an effective mom solution with nested complex source beam method for electromagnetic scattering problems a nested complex source beam ncsb method based on octree structure is proposed to accelerate the method of moments mom solution for three dimensional 3 d electromagnetic scattering analysis at the finest level the far field contribution of each basis function is expressed with that of complex source beams csbs distributed on the equivalent sphere surface for every group furthermore the far field radiated by csbs of the source group at child level can be represented by that radiated by csbs of the source group at parent level similarly the received field of csbs for each observation group at the parent level can be expressed with that of csbs belong to their child level group an equivalent relationship of csbs between every two adjacent levels is built to obtain the nested equivalent process for ncsb method and only the csbs at the finest level have the direct association with the basis functions in their group the ncsb method is shown to be o nlogn computational complexity for both matrix vector product mvp time and memory storage by numerical examples with appropriate parameters and numerical results demonstrate the better efficiency of the proposed method than the current multilevel csb method

fast iterative solution of integral equations with method of moments and matrix decomposition algorithm singular value decomposition the multilevel matrix decomposition algorithm mlmda was originally developed by michielsen and boag for 2d tmz scattering problems and later implemented in 3d by rius et al the 3d mlmda was particularly efficient and accurate for piece wise planar objects such as printed antennas however for arbitrary 3d problems it was not as efficient as the multilevel fast multipole algorithm mlfma and the matrix compression error was too large for practical applications this paper will introduce some improvements in 3d mlmda like new placement of equivalent functions and svd postcompression the first is crucial to have a matrix compression error that converges to zero as the compressed matrix size increases as a result the new mda svd algorithm is comparable with the mlfma and the adaptive cross approximation aca in terms of computation time and memory requirements remarkably in high accuracy computations the mda svd approach obtains a matrix compression error one order of magnitude smaller than aca or mlfma in less computation time like the aca the mda svd algorithm can be implemented on top of an existing mom code with most commonly used green s functions but the mda svd is much more efficient in the analysis of planar or piece wise planar objects like printed antennas

accelerated direct solution of electromagnetic scattering via characteristic basis function method with sherman morrison woodbury formula based algorithm the characteristic basis function method has been proposed to accelerate the direct solution of the method of moments by reducing the number of unknowns however with the increase in the target size the reduced matrix will still become too large to directly solve and store in this communication the algorithm based on the sherman morrison woodbury smw formula is employed to address this problem for the first time it compresses the reduced matrix into a product of block diagonal matrices with the aid of the smw formula and the adaptive cross approximation subsequently the inverse of the reduced matrix can be easily obtained by the inverses of these block diagonal matrices the numerical results about the electromagnetic scattering are shown to demonstrate the performance of the proposed method

asymptotic stability of the optimal filter with respect toits initial condition consider the problem of estimation of a diffusion signal observed in additive white noise if the solution to the filtering equations initialized with an incorrect prior distribution approaches the true conditional distribution asymptotically in time then the filter is said to be asymptotically stable with respect to perturbations of the initial condition this paper presents asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior for the linear case stability of the riccati equation of kalman filtering is used to derive almost sure asymptotic stability of linear filters for possibly non gaussian initial conditions in the nonlinear case asymptotic stability in a weak convergence sense is shown for filters of signal diffusions which converge in law to an invariant distribution

block size optimization of cbfm for scattering problems the characteristic basis function method cbfm was developed as an algorithm to accelerate the analysis based on direct matrix solver in cbfm the optimum block size greatly influences the overall analysis time in this paper we propose a guideline for setting the optimal block size that can be applied to every algorithm based on cbfm rough estimation of the optimal block size is l b rough approx 0 05 n 6 29 and the exact optimal block size can be determined through preprocessing based on rough estimation numerical results of electromagnetic scattering problems are shown to demonstrate the performance of the proposed method

the stability of conditional markov processes and markov chains in random environments we consider a discrete time hidden markov model where the signal is a stationary markov chain when conditioned on the observations the signal is a markov chain in a random environment under the conditional measure it is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate this permits a delicate exchange of the intersection and supremum of fields which is key for the stability of the nonlinear filter and partially resolves a long standing gap in the proof of a result of kunita j multivariate anal 1 1971 365 393 a similar result is obtained also in the continuous time setting the proofs are based on an ergodic theorem for markov chains in random environments in a general state space

