numerical solution to a sideways parabolic equation the inverse heat conduction problem can be considered to be a sideways parabolic equation in the quarter plane this is a model of a problem where one wants to determine the temperature on both sides of a thick wall but one side is inaccessible to measurements a numerical procedure for this severely ill posed problem is suggested which consists of two steps namely a mollification of the data and a marching difference scheme the numerical method is proved to be stable several computational results are presented and discussed copyright 2001 john wiley sons ltd

l infty error bounds in partial deconvolution of the inverse gaussian pulse when a c infty approximation to the dirac 5 function in the form of an inverse gaussian pulse is used as input into a linear time invariant system the output waveform is an approximation to that system s green s function in which the singularities have been smoothed out the ill posed deconvolution problem for the output signal aims at reconstructing these singularities by exploiting the smoothing properties of the inverse gaussian kernel we prove that partial deconvolution of the output waveform given l 2 a priori bounds on the data noise and the unknown green s function results in l infty error bounds for the regularized solution and its derivatives consequently when the l 2 norm of the output noise is sufficiently small partial deconvolution is a pointwise reliable c infty function which in turn approximates the desired green s function in many applications

identification of dynamic green s functions in structural networks this paper presents the mathematical and computational basis of a method for experimentally identifying the dynamic behavior of linear structural systems the method consists of exciting the structure with specific pulses and reconstructing the dynamic green s functions by deconvolution of the measured response the reconstruction procedure involves the solution of an ill posed integral equation in the presence of noise inverse gaussian functions which have mathematical properties that render the deconvolution problem tractable are used as probing pulses two examples are presented for which dynamic green s functions are reconstructed from synthetic response data 1 a nondispersive medium consisting of a mechanical slider with free body motion and 2 a network that may be representative of a large orbiting structure with rigidly connected torsional members and timoshenko beams it is shown that residual effects of noise that may be detectable in the reconstructed green s functions are effectively smoothed out when these functions are used to predict the response to forces such as may be induced by control systems finally an example is presented of deconvolution implemented in slow motion

infinitely divisible pulses continuous deconvolution and the characterization of linear time invariant systems this paper addresses the problem of determining the impulse response of a linear time invariant system by probing the system with a causal c infty approximation to the dirac delta function we analyze the ill posed deconvolution problem which results from a wide choice of possibly multimodal infinitely divisible probe pulses the notion of infinite divisibility is shown to play a key role when the system s response is suspected of having nondifferentiable singularities we reformulate the volterra integral equation as a cauchy problem for a linear partial differential equation in two independent variables and introduce the concepts of partial and continuous deconvolution we then show that partial deconvolution of the output waveform results in l infty error bounds for the regularized solution and its derivatives under l 2 a priori bounds on the data noise and the unknown system response using the poisson summation formula and fft algorithms we construct an efficient computational alg

impulse response acquisition as an inverse heat conduction problem the impulse response of a linear time invariant system can be obtained by deconvolution of the response to a physically realizable smooth approximation to the dirac delta function as shown in siam j app math 47 1987 pp 892 927 it is advantageous to synthesize infinitely divisible probe waveforms and recover the system s impulse response by solving the related cauchy problem for a generalized diffusion equation the present paper is concerned with perturbations of the originally designed pulse shape caused by amplifiers transducers and other interfacing devices that may destroy infinite divisibility it is shown that in a large number of cases a simple linear transformation of the noisy output data can be found that converts the deconvolution problem with the perturbed pulse back into one for which the diffusion theory applies the search for this transformation is accomplished in the fourier domain by comparing the perturbed pulse with the designed pulse as a by product it is shown

quadrature of smooth stochastic processes the integral of a stochastic process is estimated by means of classical quadrature formulae in contrast to certain conventional methods knowledge of covariances is not required and no regularity conditions are assumed explicit error representations and error bounds with respect to thel p norm are established

