stability estimate for the aligned magnetic field in a periodic quantum waveguide from dirichlet to neumann map in this article we study the boundary inverse problem of determining the aligned magnetic field appearing in the magnetic schrodinger equation in a periodic quantum cylindrical waveguide by knowledge of the dirichlet to neumann map we prove a holder stability estimate with respect to the dirichlet to neumann map by means of the geometrical optics solutions of the magnetic schrodinger equation

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

a unified formulation of gaussian versus sparse stochastic processes part i continuous domain theory we introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous time stochastic processes the cornerstone of our approach is an innovation model that is driven by some generalized white noise process which may be gaussian or not e g laplace impulsive poisson or alpha stable this allows for a conceptual decoupling between the correlation properties of the process which are imposed by the whitening operator l and its sparsity pattern which is determined by the type of noise excitation the latter is fully specified by a levy measure we show that the range of admissible innovation behavior varies between the purely gaussian and super sparse extremes we prove that the corresponding generalized stochastic processes are well defined mathematically provided that the adjoint inverse of the whitening operator satisfies some lp bound for p 1 we present a novel operator based method that yields an explicit characterization of all levy driven processes that are solutions of constant coefficient stochastic differential equations when the underlying system is stable we recover the family of stationary continuous time autoregressive moving average processes carma including the gaussian ones the approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the levy processes which are sparse and non stationary finally we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform domain statistics

an ϵ uniform hybrid scheme for singularly perturbed delay differential equations in this paper a singularly perturbed delay differential equation of first order has been considered the problem is solved by using a hybrid scheme on a shishkin mesh the difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter truncation errors are obtained finally numerical experiments are carried out on a test problem confirming the effectiveness of the proposed technique

a parameter uniform numerical method for a system of singularly perturbed ordinary differential equations in this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered a parameter robust computational method is constructed and it is proved that it gives essentially first order parameter uniform convergence in the maximum norm numerical results are presented in support of the theory

convergence analysis of fitted numerical method for a singularly perturbed nonlinear volterra integro differential equation with delay abstract in this paper the initial value problem for a quasilinear singularly perturbed delay volterra integro differential equation was considered by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder term in integral form a fitted difference scheme is constructed and analysed it is shown that the method displays first order uniform convergence in perturbation parameter some numerical results are given to confirm the theoretical analysis

optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data abstract we consider the inverse boundary value problem for operators of the form q in an infinite domain r r 1 n n 3 with a periodic potential q for dirichlet to neumann data localized on a portion of the boundary of the form 1 r 1 with 1 being the complement either of a flat or spherical portion of we prove that a log type stability estimate holds

stability estimate for an inverse problem for the magnetic schrodinger equation from the dirichlet to neumann map abstract we consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic schrodinger equation in a bounded smooth domain of r n with input dirichlet data from measured neumann boundary observations this information is enclosed in the dynamical dirichlet to neumann map associated to the solutions of the magnetic schrodinger equation we prove in dimension n 2 that the knowledge of the dirichlet to neumann map for the magnetic schrodinger equation measured on the boundary determines uniquely the magnetic field and we prove a holder type stability in determining the magnetic field induced by the magnetic potential

holder stable determination of a quantum scalar potential in unbounded cylindrical domains we consider the inverse problem of determining the time independent scalar potential of the dynamic schrodinger equation in an infinite cylindrical domain from one boundary neumann observation of the solution we prove holder stability by choosing the dirichlet boundary condition suitably

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

sparsity and infinite divisibility we adopt an innovation driven framework and investigate the sparse compressible distributions obtained by linearly measuring or expanding continuous domain stochastic models starting from the first principles we show that all such distributions are necessarily infinitely divisible this property is satisfied by many distributions used in statistical learning such as gaussian laplace and a wide range of fat tailed distributions such as student s t and stable laws however it excludes some popular distributions used in compressed sensing such as the bernoulli gaussian distribution and distributions that decay like exp o x p for 1 p 2 we further explore the implications of infinite divisibility on distributions and conclude that tail decay and unimodality are preserved by all linear functionals of the same continuous domain process we explain how these results help in distinguishing suitable variational techniques for statistically solving inverse problems like denoising

layer adapted methods for quasilinear singularly perturbed delay differential problems abstract in this work we consider a class of initial value problems for quasilinear singularly perturbed first order delay differential equations to solve this class of problems numerically we consider two finite difference schemes the backward euler scheme and a high order hybrid scheme which is a blend of the trapezoidal scheme and the backward euler scheme we derive general convergence results for both the schemes based on which a number of layer adapted meshes can be constructed and analyzed as consequences of these results we establish uniform convergence of the schemes on certain layer adapted meshes numerical experiments confirm our theoretical findings

inverse problems for the schrodinger equations with time dependent electromagnetic potentials and the aharonov bohm effect we consider the inverse boundary value problem for the schrodinger operator with time dependent electromagnetic potentials in domains with obstacles we extend the resuls of the author s works inverse probl 19 49 2003 19 985 2003 20 1497 2004 to the case of time dependent potentials we relate our results to the aharonov bohm effect caused by magnetic and electric fluxes