comments on fast direct solution of method of moments linear system in the above named work a new algorithm is presented for direct non iterative frequency domain method of moments mom calculations based on block wise compression of the impedance matrix the storage requirements and computational complexity of the algorithm are claimed to be n3 2 and n2 respectively if n is the number of unknowns for a mom surface discretization that is fixed with respect to the wavelength a more rigorous analysis explicitly using the asymptotic expression for the number of degrees of freedom dof in the interaction between groups of scatterers is presented here showing that this should be n3 2log n and n2log2 n respectively the complexity analysis applies to problems that can be formulated as a surface integral equation sie on surfaces in or between homogeneous media such as a ship or an airplane it does not apply to volume discretization mom or multilayer problems the results presented here are also relevant for block wise compression methods using iterative solvers since the compressed impedance matrix is the same consequently the above complexity analysis applies to very large problems well beyond the reach of even the largest existent computers for moderate size problems the measured complexity is lower

approximation and limit results for nonlinear filters over an infinite time interval this paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion or just a process that is approximated by a jump diffusion the observation noise can be either white or of wide bandwidth the observations can be taken in either discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval it is shown under quite reasonable conditions on the approximating filter and the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter were used analogous results are obtained with appropriate scaling if the observations are taken in discrete time and the sampling interval also goes to zero for these cases the approximating filter is a numerical approximation to the optimal filter for the presumed limit signal observation noise problem

markov property and ergodicity of the nonlinear filter in this paper we first prove under quite general conditions that the nonlinear filter and the pair signal filter are feller markov processes the state space of the signal is allowed to be nonlocally compact and the observation function h can be unbounded our proofs in contrast to those of kunita j multivariate anal 1 1971 pp 365 393 spatial stochastic processes birkhauser 1991 pp 233 256 and stettner stochastic differential equations springer verlag 1989 pp 279 292 do not depend upon the uniqueness of the solutions to the filtering equations we then obtain conditions for existence and uniqueness of invariant measures for the nonlinear filter and the pair process these results extend those of kunita and stettner which hold for locally compact state space and bounded h to our general framework finally we show that the recent results of ocone and pardoux siam j control optim 34 1996 pp 226 243 on asymptotic stability of the nonlinear filter which use the kunita stettner setup hold for the general situation considered in this paper

efficient scalable parallel higher order direct mom sie method with hierarchically semiseparable structures for 3 d scattering a novel fast scalable parallel algorithm is proposed for the solution of large 3 d scattering problems based on 1 the double geometrical and current approximation higher order dho method of moments mom in the surface integral equation sie formulation and 2 a direct solver for dense linear systems utilizing hierarchically semiseparable hss structures namely an hss matrix representation is used for compression factorization and solution of the system matrix in addition a rank revealing qr decomposition for memory compression is used with a stopping criterion in terms of the relative rank tolerance value a method for geometrical preprocessing of the scatterers based on the cobblestone distance sorting technique is employed in order to enhance the hss algorithm accuracy and parallelization numerical examples show how the accuracy of the dho hss mom sie method is easily controllable by using the relative tolerance for the matrix compression moreover the examples demonstrate low memory consumption as well as much faster simulation time when compared to the direct lu decomposition the method enables dramatically faster monostatic scattering computations than iterative solvers and reduced number of unknowns when compared to low order discretizations finally great scalability of the algorithm is demonstrated on more than one thousand processes

sparsified adaptive cross approximation algorithm for accelerated method of moments computations this paper presents a modification of the adaptive cross approximation aca algorithm for accelerated solution of the method of moments linear system for electrically large radiation and scattering problems as with aca subblocks of the impedance matrix that represent the interaction between well separated subdomains are substituted by compressed approximations allowing for reduced storage and accelerated iterative solution the modified algorithm approximates the original subblocks with products of sparse matrices constructed with the aid of the aca algorithm and of a sub sampling of the original basis functions belonging to either subdomain because of the sampling an additional error is introduced with respect to aca but this error is controllable just like ordinary aca sparsified aca is kernel independent and needs no problem specific information except for the topology of the basis functions as a numerical example rcs computations of the nasa almond are presented showing an important gain in efficiency furthermore the numerical experiment reveals a computational complexity close to n logn for sparsified aca for a target electrical size of up to 50 wavelengths

