learning overcomplete representations in an overcomplete basis the number of basis vectors is greater than the dimensionality of the input and the representation of an input is not a unique combination of basis vectors overcomplete representations have been advocated because they have greater robustness in the presence of noise can be sparser and can have greater flexibility in matching structure in the data overcomplete codes have also been proposed as a model of some of the response properties of neurons in primary visual cortex previous work has focused on finding the best representation of a signal using a fixed overcomplete basis or dictionary we present an algorithm for learning an overcomplete basis by viewing it as probabilistic model of the observed data we show that overcomplete bases can yield a better approximation of the underlying statistical distribution of the data and can thus lead to greater coding efficiency this can be viewed as a generalization of the technique of independent component analysis and provides a method for bayesian reconstruction of signals in the presence of noise and for blind source separation when there are more sources than mixtures

estimation of discretization errors using the method of nearby problems the method of nearby problems is developed as an approach for estimating numerical errors due to insufficient mesh resolution a key aspect of this approach is the generation of accurate analytic curve fits to an underlying numerical solution accurate fits are demonstrated using fifth order hermite splines that provide for solution continuity up to the third derivative which is recommended for second order differential equations this approach relies on the generation of a new problem and corresponding exact solution that is nearby the original problem of interest and the nearness requirements are discussed the method of nearby problems is demonstrated as an accurate discretization error estimator for steady state burgers s equation for a viscous shock wave at reynolds numbers of 8 and 64 a key advantage of using the method of nearby problems as an error estimator is that it requires only one additional solution on the same mesh as opposed to multiple mesh solutions required for extrapolation based error estimators furthermore the present results suggest that the method of nearby problems can produce better error estimates than other methods in the preasymptotic regime the method of nearby problems is also shown to provide a useful framework for evaluating other discretization error estimators this framework is demonstrated by the generation of exact solutions to problems nearby burgers s equation as well as a form of burgers s equation with a nonlinear viscosity variation

h 1 scattering for systems of n defocusing weakly coupled nls equations in low space dimensions abstract we prove that the scattering operators and wave operators are well defined in the energy space for the system of defocusing schrodinger equations i t u u 1 n u p 1 u p 1 u 0 1 n u 0 1 n u 0 1 n h 1 r d n with n 2 0 0 for p 2 if d 1 p 1 if d 2 and 1 p 2 if d 3

infinitely many solutions to quasilinear schrodinger system in r n in this work we study the existence of multiple solutions to the quasilinear schrodinger system 0 1 p u a x u p 2 u d 1 f u x u v u m 2 u x r n q v b x v q 2 v d 1 f v x u v v m 2 v x r n u w 1 p r n v w 1 q r n where n 3 1 p q n 0 and m d q p r the potential functions a x b x l r n are positive in r n a major point is that we use the technique in chen 2015 to verify the p s conditions and then apply a version of mountain pass lemma to prove the existence of infinitely many solutions of system 0 1

analysis and synthesis of multicomponent signals using positive time frequency distributions a new approach to the analysis and reconstruction of multicomponent nonstationary signals from their time frequency distribution tfd is presented specifically we consider a tfd based on the recently introduced minimum cross entropy principle mce this positive tfd is cross terms free and hence has an advantage over the family of bilinear distributions based on the mce tfd a new algorithm for reconstructing the phase and amplitude parameters of each component of the signal is developed to evaluate the accuracy of the algorithm monte carlo simulations are presented and compared with the corresponding cramer rao bound it is shown that the new algorithm is superior to presently available methods in both efficiency and performance it is concluded that together with the mce tfd representation the proposed approach provides a powerful tool for analysis of nonstationary multicomponent signals embedded in additive gaussian noise

blind source separation based on time frequency signal representations blind source separation consists of recovering a set of signals of which only instantaneous linear mixtures are observed thus far this problem has been solved using statistical information available on the source signals this paper introduces a new blind source separation approach exploiting the difference in the time frequency t f signatures of the sources to be separated the approach is based on the diagonalization of a combined set of spatial t f distributions in contrast to existing techniques the proposed approach allows the separation of gaussian sources with identical spectral shape but with different t f localization properties the effects of spreading the noise power while localizing the source energy in the t f domain amounts to increasing the robustness of the proposed approach with respect to noise and hence improved performance asymptotic performance analysis and numerical simulations are provided