average case complexity of multivariate integration we study the average case complexity of multivariate integration for the class of continuous functions of d variables equipped with the classical wiener sheet measure to derive the average case complexity one needs to obtain optimal sample points we prove that optimal design is closely related to discrepancy theory which has been extensively studied for many years this relation enables us to show that optimal sample points can be derived from hammersley points extending the result of roth and using the recent result of wasilkowski we conclude that the average case complexity is e 1 lne 1 d 1 2

spline smoothing and optimal rates of convergence in nonparametric regression models linear estimation is considered in nonparametric regression models of the form yi f xi ei xi e a b where the zero mean errors are uncorrelated with common variance a2 and the response function f is assumed only to have a bounded square integrable qth derivative the linear estimator which minimizes the maximum mean squared error summed over the observation points is derived and the exact minimax rate of convergence is obtained for practical problems where bounds on 11 f2q 11 and a2 may be unknown generalized cross validation is shown to give an adaptive estimator which achieves the minimax optimal rate under the additional assumption of normality 1 the model consider the nonparametric regression model

average case complexity of linear multivariate problems we study the average case complexity of a linear multivariate problem lmp defined on functions of d variables we consider two classes of information the first std consists of function values and the second all of all continuous linear functionals tractability of lmp means that the average case complexity is o 1 e p with p independent of d we prove that tractability of an lmp in std is equivalent to tractability in all although the proof is no constructive we provide a simple condition to check tractability in all we also address the optimal design problem for an lmp by using a relation to the worst case setting we find the order of the average case complexity and optimal sample points for multivariate function approximation

a mollification method for a noncharacteristic cauchy problem for a parabolic equation abstract in this paper the noncharacteristic cauchy problem ncp formula is considered with lp r p 1 it is proved that a solution of ncp exists if and only if is infinitely differentiable and n lp r c 2n s2n n n for certain constantscands ncp is well known to be severely ill posed a small perturbation in the cauchy data may cause a dramatically large error in the solution the following mollification method is suggested for solving ncp in a stable way if lp r is given inexactly by lp r then we mollify by convolutions with the dirichlet kernel and the de la vallee poussin kernel the exact solution of ncp is approximated by the solution of the mollified problem with a reasonable choice of mollification parameters which yields error estimates of the holder type by the method we can work with the data inlp r p 1 and obtain several sharp stability estimates inlp andl norms of the holder type for the solution of the problem the method can easily be implemented numerically using the fast fourier transform a stable marching difference scheme based on this method is suggested several numerical examples are given which show that the method is effective

quantum scr w algebras and elliptic algebras we define a quantum open image in new window algebra associated to mathfrak s mathfrak l n as an associative algebra depending on two parameters for special values of the parameters this algebra becomes the ordinary open image in new window algebra of mathfrak s mathfrak l n or theq deformed classical open image in new window algebra algebra of mathfrak s mathfrak l n we construct free field realizations of the quantum open image in new window algebra and the screening currents we also point out some interesting elliptic structures arising in these algebras in particular we show that the screening currents satisfy elliptic analogues of the drinfeld relations in open image in new window

cubature of random fields by product type integration rules abstract cubature of random functions with multivariate domains by means of product type integration rules is examined bounds for the approximation error with respect to the l p norm and in terms of the levy distance are established and an additional error bound involving the spectral measure is derived for homogeneous random fields

mathcal w algebras and surface operators in mathcal n 2 gauge theories a general class of algebras can be constructed from the affine sl n algebra by quantum drinfeld sokolov reduction and are classified by partitions of n surface operators in an su n 4d gauge theory are also classified by partitions of n we argue that instanton partition functions of gauge theories in the presence of a surface operator can also be computed from the corresponding algebra we test this proposal by analysing the polyakov bershadsky algebra obtaining results that are in agreement with the known partition functions for su 3 gauge theories with a so called simple surface operator as a byproduct our proposal implies relations between and algebras

