a semi circulant preconditioner for the convection diffusion equation in this paper we define and analyze a semi circulant preconditioner for the convection diffusion equation we derive analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system of equations we show that for mesh peclet numbers less than 2 the rate of convergence depends only on the mesh peclet number and the direction of the convective field and not on the spatial grid ratio or the number of unknowns

numerical approximation for functionals of reflecting diffusion processes the aim of this paper is to approximate the expectation of a large class of functionals of the solution x xi of a stochastic differential equation with normal reflection in a piecewise smooth domain of rd this also yields a monte carlo method for solving partial differential problems of parabolic type with mixed boundary conditions the approximation is based on a modified euler scheme for the stochastic differential equation the scheme can be driven by a sequence of bounded independently and identically distributed i i d random variables or when the domain is convex by a sequence of gaussian i i d random variables the order of weak convergence for both cases is given in the former case the order of convergence is 1 2 and it is shown to be exact by an example in the last section numerical tests are presented the behavior of the error as a function of the final time t for fixed values of the discretization step and as a function of the discretization step for fixed values of the final

on weak solutions of forward backward sdes in this paper we continue exploring the notion of weak solution of forward backward stochastic differential equations fbsdes and associated forward backward martingale problems fbmps the main purpose of this work is to remove the constraints on the martingale integrands in the uniqueness proofs in our previous work ma et al in ann probab 36 6 2092 2125 2008 we consider a general class of non degenerate fbsdes in which all the coefficients are assumed to be essentially only bounded and uniformly continuous and the uniqueness is proved in the space of all the square integrable adapted solutions the standard solution space in the fbsde literature a new notion of semi strong solution is introduced to clarify the relations among different definitions of weak solution in the literature and it is in fact instrumental in our uniqueness proof as a by product we also establish some a priori estimates of the second derivatives of the solution to the decoupling quasilinear pde

an iterative computation of approximations on korobov like spaces this paper treats the multidimensional application of a previous iterative monte carlo algorithm that enables the computation of approximations in l2 the case of regular functions is studied using a fourier basis on periodised functions legendre and tchebychef polynomial bases the dimensional effect is reduced by computing these approximations on korobov like spaces numerical results show the efficiency of the algorithm for both approximation and numerical integration

a filtering problem with uncertainty in observation abstract this paper is concerned with a generalized kalman bucy filtering model and corresponding robust problem under model uncertainty we find that this robust problem is equivalent to considering an estimate problem under some sublinear operator therefore we turn to obtaining the minimum mean square estimator under a sublinear operator by girsanov theorem and minimax theorem we obtain the optimal estimator x t of the signal process x t for given time t 0 t

reflected solutions of backward sde s and related obstacle problems for pde s we study reflected solutions of one dimensional backward stochastic differential equations the reflection keeps the solution above a given stochastic process we prove uniqueness and existence both by a fixed point argument and by approximation via penalization we show that when the coefficient has a special form then the solution of our problem is the value function of a mixed optimal stopping optimal stochastic control problem we finally show that when put in a markovian framework the solution of our reflected bsde provides a probabilistic formula for the unique viscosity solution of an obstacle problem for a parabolic partial differential equation

diagonalization of tensors with circulant structure the concepts of tensors with diagonal and circulant structure are defined and a framework is developed for the analysis of such tensors it is shown a tensor of arbitrary order which is circulant with respect to two particular modes can be diagonalized in those modes by discrete fourier transforms this property can be used in the efficient solution of linear systems involving contractive products of tensors with circulant structure tensors with circulant structure occur in models for image blurring with periodic boundary conditions it is shown that the new framework can be applied to such problems

geometric variance reduction in markov chains application to value function and gradient estimation we study a variance reduction technique for monte carlo estimation of functionals in markov chains the method is based on designing sequential control variates using successive approximations of the function of interest v regular monte carlo estimates have a variance of o 1 n where n is the number of sample trajectories of the markov chain here we obtain a geometric variance reduction o n with 1 up to a threshold that depends on the approximation error v av where a is an approximation operator linear in the values thus if v belongs to the right approximation space i e av v the variance decreases geometrically to zero an immediate application is value function estimation in markov chains which may be used for policy evaluation in a policy iteration algorithm for solving markov decision processes another important domain for which variance reduction is highly needed is gradient estimation that is computing the sensitivity v of the performance measure v with respect to some parameter of the transition probabilities for example in policy parametric optimization computing an estimate of the policy gradient is required to perform a gradient optimization method we show that using two approximations for the value function and the gradient a geometric variance reduction is also achieved up to a threshold that depends on the approximation errors of both of those representations

