the painleve property for partial differential equations in this paper we define the painleve property for partial differential equations and show how it determines in a remarkably simple manner the integrability the backlund transforms the linearizing transforms and the lax pairs of three well known partial differential equations burgers equation kdv equation and the modified kdv equation this indicates that the painleve property may provide a unified description of integrable behavior in dynamical systems ordinary and partial differential equations while at the same time providing an efficient method for determining the integrability of particular systems

a new integral equation method to solve highly nonlinear inverse scattering problems a family of new integral equations nie is proposed in this paper which are transformed from the original lippmann schwinger integral equation it can be shown that the nie can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects introduced by the multiple scattering behaviors in estimating the contrast equipping the previously proposed twofold subspace based optimization method with such nie the new inversion method is able to solve inverse scattering problems with strong scatterers like with high contrast and or large dimensions in terms of wavelength ones furthermore such a family of nie could provide a convenient tool to appraise reconstructed results several representative numerical tests are carried out using both synthetic and experimental data to verify the efficacy of the new inversion method

a new twist two step iterative shrinkage thresholding algorithms for image restoration iterative shrinkage thresholding 1st algorithms have been recently proposed to handle a class of convex unconstrained optimization problems arising in image restoration and other linear inverse problems this class of problems results from combining a linear observation model with a nonquadratic regularizer e g total variation or wavelet based regularization it happens that the convergence rate of these 1st algorithms depends heavily on the linear observation operator becoming very slow when this operator is ill conditioned or ill posed in this paper we introduce two step 1st twist algorithms exhibiting much faster convergence rate than 1st for ill conditioned problems for a vast class of nonquadratic convex regularizers lscrp norms some besov norms and total variation we show that twist converges to a minimizer of the objective function for a given range of values of its parameters for noninvertible observation operators we introduce a monotonic version of twist mtwist although the convergence proof does not apply to this scenario we give experimental evidence that mtwist exhibits similar speed gains over ist the effectiveness of the new methods are experimentally confirmed on problems of image deconvolution and of restoration with missing samples

an algebraic solution method for nonlinear inverse scattering by using properly designed synthetic or virtual experiments and an original approximation of the contrast sources we are able to recast the inverse scattering problem in an algebraic form in a subset of points of the imaged domain and hence to solve it by means of closed form formulas the new approximation relies on the assumption that the contrast sources induced by the different virtual experiments are focused in given points belonging to the scatterer as such the method involves a preprocessing step in which the outcome of the original scattering experiments is recombined into the new virtual ones capable of enforcing the expected contrast sources behavior examples with numerical and experimental data are provided to assess the actual possibility of setting such a virtual experiments framework and show the effectiveness of the proposed method

existence of nodal solutions of a nonlinear eigenvalue problem with indefinite weight function abstract we investigate the existence of nodal solutions of an indefinite weight boundary value problem u r h t f u 0 0 t 1 u 0 u 1 0 where h c 0 1 changes sign the proof of our main result is based upon bifurcation techniques

microwave medical imaging based on sparsity and an iterative method with adaptive thresholding we propose a new image recovery method to improve the resolution in microwave imaging applications scattered field data obtained from a simplified breast model with closely located targets is used to formulate an electromagnetic inverse scattering problem which is then solved using the distorted born iterative method dbim at each iteration of the dbim method an underdetermined set of linear equations is solved using our proposed sparse recovery algorithm imatcs our results demonstrate the ability of the proposed method to recover small targets in cases where traditional dbim approaches fail furthermore in order to regularize the sparse recovery algorithm we propose a novel l 2 based approach and prove its convergence the simulation results indicate that the l 2 regularized method improves the robustness of the algorithm against the ill posed conditions of the em inverse scattering problem finally we demonstrate that the regularized imatcs dbim approach leads to fast accurate and stable reconstructions of highly dense breast compositions

pdeptest a package for the painleve test of nonlinear partial differential equations the so called wtc kruskal algorithm is presented in order to study the painleve integrability of nonlinear partial differential equations which combines the wtc method and kruskal s simplification method based on the wtc kruskal and wtc kruskal algorithms we give an implementation in maple called pdeptest this package can handle nonlinear partial differential equations which can be transformed to polynomial ones and its effectiveness is illustrated by applying it to a variety of equations some new results are reported