mixed form nested approximation for wideband multiscale simulations we propose a mixed skeleton and equivalence nested approximation method to compress the impedance matrix of the method of moments for the wideband multiscale simulations we first introduce a nested skeleton approximation where the impedance matrix is expressed recursively by sampling the dominant basis functions e g skeletons with a fully algebraic implementation from the original basis functions the idea is to introduce the automatically constructed test surface around the interface between near and far field regions for each group the dominant rwgs are sampled recursively with the adaptive cross approximation to compress the matrix against the test surface second we introduce a mixed form algorithm of skeleton and equivalence nested approximation method at low levels the nested skeleton approximation is employed and it is smoothly transferred to standard wideband nested equivalence approximation wnesa at high levels an accurate number of skeletons can be always found with a predetermined threshold at a low level which will improve computation efficiency with respect to wnesa the computational complexity of the proposed algorithm is mathcal o n log n n is the number of unknowns numerical wideband multiscale simulations demonstrate the efficiency of the proposed algorithm

asymptotic optimality of finite model approximations for partially observed markov decision processes with discounted cost we consider finite model approximations of discrete time partially observed markov decision processes pomdps under the discounted cost criterion after converting the original partially observed stochastic control problem to a fully observed one on the belief space the finite models are obtained through the uniform quantization of the state and action spaces of the belief space markov decision process mdp under mild assumptions on the components of the original model it is established that the policies obtained from these finite models are nearly optimal for the belief space mdp and so for the original partially observed problem the assumptions essentially require that the belief space mdp satisfies a mild weak continuity condition we provide an example and introduce explicit approximation procedures for the quantization of the set of probability measures on the state space of pomdp i e belief space

approximation of boundary element matrices this article considers the problem of approximating a general asymptotically smooth function in two variables typically arising in integral formulations of boundary value problems by a sum of products of two functions in one variable from these results an iterative algorithm for the low rank approximation of blocks of large unstructured matrices generated by asymptotically smooth functions is developed this algorithm uses only few entries from the original block and since it has a natural stopping criterion the approximative rank is not needed in advance

uniqueness and robustness of solution of measure valued equations of nonlinear filtering we consider the zakai equation for the unnormalized conditional distribution when the signal process x takes values in a complete separable metric space e and when h is a continuous possibly unbounded function on e it is assumed that x is a markov process which is characterized via a martingale problem for an operator a 0 uniqueness of solution for the measure valued zakai and fujisaki kallianpur kunita equations is proved when the test functions belong to the domain of a 0 it is also shown that the conditional distributions are robust

weak feller property of non linear filters abstract weak feller property of controlled and control free markov chains leads to many desirable properties in control free setups this leads to the existence of invariant probability measures for compact spaces and applicability of numerical approximation methods for controlled setups this leads to existence and approximation results for optimal control policies we know from stochastic control theory that partially observed systems can be converted to fully observed systems by replacing the original state space with a probability measure valued state space with the corresponding kernel acting on probability measures known as the non linear filter process establishing sufficient conditions for the weak feller property for such processes is a significant problem studied under various assumptions and setups in the literature in this paper we prove the weak feller property of the non linear filter process i first under weak continuity of the transition probability of controlled markov chain and total variation continuity of its observation channel and then ii under total variation continuity of the transition probability of controlled markov chain the former result i has first appeared in feinberg et al 2016 here we present a concise and easy to follow alternative proof for this existing result the latter result ii establishes weak feller property of non linear filter process under conditions which have not been previously reported in the literature