time frequency approach to underdetermined blind source separation this paper presents a new time frequency tf underdetermined blind source separation approach based on wigner ville distribution wvd and khatri rao product to separate n non stationary sources from m m n mixtures first an improved method is proposed for estimating the mixing matrix where the negative value of the auto wvd of the sources is fully considered then after extracting all the auto term tf points the auto wvd value of the sources at every auto term tf point can be found out exactly with the proposed approach no matter how many active sources there are as long as n 2m 1 further discussion about the extraction of auto term tf points is made and finally the numerical simulation results are presented to show the superiority of the proposed algorithm by comparing it with the existing ones

positive solutions for a weakly coupled nonlinear schrodinger system abstract existence of a nontrivial solution is established via variational methods for a system of weakly coupled nonlinear schrodinger equations the main goal is to obtain a positive solution of minimal action if possible with all vector components not identically zero generalizations for nonautonomous systems are considered

special set and solutions of coupled nonlinear schrodinger equations a special set of n coupled nonlinear schrodinger equations which consists of 2n interaction types each of which is characterized by a specific array of interaction parameters is presented it is shown that every lame function of order n n is a solution for one or more components for one or more interaction types of this special set simple rules that relate interaction and solution types and interesting features of these relationships are presented

least energy solitary waves for a system of nonlinear schrodinger equations in mathbb r n in this paper we consider systems of coupled schrodinger equations which appear in nonlinear optics the problem has been considered mostly in the one dimensional case here we make a rigorous study of the existence of least energy standing waves solitons in higher dimensions we give conditions on the parameters of the system under which it possesses a solution with least energy among all multi component solutions conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie

performance analysis of minimum spl lscr sub 1 norm solutions for underdetermined source separation results of the analysis of the performance of minimum spl lscr sub 1 norm solutions in underdetermined blind source separation that is separation of n sources from m n linearly mixed observations are presented in this paper the minimum spl lscr sub 1 norm solutions are known to be justified as maximum a posteriori probability map solutions under a laplacian prior previous works have not given much attention to the performance of minimum spl lscr sub 1 norm solutions despite the need to know about its properties in order to investigate its practical effectiveness we first derive a probability density of minimum spl lscr sub 1 norm solutions and some properties we then show that the minimum spl lscr sub 1 norm solutions work best in a case in which the number of simultaneous nonzero source time samples is less than the number of sensors at each time point or in a case in which the source signals have a highly peaked distribution we also show that when neither of these conditions is satisfied the performance of minimum spl lscr sub 1 norm solutions is almost the same as that of linear solutions obtained by the moore penrose inverse our results show when the minimum spl lscr sub 1 norm solutions are reliable

new existence and symmetry results for least energy positive solutions of schrodinger systems with mixed competition and cooperation terms abstract in this paper we focus on existence and symmetry properties of solutions to the cubic schrodinger system u i i u i j 1 d i j u j 2 u i in r n i 1 d where d 2 i i i 0 i j j i r for j i n 2 3 the underlying domain is either bounded or the whole space and u i h 0 1 or u i h rad 1 r n respectively we establish new existence and symmetry results for least energy positive solutions in the case of mixed cooperation and competition coefficients as well as in the purely cooperative case

the computation of ck spline functions abstract we present in this paper an algorithm for computing ck spline functions i e spline functions which generate k time continuous and derivable approximations these functions seem to be very efficient in several applied nonlinear differential problems like nonlinear control or nonlinear simulation

positive solutions to some systems of coupled nonlinear schrodinger equations abstract we study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time independent schrodinger equations u j j u j j u j 3 k 1 k j n j k u j u k 2 u j w 1 2 r n j 1 n where n 1 2 3 j j 0 for j 1 n the coupling parameters j k k j r for j k 1 n j k precisely we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors additionally with more restrictive conditions on the coupled parameters we show that the bound states founded are positive

improved time frequency representation of multicomponent signals using exponential kernels the authors introduce a time frequency distribution of l cohen s 1966 class and examines its properties this distribution is called exponential distribution ed after its exponential kernel function first the authors interpret the ed from the spectral density estimation point of view they then show how the exponential kernel controls the cross terms as represented in the generalized ambiguity function domain and they analyze the ed for two specific types of multicomponent signals sinusoidal signals and chirp signals next they define the ed for discrete time signals and the running windowed exponential distribution rwed which is computationally efficient finally the authors present numerical examples of the rwed using the synthetically generated signals it is found that the ed is very effective in diminishing the effects of cross terms while retaining most of the properties which are useful for a time frequency distribution

