block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

monotone iterative methods for the adaptive finite element solution of semiconductor equations picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive finite element solution of semiconductor equations in terms of the slotboom variables the adaptive meshes are generated by the 1 irregular mesh refinement scheme based on these unstructured meshes and a corresponding modification of the scharfetter gummel discretization scheme it is shown that the resulting finite element stiffness matrix is an m matrix which together with the shockley read hall model for the generation recombination rate leads to an existence uniqueness comparison theorem with simple upper and lower solutions as initial iterates numerical results of simulations on a mosfet device model are given to illustrate the accuracy and efficiency of the adaptive and monotone properties of the present methods

constructive existence and uniqueness for two point boundary value problems with a linear gradient term constructive existence and uniqueness theorems are established for nonlinear two point boundary value problems governed by the equation y f x y p x y 0

a higher order compact adi method with monotone iterative procedure for systems of reaction diffusion equations this paper is concerned with an existing compact finite difference adi method published in the paper by liao et al 2002 3 for solving systems of two dimensional reaction diffusion equations with nonlinear reaction terms this method has an accuracy of fourth order in space and second order in time the existence and uniqueness of its solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear reaction terms the convergence of the finite difference solution to the continuous solution is proved an efficient monotone iterative algorithm is presented for solving the resulting discrete system and some techniques for the construction of upper and lower solutions are discussed an application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method

free vibrations for a semilinear wave equation abstract existence and regularity of nontrivial time periodic solutions are proved for semilinear wave equations under mild smoothness monotonicity and superlinearity assumptions the forced vibration case is also treated

stability and convergence of finite element approximation schemes for harmonic maps this article discusses stability and convergence of approximation schemes for harmonic maps a finite element discretization of an iterative algorithm due to f alouges is introduced and shown to be stable and convergent in general only on acute type triangulations an break a posteriori criterion is proposed which allows us to monitor sufficient conditions for weak convergence to a harmonic map on general triangulations and for adaptive mesh refinement numerical break experiments show that an adaptive strategy automatically refines triangulations in neighborhoods of typical point singularities and thereby underline its efficiency

a monotonicity procedure for the accurate efficient solution of certain heat transfer problems with variable thermal properties abstract an efficient solution procedure of high accuracy is described for the numerical solution of nonlinear boundary value problems associated with heat transfer and reaction diffusion processes constructive existence uniqueness theorems lead to newton picard iterative schemes for bounding the positive solutions of these problems the iteration schemes are discretised as matrix tridiagonal systems of error order o h2 richardson extrapolation can be exploited to produce very accurate results comparisons are made with the results of other methods on specific examples

a new class of incremental gradient methods for least squares problems the least mean squares lms method for linear least squares problems differs from the steepest descent method in that it processes data blocks one by one with intermediate adjustment of the parameter vector under optimization this mode of operation often leads to faster convergence when far from the eventual limit and to slower sublinear convergence when close to the optimal solution we embed both lms and steepest descent as well as other intermediate methods within a one parameter class of algorithms and we propose a hybrid class of methods that combine the faster early convergence rate of lms with the faster ultimate linear convergence rate of steepest descent these methods are well suited for neural network training problems with large data sets furthermore these methods allow the effective use of scaling based for example on diagonal or other approximations of the hessian matrix

method of mixed monotony for nonlinear equations with a singular linear part the existence of multiple solutions for nonlinear equations with linear singular part is proved when the nonlinear function f possesses the mixed quasimonotone property when f does not possess that property similar results are obtained by the technique of mixed monotony

accelerated monotone iterative methods for finite difference equations of reaction diffusion this paper is concerned with numerical methods for a finite difference system of reaction diffusion convection equation under nonlinear boundary condition various monotone iterative methods are presented and each of these methods leads to an existence comparison theorem as well as a computational algorithm for numerical solutions the monotone property of the iterations gives improved upper and lower bounds of the solution in each iteration and the rate of convergence of the iterations is either quadratic or nearly quadratic depending on the property of the nonlinear function application is given to a model problem from chemical engineering and some numerical results including a test problem with known analytical solution are presented to illustrate the various rates of convergence of the iterations

iterative methods for a class of complementarity problems abstract in this paper we propose and study an algorithm for a new class of complementarity problems of finding u r n such that u 0 tu a u 0 u tu a u 0 where t is a continuous mapping and a is a nonlinear transformation from r n into itself it is proved that the approximate solution obtained from the iterative scheme converges to the exact solution several special cases are also discussed

a new algorithm for computing liquid crystal stable configurations the harmonic mapping case in this article we propose a new algorithm for minimizing the energy of a nematic liquid crystal based on the equal elastic constants oseen frank model the problem reduces to finding harmonic minimizing maps that take values into the unit sphere of bbb r 3 the convergence of this algorithm is proved in a continuous setting then numerous numerical results that show its efficiency are given

a mixed boundary value problem for tft lcd analysis and numerical methods abstract we study mixed boundary value problems for lcd liquid crystal displays with chiral nematic for calculating the orientation of molecules and the light transmission existence results are established for mixed boundary value problem numerical methods for time independent and dependent problems are investigated some numerical results are also given

singular non linear two point boundary value problems existence and uniqueness abstract a general approach is presented for proving existence and uniqueness of solutions to the singular boundary value problem y x m x y x f x y x x 0 1 y 0 0 a y 1 b y 1 c a 0 b c 0 the proof is constructive in nature and could be used for numerical generation of the solution the only restriction placed on f x y is that it not be a singular function of the independent variable x singularities in y are easily avoided solutions are found in finite regions where f y 0 using an integral equation whose green s function contains an adjustable parameter that secures convergence of the picard iterative sequence methods based on the theory are developed and applied to a set of problems that have appeared previously in published works