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

a multilevel matrix decomposition algorithm for analyzing scattering from large structures a multilevel algorithm is presented for analyzing scattering from electrically large surfaces the algorithm accelerates the iterative solution of integral equations that arise in computational electromagnetics the algorithm permits a fast matrix vector multiplication by decomposing the traditional method of moment matrix into a large number of blocks with each describing the interaction between distant scatterers the multiplication of each block by a trial solution vector is executed using a multilevel scheme that resembles a fast fourier transform fft and that only relies on well known algebraic techniques the computational complexity and the memory requirements of the proposed algorithm are o n log sup 2 n

on the optimality of symbol by symbol filtering and denoising we consider the problem of optimally recovering a finite alphabet discrete time stochastic process x sub t from its noise corrupted observation process z sub t in general the optimal estimate of x sub t will depend on all the components of z sub t on which it can be based we characterize nontrivial situations i e beyond the case where x sub t z sub t are independent for which optimum performance is attained using symbol by symbol operations a k a singlet decoding meaning that the optimum estimate of x sub t depends solely on z sub t for the case where x sub t is a stationary binary markov process corrupted by a memoryless channel we characterize the necessary and sufficient condition for optimality of symbol by symbol operations both for the filtering problem where the estimate of x sub t is allowed to depend only on z sub t sub t spl les t and the denoising problem where the estimate of x sub t is allowed dependence on the entire noisy process it is then illustrated how our approach which consists of characterizing the support of the conditional distribution of the noise free symbol given the observations can be used for characterizing the entropy rate of the binary markov process corrupted by the binary symmetric channel bsc in various asymptotic regimes for general noise free processes not necessarily markov general noise processes not necessarily memoryless and general index sets random fields we obtain an easily verifiable sufficient condition for the optimality of symbol by symbol operations and illustrate its use in a few special cases for example for binary processes corrupted by a bsc we establish under mild conditions the existence of a spl delta sup 0 such that the say what you see scheme is optimal provided the channel crossover probability is less than spl delta sup finally we show how for the case of a memoryless channel the large deviations ld performance of a symbol by symbol filter is easy to obtain thus characterizing the ld behavior of the optimal schemes when these are singlet decoders and constituting the only known cases where such explicit characterization is available

multilevel fast adaptive cross approximation algorithm with characteristic basis functions this paper presents a multilevel fast adaptive cross approximation mlfaca algorithm for accelerated iterative solution of the method of moments mom matrix equation for electrically large targets the mlfaca compresses the impedance submatrices between well separated blocks into products of sparse matrices constructed with the aid of the fast adaptive cross sampling facs scheme and the butterfly algorithm as a result the mlfaca can reduce both the computational time and the storage of the mom to mathbf o mathbf n mathbf log mathbf 2 text n where mathbf n is the number of the rao wilton glisson rwg basis functions in the analyzed target meanwhile the mlfaca maintains the adaptive and kernel independent properties furthermore the characteristic basis function method cbfm is employed to decrease the size of the outer matrices of the mlfaca to further reduce the storage and iteration time numerical results are presented to demonstrate the advantages of the proposed method including a successful solution of a scattering problem involving 10 861 668 rwg basis functions

asymptotic stability of the wonham filter ergodic and nonergodic signals the stability problem of the wonham filter with respect to initial conditions is addressed the case of ergodic signals is revisited in view of a gap in the classic work of h kunita 1971 we give new bounds for the exponential stability rates which do not depend on the observations in the nonergodic case the stability is implied by identifiability conditions formulated explicitly in terms of the transition intensities matrix and the observation structure

analysis of partial geometry modification problems using the partitioned inverse formula and sherman morrison woodbury formula based method in this paper we present an efficient method for accelerated analysis of the partial geometry modification problem involving an original structure with small modifications the conventional method of moments can be very time consuming for this type of problem because the impedance matrix equation has to be solved anew each time the original structure is modified however the proposed method only requires the solution of the impedance matrix equation of the original structure any small modification of the original object is handled into two steps first by subtracting a small part from the original structure and then by adding a small part to the rest structure both of these steps can be efficiently computed by using the partitioned inverse and sherman morrison woodbury formulas with the solution of the original structure when the structure is modified the proposed method only requires additional operations whose computational burden is o n 2 as opposed to o n 3 where n is the number of unknowns in the original structure furthermore the presented method is purely algebraic and rigorous rather than based on approximations numerical results for electromagnetic scattering are included in this paper to demonstrate the efficiency and accuracy of this method