existence and nonexistence of bound state solutions for schrodinger systems with linear and nonlinear couplings abstract we study the schrodinger systems with linear and nonlinear coupling terms doubly coupled nonlinear schrodinger system for short which arise naturally in nonlinear optics and in the hartree fock theory for bose einstein condensates among other physical problems first for small linear coupling constant we get existence of a nontrivial bound state solution to the system via perturbation method furthermore we prove each component of the bound state solution is nonnegative by energy estimate second we establish a version of pohozaev nehari identity and prove a nonexistence result for the more general system when the spatial dimension n 4

semitrivial vs fully nontrivial ground states in cooperative cubic schrodinger systems with d 3 equations abstract in this work we consider the weakly coupled schrodinger cubic system u i i u i i u i 3 u i j i b i j u j 2 u i h 1 r n r i 1 d where 1 n 3 i i 0 and b i j b j i 0 for i j this system admits semitrivial solutions that is solutions u u 1 u d with null components we provide optimal qualitative conditions on the parameters i i and b i j under which the ground state solutions have all components nontrivial or conversely are semitrivial this question had been clarified only in the d 2 equations case for d 3 equations prior to the present paper only very restrictive results were known namely when the above system was a small perturbation of the super symmetrical case i and b i j b we treat the general case uncovering in particular a much more complex and richer structure with respect to the d 2 case

ground state of n coupled nonlinear schrodinger equations in mathbb r n n leq 3 we establish some general theorems for the existence and nonexistence of ground state solutions of steady state n coupled nonlinear schrodinger equations the sign of coupling constants ij s is crucial for the existence of ground state solutions when all ij s are positive and the matrix is positively definite there exists a ground state solution which is radially symmetric however if all ij s are negative or one of ij s is negative and the matrix is positively definite there is no ground state solution furthermore we find a bound state solution which is non radially symmetric when n 3

blind source separation by sparse decomposition in a signal dictionary the blind source separation problem is to extract the underlying source signals from a set of linear mixtures where the mixing matrix is unknown this situation is common in acoustics radio medical signal and image processing hyperspectral imaging and other areas we suggest a two stage separation process a priori selection of a possibly overcomplete signal dictionary for instance a wavelet frame or a learned dictionary in which the sources are assumed to be sparsely representable followed by unmixing the sources by exploiting the their sparse representability we consider the general case of more sources than mixtures but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques

exploitation of source nonstationarity in underdetermined blind source separation with advanced clustering techniques the problem of blind source separation bss is investigated following the assumption that the time frequency tf distributions of the input sources do not overlap quadratic tf representation is used to exploit the sparsity of the statistically nonstationary sources however separation performance is shown to be limited by the selection of a certain threshold in classifying the eigenvectors of the tf matrices drawn from the observation mixtures two methods are therefore proposed based on recently introduced advanced clustering techniques namely gap statistics and self splitting competitive learning sscl to mitigate the problem of eigenvector classification the novel integration of these two approaches successfully overcomes the problem of artificial sources induced by insufficient knowledge of the threshold and enables automatic determination of the number of active sources over the observation the separation performance is thereby greatly improved practical consequences of violating the tf orthogonality assumption in the current approach are also studied which motivates the proposal of a new solution robust to violation of orthogonality in this new method the tf plane is partitioned into appropriate blocks and source separation is thereby carried out in a block by block manner numerical experiments with linear chirp signals and gaussian minimum shift keying gmsk signals are included which support the improved performance of the proposed approaches

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

dictionary learning algorithms for sparse representation algorithms for data driven learning of domain specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of bayesian models with concave schur concave csc negative log priors such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen environmentally matched dictionary the elements of the dictionary can be interpreted as concepts features or words capable of succinct expression of events encountered in the environment the source of the measured signals this is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries the proverbial 25 words or less but not necessarily as succinct as one entry to learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment we develop algorithms that iterate between a representative set of sparse representations found by variants of focuss and an update of the dictionary using these sparse representations experiments were performed using synthetic data and natural images for complete dictionaries we demonstrate that our algorithms have improved performance over other independent component analysis ica methods measured in terms of signal to noise ratios of separated sources in the overcomplete case we show that the true underlying dictionary and sparse sources can be accurately recovered in tests with natural images learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries that is images encoded with an overcomplete dictionary have both higher compression fewer bits per pixel and higher accuracy lower mean square error

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing