral method for gaussian type functions although hermite functions were widely used for many practical problems numerical experiments with the standard normalized hermite functions psi n v worked poorly in the sense that too many hermite functions are required to solve differential equations in order to obtain accurate numerical solutions it is necessary to choose a scaling factor alpha and use psi n alpha v as the basis functions in this paper the scaling factors are given for functions that are of gaussian type which have finite supports m m the scaling factor used is max 0 leqq j leqq n gamma j m where gamma j j 0 n are the roots of psi n 1 v and n 1 is the number of the truncated terms used the numerical results show that after using this scaling factor only reasonable numbers of the hermite functions are required to solve differential equations

generalized hermite spectral method matching asymptotic behaviors in this paper we propose the generalized hermite spectral method by using a family of new generalized hermite functions which are mutually orthogonal with the weight function 1 x 2 c c being an arbitrary real number we establish the basic results on the corresponding orthogonal approximation and interpolation which simulate the asymptotic behaviors of approximated functions at infinity reasonably as examples of applications the spectral schemes are provided for two model problems numerical results demonstrate their spectral accuracy in space

semi orthogonal wavelet packet bases for parallel least squares approximation a generalization to the wavelet packet case of the semi orthogonal wavelets given by c k chui is presented and a simple numerical algorithm for their practical evaluation is developed to provide a motivation it is shown how using these new bases the least squares approximation problem in a multiresolution space can easily be solved by means of an efficient parallel algorithm

combined hermite spectral finite difference method for the fokker planck equation the convergence of a class of combined spectral finite difference methods using hermite basis applied to the fokker planck equation is studied it is shown that the hermite based spectral methods are convergent with spectral accuracy in weighted sobolev space numerical results indicating the spectral convergence rate are presented a velocity scaling factor is used in the hermite basis and is shown to improve the accuracy and effectiveness of the hermite spectral approximation with no increase in workload some basic analysis for the selection of the scaling factors is also presented

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

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

analysis of the spectral vanishing viscosity method for periodic conservation laws the convergence of the spectral vanishing method for both the spectral and pseudospectral discretizations of the inviscid burgers equation is analyzed it is proved that this kind of vanishing viscosity is responsible for spectral decay of those fourier coefficients located toward the end of the computed spectrum consequently the discretization error is shown to be spectrally small independently of whether or not the underlying solution is smooth this in turn implies that the numerical solution remains uniformly bounded and convergence follows by compensated compactness arguments

convergence and regularization for monotonicity based shape reconstruction in electrical impedance tomography the inverse problem of electrical impedance tomography is severely ill posed meaning that only limited information about the conductivity can in practice be recovered from boundary measurements of electric current and voltage recently it was shown that a simple monotonicity property of the related neumann to dirichlet map can be used to characterize shapes of inhomogeneities in a known background conductivity in this paper we formulate a monotonicity based shape reconstruction scheme that applies to approximative measurement models and regularizes against noise and modelling error we demonstrate that for admissible choices of regularization parameters the inhomogeneities are detected and under reasonable assumptions asymptotically exactly characterized moreover we rigorously associate this result with the complete electrode model and describe how a computationally cheap monotonicity based reconstruction algorithm can be implemented numerical reconstructions from both simulated and real life measurement data are presented

electrical conductivity reconstruction using nonlocal boundary conditions in this paper we study existence uniqueness and solution estimates to the mixed problem nabla cdot sigma nabla u 0 with dirichlet to neumann map boundary conditions and neumann boundary conditions we then show how this can be used in the reconstruction of sigma given the relationship between u and its normal derivative on the boundary portion where we do not apply the dirichlet to neumann map a numerical reconstruction scheme is also derived