piecewise smoothed value picking regularization applied to 2 d tm and te inverse scattering the stepwise relaxed value picking srvp regularization technique proposed earlier for the iterative reconstruction of piecewise quasi homogeneous objects is a non spatial technique whereby the reconstruction unknowns are clustered around a limited number of a priori unknown reference values artifacts have been observed in some 2 d and 3d complex permittivity reconstructions this paper therefore combines the non spatial srvp technique with a spatial smoothing technique whereby the reference values provided by the former in each iteration are employed by the latter to define separate smoothing regions this way edges are preserved since the spatial smoothing constraints in the cost function are active within but not across the region boundaries this combined technique denoted as stepwise relaxed piecewise smoothed value picking srpsvp regularization is formulated for the 2 5d microwave inverse scattering problem and is illustrated with reconstructions from the institut fresnel 2 d scattering database

classification of certain integrable coupled potential kdv and modified kdv type equations in this article we describe the classification of integrable symmetrically coupled potential kdv and modified kdv type equations that possess higher symmetries restricting our attention to the systems that cannot be decoupled by a change of dependent variables we obtain 11 previously unknown classes of integrable equations in some cases we present hamiltonian or bi hamiltonian formulations

microwave imaging within the first order born approximation by means of the contrast field bayesian compressive sensing a new approach based the contrast field cf formulation of the microwave imaging problem that exploits the bayesian compressive sampling bcs paradigm is proposed for the reconstruction of sparse distributions of weak scatterers towards this end the original inverse scattering problem is recast to a probabilistic sparseness constrained optimization by introducing suitable hierarchical priors as sparsity constraints a fast relevance vector machine rvm is then employed to reconstruct the scatterers as well as to estimate the confidence level of the inversion representative numerical results are presented to illustrate the method as well as to assess its potentialities and limitations in terms of inversion accuracy computational efficiency and robustness comparisons with state of the art deterministic and stochastic reconstruction methodologies still within the born approximation ba are discussed as well

a hybrid quantitative method for inverse scattering of multiple dielectric objects a hybrid method is proposed to locate the boundaries and determine the dielectric constants of multiple homogeneous scatterers by postprocessing the results of the linear sampling method lsm the hybrid method is composed of three steps combining the capabilities of the lsm the multiple signal classification music algorithm and the generalized multipole technique gmt initially a new measure of the noise level of the multistatic response msr matrix is proposed subsequently the lsm is applied using the tikhonov regularization then the scattered field is back propagated into the investigation domain using the gmt where in order to ensure the stability a music based approach is exploited to determine the optimum reconstructing multipole sources the case of limited scattered field data is also addressed by introducing a modification of the papoulis gerchberg algorithm pga finally the boundary and the dielectric constant of each scatterer are obtained simultaneously by solving an inverse boundary value problem through an optimization the proposed hybrid approach is capable of efficiently approximating the dielectric constant of multiple homogeneous dielectric scatterers in a parallel manner the effectiveness of the proposed method and the accuracy of the solutions are demonstrated by applying the method to both numerical and measured data

a numerical investigation of sign changing solutions to superlinear elliptic equations on symmetric domains in this paper we investigate numerically sign changing solutions of superlinear elliptic equations on symmetric domains based upon the symmetric criticality principle of palais the existence of sign changing solutions which re2ect the symmetry

numerical simulation of singularly perturbed non linear elliptic boundary value problems using finite element method the present paper presents the finite element solution of two dimensional non linear singularly perturbed elliptic partial differential equation subject to appropriate dirichlet boundary conditions a new fifth order convergent newton type iterative method has been described and used to linearize the non linear problem the inclusion of this newton s method of fifth order convergence in finite element method for solving non linear system of equations reduces the number of iterations and hence the cost of computation to demonstrate the usefulness of the proposed scheme a non convex variational ginzburg landau equation is considered

positive mountain pass solutions for a semilinear elliptic equation with a sign changing weight function abstract we use the mountain pass theorem to study the existence and multiplicity of positive solutions of the generalisation of the well known logistic equation u g x u x 1 u x with dirichlet boundary conditions to the case where g changes sign

global bifurcation of positive solutions of a second order periodic boundary value problem with indefinite weight abstract in this paper we are concerned with the global structure and stability of positive solutions of periodic boundary value problem u t q t u t a t f u t 0 t 2 u 0 u 2 u 0 u 2 where q c r 0 is of periodic 2 and q t 0 t 0 2 a c r r is of periodic 2 and changes sign the proof of our main results are based on bifurcation techniques

a finite element approach for finding positive solutions of a logistic equation with a sign changing weight function in this article we describe the mathematical modeling and use finite element approximation to study the existence and multiplicity of numerical positive solutions of well known generalized logistic equation with sign changing weight function we investigate the range of the real parameter involved in this problem to achieve the numerical solutions and discuss the behavior of the branch of the solutions using matlab 7 0