a fourier cosine method for an efficient computation of solutions to bsdes we develop a fourier method to solve backward stochastic differential equations bsdes a general theta discretization of the time integrands leads to an induction scheme with conditional expectations these are approximated by using fourier cosine series expansions relying on the availability of a characteristic function the method is applied to bsdes with jumps numerical experiments demonstrate the applicability of bsdes in financial and economic problems and show fast convergence of our efficient probabilistic numerical method

a stable multistep scheme for solving backward stochastic differential equations in this paper we propose a stable multistep scheme on time space grids for solving backward stochastic differential equations in our scheme the integrands which are conditional mathematical expectations derived from the original equations are approximated by using lagrange interpolating polynomials with values of the integrands at multiple time levels they are then numerically evaluated using the gauss hermite quadrature rules and polynomial interpolations on the spatial grids error estimates are rigorously proved for the semidiscrete version of the proposed scheme for backward stochastic differential equations with certain types of simplified generator functions finally various numerical examples and comparisons with some other methods are presented to demonstrate high accuracy of the proposed multistep scheme

a new kurtosis matrix with statistical applications abstract the number of fourth order moments which can be obtained from a random vector rapidly increases with the vector s dimension scalar measures of multivariate kurtosis may not satisfactorily capture the fourth order structure and matrix measures of multivariate kurtosis are called for in this paper we propose a kurtosis matrix derived from the dominant eigenpair of the fourth standardized moment we show that it is the best symmetric positive semidefinite kronecker square root approximation to the fourth standardized moment additional properties are derived for realizations from garch and reversible random processes statistical applications include independent component analysis and projection pursuit the star product of matrices highlights the connection between the proposed kurtosis matrix and other kurtosis matrices which appeared in the statistical literature a simulation study assesses the practical relevance of theoretical results in the paper

on semi linear degenerate backward stochastic partial differential equations in this paper we study a class of one dimensional degenerate semilinear backward stochastic partial differential equations bspdes for short of parabolic type by establishing some new a priori estimates for both linear and semilinear bspdes we show that the regularity and uniform boundedness of the adapted solution to the semilinear bspde can be determined by those of the coefficients a special feature that one usually does not expect from a stochastic differential equation the proof follows the idea of the so called bootstrap method which enables us to analyze each of the derivatives of the solution under consideration some related results including some comparison theorems of the adapted solutions for semilinear bspdes as well as a nonlinear stochastic feynman kac formula are also given

solving bsde with adaptive control variate we present and analyze an algorithm to solve numerically bsdes based on picard s iterations and on a sequential control variate technique its convergence is geometric moreover the solution provided by our algorithm is regular both w r t time and space

quasi linear pdes and forward backward stochastic differential equations weak solutions abstract in this paper we study the existence uniqueness and the probabilistic representation of the weak solutions of quasi linear parabolic and elliptic partial differential equations pdes in the sobolev space h 1 r d for this we study first the solutions of forward backward stochastic differential equations fbsdes with smooth coefficients regularity of solutions and their connection with classical solutions of quasi linear parabolic pdes then using the approximation procedure we establish their convergence in the sobolev space to the solutions of the fbsdes in the space l 2 r d r d l 2 r d r k l 2 r d r k d this gives a connection with the weak solutions of quasi linear parabolic pdes finally we study the unique weak solutions of quasi linear elliptic pdes using the solutions of the fbsdes on infinite horizon

on numerical approximations of forward backward stochastic differential equations a numerical method for a class of forward backward stochastic differential equations fbsdes is proposed and analyzed the method is designed around the four step scheme j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 but with a hermite spectral method to approximate the solution to the decoupling quasi linear pde on the whole space a rigorous synthetic error analysis is carried out for a fully discretized scheme namely a first order scheme in time and a hermite spectral scheme in space of the fbsdes equally important a systematical numerical comparison is made between several schemes for the resulting decoupled forward sde including a stochastic version of the adams bashforth scheme it is shown that the stochastic version of the adams bashforth scheme coupled with the hermite spectral method leads to a convergence rate of frac 32 in time which is better than those in previously published work for the fbsdes

evolutionary design of digital filters with application to subband coding and data transmission in this paper two evolutionary programming ep algorithms classical ep and fast ep are applied to design prototype lowpass finite impulse response filters for use in a modulated filterbank the chosen filter design technique is based on frequency sampling where the fourier transform magnitudes of the filter are the objective variables design is simplified by constraining most of these values leaving only a small number of values in the filter transition band to be optimized the ep algorithms were used to determine the optimum values for this subset of values since there is an additional monotonic constraint on the transition band values a modification to the ep algorithms was developed called variable limits evolutionary programming results indicate that a both ep algorithms were insensitive to initial conditions and reliably found the minimum values of the chosen objective functions and b the designed prototype filters are suitable to obtain near perfect reconstruction filter banks offering quality parameters comparable or better than systems obtained using other techniques