multilevel adaptive cross approximation mlaca the multilevel adaptive cross approximation mlaca is proposed as a fast method to accelerate the matrix vector products in the iterative solution of the linear system that results from the discretization of electromagnetic integral equations ie with the method of moments mom the single level aca already described in the literature is extended with a multilevel recursive algorithm in order to improve the asymptotical complexity of both the computational cost and the memory requirements the main qualities of aca are maintained it is purely algebraic and independent of the integral equation kernel green s function as long as it produces pseudo rank deficient matrix blocks the algorithm is presented in such a way that it can be easily implemented on top of an existing mom code with most commonly used green s functions

molecular modeling of proteins and mathematical prediction of protein structure this paper discusses the mathematical formulation of and solution attempts for the so called protein folding problem the static aspect is concerned with how to predict the folded native tertiary structure of a protein given its sequence of amino acids the dynamic aspect asks about the possible pathways to folding and unfolding including the stability of the folded protein from a mathematical point of view there are several main sides to the static problem the selection of an appropriate potential energy function the parameter identification by fitting to experimental data and the global optimization of the potential the dynamic problem entails in addition the solution of because of multiple time scales very stiff ordinary or stochastic differential equations molecular dynamics simulation or in case of constrained molecular dynamics of differential algebraic equations a theme connecting the static and dynamic aspect is the determination and formation of secondary structure motifs the present paper gives a self contained introduction to the necessary background from physics and chemistry and surveys some of the literature it also discusses the various mathematical problems arising some deficiencies of the current models and algorithms and possible past and future attacks to arrive at solutions to the protein folding problem

error bound of the multilevel adaptive cross approximation mlaca an error bound of the multilevel adaptive cross approximation mlaca which is a multilevel version of the adaptive cross approximation singular value decomposition aca svd is rigorously derived for compressing an off diagonal submatrix of the method of moments mom impedance matrix with a binary tree the l level mlaca includes l 1 steps and each step includes 2 l aca svd decompositions if the relative frobenius norm error of the aca svd used in the mlaca is smaller than varepsilon the rigorous proof in this communication shows that the relative frobenius norm error of the l level mlaca is smaller than 1 varepsilon l 1 1 in practical applications the error bound of the mlaca can be approximated as varepsilon l 1 because varepsilon is always ll 1 the error upper bound can be used to control the accuracy of the mlaca to ensure an error of the l level mlaca smaller than varepsilon for different l the aca svd threshold can be set to 1 varepsilon tfrac 1 l 1 1 which approximately equals varepsilon l 1 for practical applications

approximation and limit results for nonlinear filters over an infinite time interval part ii random sampling algorithms the paper is concerned with approximations to nonlinear filtering problems that are of interest over a very long time interval since the optimal filter can rarely be constructed one needs to compute with numerically feasible approximations the signal model can be a jump diffusion reflected or not the observations can be taken either in discrete or continuous time the cost of interest is the pathwise error per unit time over a long time interval in a previous paper of the authors a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 it was shown under quite reasonable conditions on the approximating filter and on the signal and noise processes that as time bandwidth process and filter approximation etc go to their limit in any way at all the limit of the pathwise average costs per unit time is just what one would get if the approximating processes were replaced by their ideal values and the optimal filter was used when suitable approximating filters cannot be readily constructed due to excessive computational requirements or to problems associated with a high signal dimension approximations based on random sampling methods or perhaps combinations of sampling and analytical methods become attractive and are the subject of a great deal of attention the work of the previous paper is extended to a wide class of such algorithms under quite broad conditions covering virtually all the cases considered to date it is shown that the pathwise average errors converge to the same limit that would be obtained if the optimal filter was used as time goes to infinity and the approximation parameter goes to its limit in any way at all all the extensions e g wide bandwidth observation or system driving noise in a budhiraja and h j kushner siam j control optim 37 1999 pp 1946 1979 hold for our random sampling algorithms as well