algebraic geometrical solutions of some multidimensional nonlinear evolution equations the known 2 1 dimensional breaking soliton equation the coupled kp equation with three potentials and a new 3 1 dimensional nonlinear evolution equation are decomposed into systems of solvable ordinary differential equations with the help of the 1 1 dimensional akns equations the abel jacobi coordinates are introduced to straighten out the associated flows from which algebraic geometrical solutions of the 2 1 dimensional breaking soliton equation the coupled kp equation and the 3 1 dimensional evolution equation are explicitly given in terms of the riemann theta functions

2 1 dimensional m n component akns system painleve integrability infinitely many symmetries similarity reductions and exact solutions the 2 1 dimensional m n component akns system that is derived from the inner parameter dependent symmetry constraint of the kp equation is studied in detail first the painleve integrability of the model is proved by using the standard wtc and kruskal approach using the formal series symmetry approach the generalized kmv symmetry algebra and the related symmetry group are found the two dimensional similarity partial differential equation reductions and the ordinary differential equation reductions are obtained from the generalized kmv symmetry algebra and the direct method abundant localized coherent structures are revealed by the variable separation approach some special types of the localized excitations like the multiple solitoffs dromions lumps ring solitons breathers and instantons are plotted also

microwave imaging via distorted iterated virtual experiments the linearity of the scattering phenomenon with respect to primary sources allows to recombine a posteriori the available experiments and build in a synthetic fashion new virtual experiments starting from this circumstance an iterative procedure is proposed as an effective approach to tackle nonlinear inverse scattering problems in this procedure the virtual experiments the green s function and the corresponding physical inspired field approximations are updated at each iteration the structure and the complexity of the approach are comparable with those of the widely adopted distorted born iterative method but its performances are remarkably better thanks to extended validity of the exploited field approximation the overall approach also takes advantage of a compressive sensing inspired regularization scheme to promote sparsity in the search of piecewise constant dielectric profiles and further improve the accuracy of the imaging results examples with numerical and experimental data are given to assess the method

reconstruction of two dimensional permittivity distribution using the distorted born iterative method the distorted born iterative method dbim is used to solve two dimensional inverse scattering problems thereby providing another general method to solve the two dimensional imaging problem when the born and the rytov approximations break down numerical simulations are performed using the dbim and the method proposed previously by the authors int j imaging syst technol vol 1 no 1 p 100 8 1989 called the born iterative method bim for several cases in which the conditions for the first order born approximation are not satisfied the results show that each method has its advantages the dbim shows faster convergence rate compared to the bim while the bim is more robust to noise contamination compared to the dbim

the linear sampling method as a way to quantitative inverse scattering the linear sampling method lsm is a simple and effective approach to image the shape of unknown targets via the solution of a linear inverse problem in this paper we show that the lsm can also be exploited to devise a novel effective approximation of the scattering phenomenon that leads to a new noniterative linear inversion method for the estimation of the target s electric contrast since the introduced approximation relies on the broad applicability of the lsm the proposed inversion method is suitable to tackle inverse scattering problems involving nonweak scatterers as such it represents an innovative yet effective way to tackle quantitative imaging examples with numerical and experimental data are given to show the performance of the approach in particular results obtained with fresnel data sets show that the proposed method is capable of successfully imaging targets which have been so far processed using nonlinear iterative schemes and taking advantage of frequency diversity

inverse scattering using a joint l1 l2 norm based regularization inverse scattering problems suffer from ill posedness and ill conditioning necessitating the use of regularization methods to get meaningful solutions commonly used regularizations are l2 norm based but these generate over smooth solutions we propose a regularization method using both the l1 and l2 norms to obtain sharp object boundaries while also achieving good interior reconstruction of the object permittivity knowledge about the permittivity can also be used as a priori information the applicability of the method is demonstrated on synthetically generated data for two dimensional 2 d microwave imaging using the born iterative method bim the optimization routine systematically estimates all parameters while minimizing the cost function different objects chosen to represent realistic features have been considered to evaluate the performance the reconstructed images indicate that the method can produce accurate object localization shape identification and good permittivity estimation

