a quintic b spline finite elements scheme for the kdvb equation abstract an algorithm for solving the korteweg de vries burgers kdvb equation based on the collocation method with quintic b spline finite elements is set up to simulate the solutions of the kdv burgers and the kdvb equations the migration of solitary waves and temporal evolution of a maxwellian initial pulse are studied for the kdv equation burgers equation is solved for different values of reynolds number the time evaluation of the solutions of the kdvb equation with different values for the diffusion and dispersion coefficients is studied invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm a linear stability analysis shows the scheme to be unconditionally stable

tomographic imaging of dynamic objects with the ensemble kalman filter we address the image formation of a dynamic object from projections by formulating it as a state estimation problem the problem is solved with the ensemble kalman filter enkf a monte carlo algorithm that is computationally tractable when the state dimension is large in this paper we first rigorously address the convergence of the enkf then the effectiveness of the enkf is demonstrated in a numerical experiment where a highly variable object is reconstructed from its projections an imaging modality not yet explored with the enkf the results show that the enkf can yield estimates of almost equal quality as the optimal kalman filter but at a fraction of the computational effort further experiments explore the rate of convergence of the enkf its performance relative to an idealized particle filter and implications of modeling the system dynamics as a random walk

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

b spline solution of singular boundary value problems homogeneous and non homogenous singular boundary value problems special case are solved using b splines the original differential equation is modified at singular point then the boundary value problem is treated by using b spline approximation the method is tested on some model problems from the literature and the numerical results are compared with exact solution

fractional calculus a new approach to the analysis of generalized fourth order diffusion wave equations the homotopy perturbation method is applied to the generalized fourth order fractional diffusion wave equations the problem is formulated in the caputo sense moreover a reliable scheme for calculating nonlinear operators is proposed the results reveal that the present method is very effective and convenient

analytic treatment for variable coefficient fourth order parabolic partial differential equations the variable coefficient fourth order parabolic partial differential equation in one space variable that arise in the study of the transverse vibrations of a uniform flexible beam is examined the analytic solution is obtained by using adomian decomposition method the nonhomogeneous problem is solved by using the noise terms phenomenon the method is tested on three problems which have appeared in physical applications

b spline method for solving linear system of second order boundary value problems the b spline method is used for the numerical solution of a linear system of second order boundary value problems two examples are considered for the numerical illustration and the method is also compared with the method proposed by j lu j lu variational iteration method for solving a nonlinear system of second order boundary value problems comput math appl 54 2007 1133 1138 the comparison shows that the b spline method is a more efficient and effective tool and yields better results

quintic b spline collocation method for fourth order partial integro differential equations with a weakly singular kernel quintic b spline collocation method for the numerical solution of fourth order partial integro differential equations is considered the scheme is based on the second order backward differential formula in the time direction and the quintic b spline method for the spatial derivative some numerical experiments are included to demonstrate the effectiveness of the discrete technique from the numerical experiments we can find that the present scheme possesses high accuracy and efficiency and the numerical results are in good agreement with the exact solutions

galerkin method for the numerical solution of the rlw equation using quintic b splines the regularized long wave equation rlw is solved numerically by using the quintic b spline galerkin finite element method the same method is applied to the time split rlw equation comparison is made with both analytical solutions and some previous results propagation of solitary waves interaction of two solitons are studied

regularized estimation of large covariance matrices this paper considers estimating a covariance matrix of p variables from n observations by either banding the sample covariance matrix or estimating a banded version of the inverse of the covariance we show that these estimates are consistent in the operator norm as long as logp 2 n 0 and obtain explicit rates the results are uniform over some fairly natural well conditioned families of covariance matrices we also introduce an analogue of the gaussian white noise model and show that if the population covariance is embeddable in that model and well conditioned then the banded approximations produce consistent estimates of the eigenvalues and associated eigenvectors of the covariance matrix the results can be extended to smooth versions of banding and to non gaussian distributions with su ciently short tails a resampling approach is proposed for choosing the banding parameter in practice this approach is illustrated numerically on both simulated and real data

a sixth order numerical method and its convergence for generalized black scholes pde abstract the main aim of this paper is to construct a new computational approach for the numerical solution of generalized black scholes equation in this approach the temporal variable is discretized using crank nicolson scheme and spatial variable is discretized using sextic b spline collocation method convergence analysis of the method is discussed the proposed method is proved to be stable and has second order convergence with respect to time variable and sixth order convergence with respect to space variable to illustrate the applicability and efficiency of the proposed method we consider some benchmark problems describing european call options numerical results verify the orders of convergence predicted by the analysis numerical results reveal that the present method provides better results as compared to some existing numerical methods based on b spline collocation approach

on the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel ii nonsmooth initial data abstract backward euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in error analyses the stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the galerkin finite element methods in the space variables

spline methods for the solution of fourth order parabolic partial differential equations in this paper a fourth order non homogeneous parabolic partial differential equation that governs the behaviour of a vibrating beam is solved by using a new three level method based on parametric quintic spline in space and finite difference discretization in time stability analysis of the method has been carried out it has been shown that by suitably choosing the parameters most of the previous known methods for homogeneous and non homogeneous cases can be derived from our method we also obtain two new high accuracy schemes of o k 4 h 6 and o k 4 h 8 and two new schemes which are analogues of jain s formula for the non homogeneous case comparison of our results with those of some known methods show the superiority of the present approach

legendre laguerre coupled spectral element methods for second and fourth order equations on the half line some legendre spectral element laguerre spectral coupled methods are proposed to numerically solve second and fourth order equations on the half line the proposed methods are based on splitting the infinite domain into two parts then using the legendre spectral element method in the finite subdomain and laguerre method in the infinite subdomain c 0 or c 1 continuity according to the problem under consideration is imposed to couple the two methods rigorous error analysis is carried out to establish the convergence of the method more importantly an efficient computational process is introduced to solve the discrete system several numerical examples are provided to confirm the theoretical results and the efficiency of the method

analysis of a local discontinuous galerkin method for linear time dependent fourth order problems we analyze a local discontinuous galerkin method for fourth order time dependent problems optimal error estimates are obtained in one dimension and in multidimensions for cartesian and triangular meshes we extend the analysis to higher even order equations and the linearized cahn hilliard type equations numerical experiments are displayed to verify the theoretical results

particle filtering for large dimensional state spaces with multimodal observation likelihoods we study efficient importance sampling techniques for particle filtering pf when either a the observation likelihood ol is frequently multimodal or heavy tailed or b the state space dimension is large or both when the ol is multimodal but the state transition pdf stp is narrow enough the optimal importance density is usually unimodal under this assumption many techniques have been proposed but when the stp is broad this assumption does not hold we study how existing techniques can be generalized to situations where the optimal importance density is multimodal but is unimodal conditioned on a part of the state vector sufficient conditions to test for the unimodality of this conditional posterior are derived our result is directly extendable to testing for unimodality of any posterior the number of particles n to accurately track using a pf increases with state space dimension thus making any regular pf impractical for large dimensional tracking problems but in most such problems most of the state change occurs in only a few dimensions while the change in the rest of the dimensions is small using this property we propose to replace importance sampling from a large part of the state space whose conditional posterior is narrow enough by just tracking the mode of the conditional posterior this introduces some extra error but it also greatly reduces the importance sampling dimension the net effect is much smaller error for a given n especially when the available n is small an important class of large dimensional problems with multimodal ol is tracking spatially varying physical quantities such as temperature or pressure in a large area using a network of sensors which may be nonlinear and or may have nonnegligible failure probabilities improved performance of our proposed algorithms over existing pfs is demonstrated for this problem