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

solving a partially singularly perturbed initial value problem on shishkin meshes a coupled first order system of one singularly perturbed and one non perturbed ordinary differential equation with prescribed initial conditions is considered a shishkin piecewise uniform mesh is constructed and used in conjunction with a classical finite difference operator to form a new numerical method for solving this problem it is proved that the numerical approximations generated by this method are essentially first order convergent in the maximum norm at all points of the domain uniformly with respect to the singular perturbation parameter numerical results are presented in support of the theory

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

mmse estimation of sparse levy processes we investigate a stochastic signal processing framework for signals with sparse derivatives where the samples of a levy process are corrupted by noise the proposed signal model covers the well known brownian motion and piecewise constant poisson process moreover the levy family also contains other interesting members exhibiting heavy tail statistics that fulfill the requirements of compressibility we characterize the maximum a posteriori probability map and minimum mean square error mmse estimators for such signals interestingly some of the map estimators for the levy model coincide with popular signal denoising algorithms e g total variation tv regularization we propose a novel non iterative implementation of the mmse estimator based on the belief propagation bp algorithm performed in the fourier domain our algorithm takes advantage of the fact that the joint statistics of general levy processes are much easier to describe by their characteristic function as the probability densities do not always admit closed form expressions we then use our new estimator as a benchmark to compare the performance of existing algorithms for the optimal recovery of gradient sparse signals

bayesian estimation for continuous time sparse stochastic processes we consider continuous time sparse stochastic processes from which we have only a finite number of noisy noiseless samples our goal is to estimate the noiseless samples denoising and the signal in between interpolation problem by relying on tools from the theory of splines we derive the joint a priori distribution of the samples and show how this probability density function can be factorized the factorization enables us to tractably implement the maximum a posteriori and minimum mean square error mmse criteria as two statistical approaches for estimating the unknowns we compare the derived statistical methods with well known techniques for the recovery of sparse signals such as the l1 norm and log l1 l0 relaxation regularization methods the simulation results show that under certain conditions the performance of the regularization techniques can be very close to that of the mmse estimator

the exponential asymptotic stability of singularly perturbed delay differential equations with a bounded lag abstract this paper is concerned with the exponential stability of singularly perturbed delay differential equations with a bounded state independent lag a generalized halanay inequality is derived in section 2 and in section 3 a sufficient condition will be provided to ensure that any solution of the singularly perturbed delay differential equations ddes with a bounded lag is exponentially stable uniformly for sufficiently small e 0 this type of exponential asymptotic stability can obviously be applied to general delay differential equations with a bounded lag

a unified formulation of gaussian versus sparse stochastic processes part ii discrete domain theory this paper is devoted to the characterization of an extended family of continuous time autoregressive moving average carma processes that are solutions of stochastic differential equations driven by white levy innovations these are completely specified by 1 a set of poles and zeros that fixes their correlation structure and 2 a canonical infinitely divisible probability distribution that controls their degree of sparsity with the gaussian model corresponding to the least sparse scenario the generalized carma processes are either stationary or nonstationary depending on the location of the poles in the complex plane the most basic nonstationary representatives with a single pole at the origin are the levy processes which are the non gaussian counterparts of brownian motion we focus on the general analog to discrete conversion problem and introduce a novel spline based formalism that greatly simplifies the derivation of the correlation properties and joint probability distributions of the discrete versions of these processes we also rely on the concept of generalized increment process which suppresses all long range dependencies to specify an equivalent discrete domain innovation model a crucial ingredient is the existence of a minimally supported function associated with the whitening operator l this b spline which is fundamental to our formulation appears in most of our formulas both at the level of the correlation and the characteristic function we make use of these discrete domain results to numerically generate illustrative examples of sparse signals that are consistent with the continuous domain model

nonasymptotic gaussian approximation for inference with stable noise the results of a series of theoretical studies are reported examining the convergence rate for different approximate representations of alpha stable distributions although they play a key role in modelling random processes with jumps and discontinuities the use of alpha stable distributions in inference often leads to analytically intractable problems the lepage series which is a probabilistic representation employed in this work is used to transform an intractable infinite dimensional inference problem into a finite dimensional conditionally gaussian parametric problem a major component of our approach is the approximation of the tail of this series by a gaussian random variable standard statistical techniques such as expectation maximization em markov chain monte carlo and particle filtering can then be readily applied in addition to the asymptotic normality of the tail of this series we establish explicit nonasymptotic bounds on the approximation error their proofs follow classical fourier analytic arguments using esseen s smoothing lemma specifically we consider the distance between the distributions of i the tail of the series and an appropriate gaussian ii the full series and the truncated series and iii the full series and the truncated series with an added gaussian term in all three cases sharp bounds are established and the theoretical results are compared with the actual distances computed numerically in specific examples of symmetric alpha stable distributions this analysis facilitates the selection of appropriate truncations in practice and offers theoretical guarantees for the accuracy of resulting estimates one of the main conclusions obtained is that for the purposes of inference the use of a truncated series together with an approximately gaussian error term has superior statistical properties and is likely a preferable choice in practice

uniform numerical method for singularly perturbed delay differential equations this paper deals with the singularly perturbed initial value problem for a linear first order delay differential equation a numerical method is constructed for this problem which involves an appropriate piecewise uniform mesh on each time subinterval the difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter numerical results are presented