a unified approach to painleve expansions abstract the painleve test for partial differential equations developed by weiss tabor and carnevale wtc is examined in detail and shown to provide a unified approach to the integrable properties of both ordinary and partial differential equations a simple modification of the wtc procedure used for partial differential equations enables us to determine the lax pairs hirota equations and auto backlund transformations for ordinary differential equations including a new lax pair for an integrable case of the henon heiles system a detailed study of the kdv hierarchy is made and a complete picture of the pattern of resonances for all solution branches is obtained the role of the singular branches is examined in detail and important new insights obtained in particular we find that each singular branch is simply a re expansion of the principal branch about a point on the pole manifold at which several isolated poles coalesce a parallel analysis is carried out for the akns hierarchy

l_ 2 regularized iterative weighted algorithm for inverse scattering we propose a new inverse scattering technique based on sparsity for the application of microwave imaging the underdetermined inverse problem appeared in the distorted born iterative method dbim technique is solved using the suggested l 2 regularized iterative weighted algorithm l 2 iwa the l 2 regularizer has been introduced to stabilize the algorithm against nonlinear approximations and the sparsity is enforced with the aid of another reweighted l 2 norm regularizer to address the ill posedness of the inverse problem the derived algorithm is a three step iterative technique which solves the underdetermined set of equations at each dbim iteration moreover the convergence of the l 2 iwa technique is proved analytically the suggested method outperforms its other counterparts in various scenarios of homogeneous and heterogeneous breast models besides improving the resolution of the breast tumors the l 2 iwa technique is shown to be robust against additive noise

existence and multiplicity of positive solutions of a nonlinear eigenvalue problem with indefinite weight function this paper is concerned with the existence multiplicity and stability of positive solutions of an indefinite weight boundary value problemu email protected t f u 0 0

a sparsity regularization approach to the electromagnetic inverse scattering problem we investigate solving the electromagnetic inverse scattering problem using the distorted born iterative method dbim in conjunction with a variable selection approach known as the elastic net the elastic net applies both l 1 and l 2 penalties to regularize the system of linear equations that result at each iteration of the dbim the elastic net thus incorporates both the stabilizing effect of the l 2 penalty with the sparsity encouraging effect of the l 1 penalty the dbim with the elastic net outperforms the commonly used l 2 regularizer when the unknown distribution of dielectric properties is sparse in a known set of basis functions we consider two very different 3 d examples to demonstrate the efficacy and applicability of our approach for both examples we use a scalar approximation in the inverse solution in the first example the actual distribution of dielectric properties is exactly sparse in a set of 3 d wavelets the performances of the elastic net and l 2 approaches are compared to the ideal case where it is known a priori which wavelets are involved in the true solution the second example comes from the area of microwave imaging for breast cancer detection for a given set of 3 d gaussian basis functions we show that the elastic net approach can produce a more accurate estimate of the distribution of dielectric properties in particular the effective conductivity within an anatomically realistic 3 d numerical breast phantom in contrast the dbim with an l 2 penalty produces an estimate which suffers from multiple artifacts

numerical solution of the three dimensional advection diffusion equation the study of advection diffusion equation continues to be an active field of research the subject has important applications to fluid dynamics as well as many other branches of science and engineering in this paper several different numerical techniques will be developed and compared for solving the three dimensional advection diffusion equation with constant coefficient these techniques are based on the two level fully explicit and fully implicit finite difference approximations the basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach developed from the 1974 work of warming and hyett this allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes another nice feature of the modified equivalent partial differential equation approach is that a high order of accuracy can be combined with excellent stability properties the new second order accurate methods are free of numerical diffusion the results of a numerical experiment are presented and the accuracy and central processor cpu time needed are discussed and compared

shrinkage thresholding enhanced born iterative method for solving 2d inverse electromagnetic scattering problem a numerical framework that incorporates recently developed iterative shrinkage thresholding ist algorithms within the born iterative method bim is proposed for solving the two dimensional inverse electromagnetic scattering problem ist algorithms minimize a cost function weighted between measurement data misfit and a zeroth first norm penalty term and therefore promote sharpness in the solution consequently when applied to domains with sharp variations discontinuities or sparse content the proposed framework is more efficient and accurate than the classical bim that minimizes a cost function with a second norm penalty term indeed numerical results demonstrate the superiority of the ist bim over the classical bim when they are applied to sparse domains permittivity and conductivity profiles recovered using the ist bim are sharper and more accurate and converge faster

comparison of an enhanced distorted born iterative method and the multiplicative regularized contrast source inversion method for 2d transverse magnetic tm microwave inversion multiplicative regularized contrast source inversion mr csi and the distorted born iterative method dbim are compared the comparison is based on a computational resource analysis inversion of synthetic data and inversion of experimentally collected data from both the fresnel and upc barcelona data sets all inversion results are blind but appropriate physical values for the reconstructed contrast are maintained the data sets used to test the algorithms vary widely in terms of the background media antennas and far near field considerations to ensure that the comparison is replicable an automatic regularization parameter selection method is used for the additive regularization within the dbim which utilizes a fast implementation of the l curve method and the laplacian regularizer while not used in the classical dbim we introduce an mr term to the dbim in order to provide comparable results to mr csi the introduction of this mr term requires only slight modifications to the classical dbim algorithm and adds little computational complexity the results show that with the addition of the mr term in the dbim the two algorithms provide very similar inversion results but with the mr csi method providing advantages for both computational resources and ease of implementation

a comparison between the born and rytov approximations for the inverse backscattering problem we compare the born and rytov approximations in solving the inverse acoustic backscattering problem i e determining medium properties from reflections for the one dimensional problem we show that the rytov approximation is generally better than the born approximation in predicting sound speed changes while both methods have the same error in determining the positions of reflectors this is shown analytically for simple models and numerically for more general models the performance of the rytov approximation is degraded when low velocity regions are present in the medium being probed the accuracy of the inversion depends on the manner in which the sound speed perturbation is linearized the location of the receiver affects the accuracy of the inversion and in the case of the rytov approximation best results are obtained when the receiver is at the interface between the known and unknown regions furthermore the rytov method is less sensitive to the choice of reference sound speed used in the in

numerical procedures for a boundary value problem with a non linear boundary condition numerical techniques based on finite difference schemes leading to parallel algorithms have been developed for obtaining approximate solutions to an initial boundary value problem for the two dimensional parabolic partial differential equation pde with a non linear boundary condition this class of parabolic pdes plays a very important role in many branches of science and engineering the non linear condition is in the form of a double integral giving the specification of mass in the solution domain not only the problem has both neumann and dirichlet boundary conditions but the dirichlet boundary condition is in a non standard form while sharing some common features with the one dimensional models the solution of two dimensional equations are substantially more difficult thus some considerations are taken to be able to extend some ideas of one dimensional case due to the structure of the boundary conditions a reduced one dimensional parabolic equation for the new unknown v y t email protected 1 0u x y t dx can be formulated the resulting problem has a non local boundary condition the new two dimensional parabolic pde with neumann s boundary conditions will be solved numerically by using the method of lines semi discretization approach the space derivatives in the pde are approximated by finite difference replacements the solution of the resulting system of first order linear ordinary differential equations satisfies a recurrence relation which involves a matrix exponential function the accuracy in time is controlled by choosing several subdiagonal pade approximants to replace this matrix exponential term numerical techniques are developed to compute the required solution using a splitting method leading to algorithms for sequential and parallel implementation the algorithms are tested on a model problem from the literature the article concludes with the results of some numerical experiments

quantitative microwave imaging method in lebesgue spaces with nonconstant exponents an innovative nonlinear inverse scattering approach in variable exponent lebesgue spaces is proposed for microwave imaging purposes the main objective of the approach is to overcome one of the main problems associated with reconstruction procedures developed in lebesgue spaces with constant exponent i e the need for the selection of the optimum lebesgue space norm parameter which is critical for obtaining accurate reconstructions and no exact rules exist for this choice the approach proposed in this paper is oriented to tomographic imaging applications at microwave frequencies for which the adopted inversion procedure is based on a gauss newton method which has been reformulated in the unconventional variable exponent lebesgue spaces the capabilities of the approach are demonstrated by a set of numerical simulations in which cylindrical dielectric targets are inspected moreover an experimental result is reported

implementation of the painleve test for ordinary differential systems abstract by definition the painleve test is the set of all techniques which enables one to generate necessary although generally not sufficient conditions for the integrability of ordinary or partial non linear differential systems in the sense of painleve in conclusive cases the results of painleve local singularity analysis provide the most fundamental knowledge related with the integrability of such systems moreover the qualitative behaviour of dynamical systems may also be probed in terms of their analytic structure the painleve test may then provide in some cases quite pertinent indications in favour of possibly chaotic dynamical behaviours in the general solution of such non linear systems the implementation of the painleve test in the algebraic programming language reduce is presented in the case of systems of non linear ordinary differential equations which are polynomials in the dependent variables and their derivatives the implementation includes a routine for the automatic search for movable singularity families and a second routine for the process of the painleve test itself either with the help of the classical and widely used kowalevskaya and gambier method or within the perturbative framework developed by conte fordy and pickering

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series