a novel numerical scheme for solving burgers equation abstract in this paper we propose a novel numerical scheme for solving burgers equation the scheme is based on a cubic spline quasi interpolant and multi node higher order expansion which make the algorithm simple and easy to implement the numerical experiments show that the proposed method produces high accurate results

space time spectral method for a weakly singular parabolic partial integro differential equation on irregular domains abstract the spectral method is proposed for the partial integro differential equations with a weakly singular kernel on irregular domains the space discretization is based on the nodal spectral element method using the lagrange polynomials basis associated with the gauss lobatto legendre quadrature nodes also the model is discretized in time with the legendre spectral galerkin method the discretization leads to conversion of the problem to a sylvester matrix equation which can be solved efficiently by the qz algorithm gardiner et al 1992 the convergence of the method is proven by providing a priori l 2 error estimate numerical results illustrate the efficiency and spectral accuracy of the proposed method

augmented particle filters abstractparticle filters have been widely used for online filtering problems in state space models ssms the current available proposal distributions depend either only on the state dynamics or only on the observation or on both sources of information but are not available for general ssms in this article we develop a new particle filtering algorithm called the augmented particle filter apf for online filtering problems in ssms the apf combines two sets of particles from the observation equation and the state equation and the state space is augmented to facilitate the weight computation theoretical justification of the apf is provided and the connection between the apf and the optimal particle filter opf in some special ssms is investigated the apf shares similar properties as the opf but the apf can be applied to a much wider range of models than the opf simulation studies show that the apf performs similarly to or better than the opf when the opf is available and the apf can perform

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

fifth degree b spline solution for a fourth order parabolic partial differential equations in this paper we find the numerical solution of a fourth order parabolic partial differential equation a family of b spline methods has been considered for the numerical solution of the problems the results show that the present method is an applicable technique and approximates the exact solution very well

theory and application of covariance matrix tapers for robust adaptive beamforming we unify several seemingly disparate approaches to robust adaptive beamforming through the introduction of the concept of a covariance matrix taper cmt this is accomplished by recognizing that an important class of adapted pattern modification techniques are realized by the application of a conformal matrix taper to the original sample covariance matrix from the schur product theorem for positive semi definite matrices and kolmogorov s existence theorem we further establish that cmts are in fact the solution to a minimum variance optimum beamformer associated with an auxiliary stochastic process that is related to the original by a hadamard schur product this allows us to gain deeper insight into the design of both existing pattern modification techniques and new cmts that can for example simultaneously address several different design constraints such as pattern distortion due to insufficient sample support and weights mismatch due to nonstationary interference a new two dimensional 2 d cmt for space time adaptive radar applications designed to provide more robust clutter cancellation is also introduced since the cmt approach only involves a single matrix haddamard product it is also inherently low complexity the practical utility of the cmt approach is illustrated through its application to both spatial and spatio temporal adaptive beamforming examples

a tutorial on particle filters for online nonlinear non gaussian bayesian tracking increasingly for many application areas it is becoming important to include elements of nonlinearity and non gaussianity in order to model accurately the underlying dynamics of a physical system moreover it is typically crucial to process data on line as it arrives both from the point of view of storage costs as well as for rapid adaptation to changing signal characteristics in this paper we review both optimal and suboptimal bayesian algorithms for nonlinear non gaussian tracking problems with a focus on particle filters particle filters are sequential monte carlo methods based on point mass or particle representations of probability densities which can be applied to any state space model and which generalize the traditional kalman filtering methods several variants of the particle filter such as sir asir and rpf are introduced within a generic framework of the sequential importance sampling sis algorithm these are discussed and compared with the standard ekf through an illustrative example

a high order numerical method and its convergence for time fractional fourth order partial differential equations abstract this paper is concerned with design and analysis of a high order numerical approach based on a uniform mesh to approximate the solution of time fractional fourth order partial differential equations in this approach we first approximate the time fractional derivative appearing in the governing equation by means of caputo s definition and then construct a sextic b spline collocation method for solving the resulting equation it is proved that the present method is unconditionally stable convergence analysis of the method is discussed we prove that the method is 2 th order convergence with respect to time variable and fourth order convergence with respect to space variable three test problems are considered to demonstrate the applicability and efficiency of the new method it is shown that the rate of convergence predicted theoretically is the same as that obtained experimentally numerical results have been compared with those reported previously in literature it is shown that our method yields more accurate results when compared to the existing methods

optimal scan for time varying tomography ii efficient design and experimental validation for pt i see ibid vol 4 no 5 p 642 53 1995 in pt i the authors presented a theoretical analysis of tomographic reconstruction of objects with spatially localized temporal variation such as a thorax cross section with a beating heart that analysis showed that by using an optimally scrambled angular sampling order the scan rate required to avoid motion artifacts in the reconstructed images can be lowered as much as four times while preserving image quality here the authors present a simple design procedure for the optimum choice of angular sampling pattern which depends only on pre specified geometric and spectral parameters and the desired spatial resolution the resulting patterns have a simple congruential structure reconstruction is accomplished by interpolation to standard time invariant scan format followed by conventional reconstruction the interpolation only requires linear shift invariant separable filtering at a negligible computational cost simulation results demonstrate the technique and validate the analysis for both bandlimited and approximately bandlimited objects

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

efficient spectral galerkin method ii direct solvers of second and fourth order equations using chebyshev polynomials efficient direct solvers based on the chebyshev galerkin methods for second and fourth order equations are presented they are based on appropriate base functions for the galerkin formulation that lead to discrete systems with special structured matrices that can be efficiently inverted numerical results indicate that the direct solvers presented in this paper are significantly more accurate and efficient than those based on the chebyshev tau method

the submartingal problem for brownian motion in a cone with non constant oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties 1 the state space is a cone ind dimensions d 3 and the process behaves in the interior of the cone like ordinary brownian motion 2 the process reflects instantaneously at the boundary of the cone the direction of reflection is continuous except at the vertex and has a limit which is fixed on each radial line emanating from the vertex 3 the amount of time that the process spends at the vertex of the cone is zero i e the set of times for which the process is at the vertex has zero lebesgue measure

adomian decomposition method for solving bvps for fourth order integro differential equations the adomian decomposition method adm is applied to solve both linear and nonlinear boundary value problems bvps for fourth order integro differential equations the numerical results obtained with minimum amount of computation or mathematics compare reasonably well with exact solutions

finite element methods of the two nonlinear integro differential equations abstract transformation of dependent variables such as the kirchhoff transformation is a classical tool for solving nonlinear partial differential equations in larsson et al this approach was used in connection with the finite element method zlamal used it in galerkin approximation in space the euler method in time and applied it to the solution of nonlinear heat equations here we give justification of the methods for the two nonlinear integro heat equations in the case that the discretization is carried through by piecewise linear polynomials in space and by the implicit euler method in time with quadrature rules which require the storage of relatively few values of the solution the estimate of the discretization error in the maximum norm is introduced the results depend on superconvergence in the gradient for a related elliptic problem which is shown to hold only for very special meshes

variational iteration method and homotopy perturbation method for fourth order fractional integro differential equations in this article linear and nonlinear boundary value problems for fourth order fractional integro differential equations are solved by variational iteration method and homotopy perturbation method the fractional derivatives are described in the caputo sense the solutions of both problems are derived by infinite convergent series which are easily computable and then graphical representation shows that both methods are most effective and convenient one to solve linear and nonlinear boundary value problems for fourth order fractional integro differential equations

rational spline nonstandard finite difference scheme for the solution of time fractional swift hohenberg equation abstract in this paper we introduce a new scheme based on rational spline function and nonstandard finite difference technique to solve the time fractional swift hohenberg equation in the sense of riemann liouville derivative via fourier method the method is convergent and unconditionally stable also we investigated the existence and uniqueness of the proposed method numerical results are demonstrated to validate the applicability and the theoretical results