adapted solution of a backward stochastic differential equation abstract let wt t 0 1 be a standard k dimensional weiner process defined on a probability space f p and let ft denote its natural filtration given a f1 measurable d dimensional random vector x we look for an adapted pair of processes x t y t t 0 1 with values in rd and rd k respectively which solves an equation of the form x t t 1 f s x s y s ds t 1 g s x s y s dw s x a linearized version of that equation appears in stochastic control theory as the equation satisfied by the adjoint process we also generalize our results to the following equation x t t 1 f s x s y s ds t 1 g s x s y s dw s x under rather restrictive assumptions on g

independent component analysis and simultaneous third order tensor diagonalization comon s 1994 well known scheme for independent component analysis ica is based on the maximal diagonalization in a least squares sense of a higher order cumulant tensor in a previous paper we proved that for fourth order cumulants the computation of an elementary jacobi rotation is equivalent to the computation of the best rank 1 approximation of a fourth order tensor in this paper we show that for third order tensors the computation of an elementary jacobi rotation is again equivalent to a best rank 1 approximation however here it is a matrix that has to be approximated this favorable computational load makes it attractive to do something third order like for fourth order cumulant tensors as well we show that simultaneous optimal diagonalization of third order tensor slices of the fourth order cumulant is a suitable strategy this simultaneous third order tensor diagonalization approach stotd is similar in spirit to the efficient jade algorithm

a general formulation of modulated filter banks this paper presents a general framework for maximally decimated modulated filter banks the theory covers the known classes of cosine modulation and relates them to complex modulated filter banks the prototype filters have arbitrary lengths and the overall delay of the filter bank is arbitrary within fundamental limits necessary and sufficient conditions for perfect reconstruction pr are derived using the polyphase representation it is shown that these pr conditions are identical for all types of modulation modulation based on the discrete cosine transform dct both dct iii dct iv and dct i dct ii and modulation based on the modified discrete fourier transform mdft a quadratic constrained design method for prototype filters yielding pr with arbitrary length and system delay is derived and design examples are presented to illustrate the tradeoff between overall system delay and stopband attenuation subchannelization

sequential monte carlo methods for diffusion processes in this paper we show how to use sequential monte carlo methods to compute expectations of functionals of diffusions at a given time and the gradients of these quantities w r t the initial condition of the process in some cases via the exact simulation of the diffusion there is no time discretization error otherwise the methods use euler discretization we illustrate our approach on both high and low dimensional problems from optimal control and establish that our approach substantially outperforms standard monte carlo methods typically adopted in the literature the methods developed here are appropriate for solving a certain class of partial differential equations as well as for option pricing and hedging

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

a partially reflecting random walk on spheres algorithm for electrical impedance tomography in this work we develop a probabilistic estimator for the voltage to current map arising in electrical impedance tomography this novel so called partially reflecting random walk on spheres estimator enables monte carlo methods to compute the voltage to current map in an embarrassingly parallel manner which is an important issue with regard to the corresponding inverse problem our method uses the well known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions we analyze the global bias and the variance of the new estimator both theoretically and experimentally subsequently the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms

factorization strategies for third order tensors abstract operations with tensors or multiway arrays have become increasingly prevalent in recent years traditionally tensors are represented or decomposed as a sum of rank 1 outer products using either the candecomp parafac cp or the tucker models or some variation thereof such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array the specifics of the approximate representation such as how many terms to use in the sum orthogonality constraints etc depend on the application in this paper we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem reminiscent of matrix factorizations we present a new factorization of a tensor as a product of tensors to derive the new factorization we define a closed multiplication operation between tensors a major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications specifically this new multiplication allows us to introduce concepts such as tensor transpose inverse and identity which lead to the notion of an orthogonal tensor the multiplication also gives rise to a linear operator and the null space of the resulting operator is identified we extend the concept of outer products of vectors to outer products of matrices all derivations are presented for third order tensors however they can be easily extended to the order p p 3 case we conclude with an application in image deblurring

variance reduction for simulated diffusions this article develops some variance reduction techniques for the monte carlo integration of functionals of the solutions of ito stochastic differential equations sdes the monte carlo method for sdes offers a means of calculating solutions to certain types of parabolic partial differential equation and so has applications in various fields including stochastic control particle physics and econometrics it involves the representation of the required integrals as means of random variables defined on infinite dimensional wiener spaces which cannot be simulated directly sthey must at some stage be approximated by variables defined on high but finite dimensional spaces the approach taken here is to construct variance reduced random variables on the infinite dimensional spaces which can subsequently be approximated by any of a number of known finite difference methods the methods of control variates and importance sampling are developed in both cases a perfect variate i e one which is unbiased and h