probe waveforms and deconvolution in the experimental determination of elastic green s functions we propose a new time domain method for the experimental determination of the impulse response of linear systems the technique centers around the use of specifically designed probe waveforms these waveforms are particular c infty approximations to the dirac delta function and the heaviside unit step function and lead to a subsequent time domain deconvolution problem which can be implemented as a cauchy initial value problem this approach allows for continuous deconvolution a powerful option in the presence of noise we orient the discussion to the context of acoustic emission and elastic green s functions and present several numerical reconstructions of sharp signals from smooth synthetic data

a mollification method for ill posed problems a mollification method for a class of ill posed problems is suggested the idea of the method is very simple and natural if the data are given inexactly then we try to find a sequence of mollification operators which map the improper data into well posedness classes of the problem mollify the improper data within these mollified data our problem becomes well posed and when these facts are in hand we try to obtain error estimates and optimal or quasi optimal mollification parameters the method is working not only for problems in hilbert spaces but also for problems in banach spaces applications of the method to concrete problems like numerical differentiation parabolic equations backwards in time the cauchy problem for the laplace equation one and multidimensional non characteristic cauchy problems for parabolic equations in infinite or finite domains give us very sharp stability estimates of h older continuous type in these cases the method is optimal in the sense that it gives the same order of h older continuous dependence on the data as for the regularized problems furthermore the method may be implemented numerically using fast fourier transforms for the first time a uniform stability estimate of h older continuous type of the solution of the heat equation backwards in time in the space l p bbb r for all p in 1 infty could be established by our mollification method a new simple sharp pointwise estimate of h older type for the weak solution of a non characteristic cauchy problem for parabolic equations in a finite domain is established

distribution of event times in time resolved fluorescence the exponential series approach algorithm regularization analysis abstract in time resolved fluorescence spectroscopy a distribution of fluorescence lifetimes resulting from static and dynamic disorder of a polychromophoric ensemble is to be determined from the molecular fluorescence response to the optical probe pulse to do this one has to solve a convolution integral equation of the first kind and then invert a laplace transform both problems are ill posed in the sense of hadamard we describe in detail an algorithm that combines coarse discretization for inverting the laplace transform with a nonlinear least squares approach based on newton and quasi newton techniques for solving the convolution equation while this algorithm works well in many cases it does not completely remove the instabilities due to the ill posedness thus we also propose an algorithm that combines the approach described above with tikhonov regularization several examples both with synthetic and with real data show the performance of our algorithms

integration and approximation of multivariate functions average case complexity with isotropic wiener measure we study the average case complexity of multivariate integration and l 2 function approximation for the class f c 0 1 d of continuous functions of d variables the class f is endowed with the isotropic wiener measure brownian motion in levy s sense furthermore for both problems only function values are used as data

asymptotic optimality of regular sequence designs we study linear estimators for the weighted integral of a stochastic process the process may only be observed on a finite sampling design the error is defined in a mean square sense and the process is assumed to satisfy sacks ylvisaker regularity conditions of order r n o we show that sampling at the quantiles of a particular density already yields asymptotically optimal estimators hereby we extend the results of sacks and ylvisaker for regularity r o or 1 and we confirm a conjecture by eubank smith and smith

uniqueness and eventual uniqueness of optimal designs in some time series models abstract earlier results on the uniqueness and eventual uniqueness of optimal designs for certain time series models are extended to a wider class of processes which includes those with covariance structures such as that of multiple integrals of brownian motion and brownian bridge processes the relationship between the problems of regression design for time series and piecewise polynomial approximation with free breakpoints is discussed and consequently asymptotic results obtained by sacks and ylvisaker 1970 are seen to hold under weaker assumptions for these processes

a numerical method for the solution of two dimensional inverse heat conduction problems a numerical method for the solution of inverse heat conduction problems in two dimensional rectangular domains is established and its performance is demonstrated by computational results the present method extends beck s8 method to two spatial dimensions and also utilizes future times in order to stabilize the ill posedness of the underlying problems the approach relies on a line approximation of the elliptic part of the parabolic differential equation leading to a system of one dimensional problems which can be decoupled