linear quadratic stochastic differential games open loop and closed loop saddle points in this paper we consider a linear quadratic stochastic two person zero sum differential game the controls for both players are allowed to appear in both drift and diffusion of the state equation the weighting matrices in the performance functional are not assumed to be definite non singular the existence of an open loop saddle point is characterized by the existence of an adapted solution to a linear forward backward stochastic differential equation with constraints together with a convexity concavity condition and the existence of a closed loop saddle point is characterized by the existence of a regular solution to a riccati differential equation it turns out that there is a significant difference between open loop and closed loop saddle points also it is found that there is an essential feature that prevents a linear quadratic optimal control problem from being a special case of linear quadratic two person zero sum differential games

efficient design of oversampled npr gdft filterbanks we propose a flexible efficient design technique for the prototype filter of an oversampled near perfect reconstruction npr generalized discrete fourier transform gdft filterbank such filterbanks have several desirable properties for subband processing systems that are sensitive to aliasing such as subband adaptive filters the design criteria for the prototype filter are explicit bounds derived herein on the aliased components in the subbands and the output the distortion induced by the filterbank and the imaged subband errors in the output it is shown that the design of an optimal prototype filter can be transformed into a convex optimization problem which can be efficiently solved the proposed design technique provides an efficient and effective tool for exploring many of the inherent tradeoffs in the design of the prototype filter including the tradeoff between aliasing in the subbands and the distortion induced by the filterbank we calculate several examples of these tradeoffs and demonstrate that the proposed method can generate filters with significantly better performance than filters obtained using current design methods

analysis of post combiner equalizers in cosine modulated filterbank based transmultiplexer systems in this paper we present an analysis of the post combiner equalizers used to compensate for channel distortion in cosine modulated based transmultiplexer systems such equalizers have been widely used in multicarrier modulation mcm systems that use cosine modulated filterbanks cmfb for signal modulation and demodulation by making the reasonable assumption that the number of subchannels in the system is large enough such that each subchannel can be approximated by a constant complex gain we derive close form equations for optimum post combiner transfer functions these transfer functions are found to be finite impulse response fir and closely related to the prototype filter in cmfb analysis and synthesis blocks moreover by making use of the results of our analysis we propose a new post combiner structure in which the number of adaptive parameters are about one third of those in the earlier reports this reduction in the number of adaptive parameters results in a three fold increase in convergence speed of the least mean square lms algorithm when it is used for adaptation of the post combiners we also present a convergence analysis of the post combiner when the lms algorithm is used for adaptation we study the correlation matrix whose eigenvalues determine convergence behavior of the lms algorithm we show that all eigenvalues of this matrix are equal this implies that in the case of the post combiner the lms algorithm experiences no slow mode of convergence as is the case in many other applications computer simulations that corroborate our theoretical findings are also presented

exact asymptotics for the probability of exit from a domain and applications to simulation we study the asymptotics of the exit probability p x s e t where is the exit time from an open set and p x s e is the law of a diffusion process with a small parameter e multiplying the diffusion coefficient we consider the case of the brownian bridge in many dimensions this choice being motivated by applications to numerical simulation the method uses recent results reducing the problem to the solution of a system of linear first order pde s

the maximum principle for global solutions of stochastic stackelberg differential games national science foundation dms 1303775 research grant council of the hksar cityu 500113

a novel fast tensor based preconditioner for image restoration image restoration is one of the main parts of image processing mathematically this problem can be modeled as a large scale structured ill posed linear system ill posedness of this problem results in low convergence rate of iterative solvers for speeding up the convergence preconditioning usually is used despite the existing preconditioners for image restoration which are constructed based on approximations of the blurring matrix in this paper we propose a novel preconditioner with a different viewpoint here we show that image restoration problem can be modeled as a tensor contractive linear equation this modeling enables us to propose a new preconditioner based on an approximation of the blurring tensor operator due to the particular structure of the blurring tensor for zero boundaries we show that the truncated higher order singular value decomposition of the blurring tensor is obtained very fast and so could be used as a preconditioner experimental results confirm the efficiency of this new preconditioner in image restoration and its outperformance in comparison with the other well known preconditioners

sequential control variates for functionals of markov processes using a sequential control variates algorithm we compute monte carlo approximations of solutions of linear partial differential equations connected to linear markov processes by the feynman kac formula it includes diffusion processes with or without absorbing reflecting boundary and jump processes we prove that the bias and the variance decrease geometrically with the number of steps of our algorithm numerical examples show the efficiency of the method on elliptic and parabolic problems