higher order compact finite difference method for systems of reaction diffusion equations this paper is concerned with a compact finite difference method for solving systems of two dimensional reaction diffusion equations this method has the accuracy of fourth order in both space and time the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term three monotone iterative algorithms are provided for solving the resulting discrete system efficiently and the sequences of iterations converge monotonically to a unique solution of the system a theoretical comparison result for the various monotone sequences is given the convergence of the finite difference solution to the continuous solution is proved and richardson extrapolation is used to achieve fourth order accuracy in time an application is given to an enzyme substrate reaction diffusion problem and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach

eigenfunction methods and nonlinear hyperbolic boundary value problems at resonance the abstract boundary value problem lu gu f u dom l h is considered here h is used to denote a real separable hilbert space l a closed symmetric linear operator and g a nonlinear operator assumed to be lipschitz continuous and strongly monotone in addition l is assumed to have a complete set of eigenfunctions in h and is allowed to have an infinite dimensional null space the existence of unique solutions depending continuously on f is established by a constructive approach galerkin approximations are considered and error estimates are given as an application of the main result the existence of time periodic weak solutions of the n dimensional wave equation is shown

distributed parameter estimation in sensor networks nonlinear observation models and imperfect communication the paper studies distributed static parameter vector estimation in sensor networks with nonlinear observation models and noisy intersensor communication it introduces separably estimable observation models that generalize the observability condition in linear centralized estimation to nonlinear distributed estimation it studies two distributed estimation algorithms in separably estimable models the nu with its linear counterpart lu and the nlu their update rule combines a consensus step where each sensor updates the state by weight averaging it with its neighbors states and an innovation step where each sensor processes its local current observation this makes the three algorithms of the consensus innovations type very different from traditional consensus this paper proves consistency all sensors reach consensus almost surely and converge to the true parameter value efficiency and asymptotic unbiasedness for lu and nu it proves asymptotic normality and provides convergence rate guarantees the three algorithms are characterized by appropriately chosen decaying weight sequences algorithms lu and nu are analyzed in the framework of stochastic approximation theory algorithm nlu exhibits mixed time scale behavior and biased perturbations and its analysis requires a different approach that is developed in this paper

improving convergence rate in the method of successive approximations received july 10 1979 revised december 5 1979 and november 11 1980 ams mos subject classifications 1970 primary 65r05 45g99 65l10 34b15

information driven distributed maximum likelihood estimation based on gauss newton method in wireless sensor networks in this paper we develop an energy efficient distributed estimation method that can be used in applications such as the estimation of a diffusive source and the localization and tracking of an acoustic target in wireless sensor networks we first propose a statistical measurement model in which we separate the linear and nonlinear parameters this modeling strategy reduce the processing complexity we then study the distributed implementation of the gauss newton method in the maximum likelihood estimation after that we propose a fully distributed estimation approach based on an incremental realization of the gauss newton method we derive three modifications of the basic algorithm to improve the distributed processing performance while still considering the energy restriction we implement the idea of information driven collaborative signal processing and provide a sensor node scheduling scheme in which the cramer rao bound crb is used as the performance and information utility measure to select the next sensor node numerical examples are used to study the performance of the distributed estimation and we show that of the methods considered here the proposed multiple iteration kalman filtering method has the most advantages for wireless sensor networks

a structure preserving fem for the uniaxially constrained mathbf q tensor model of nematic liquid crystals we consider the one constant landau de gennes model for nematic liquid crystals the order parameter is a traceless tensor field mathbf q which is constrained to be uniaxial mathbf q s mathbf n otimes mathbf n d 1 mathbf i where mathbf n is a director field s in mathbb r is the degree of orientation and d ge 2 is the dimension building on similarities with the one constant ericksen energy we propose a structure preserving finite element method for the computation of equilibrium configurations we prove stability and consistency of the method without regularization and gamma convergence of the discrete energies towards the continuous one as the mesh size goes to zero we design an alternating direction gradient flow algorithm for the solution of the discrete problems and we show that such a scheme decreases the energy monotonically finally we illustrate the method s capabilities by presenting some numerical simulations in two and three dimensions including non orientable line fields

incremental subgradient methods for nondifferentiable optimization we consider a class of subgradient methods for minimizing a convex function that consists of the sum of a large number of component functions this type of minimization arises in a dual context from lagrangian relaxation of the coupling constraints of large scale separable problems the idea is to perform the subgradient iteration incrementally by sequentially taking steps along the subgradients of the component functions with intermediate adjustment of the variables after processing each component function this incremental approach has been very successful in solving large differentiable least squares problems such as those arising in the training of neural networks and it has resulted in a much better practical rate of convergence than the steepest descent method in this paper we establish the convergence properties of a number of variants of incremental subgradient methods including some that are stochastic based on the analysis and computational experiments the methods appear very promising and effective for important classes of large problems a particularly interesting discovery is that by randomizing the order of selection of component functions for iteration the convergence rate is substantially improved

existence and approximation of the solutions of some nonlinear problems nonlinear equations defined by a positive definite elliptic operator and nonlinear functions are considered it is proved that a unique solution exists for a wider class of problems than previously determined which may be approximated by an iterative method these results are shown to hold for an equation arising in some reaction diffusion phenomena

numerical analysis of a finite element scheme for the approximation of harmonic maps into surfaces this article studies the numerical approximation of harmonic maps mto surfaces i e critical pomts for the dirichlet energy among weakly differentiable vector fields that are constrained to attain their pointwise values in a given manifold an iterative algorithm that is based on a linearization of the constraint about the current iterate at the nodes of a triangulation is devised and its global convergence to a discrete harmonic map is proved under general conditions weak accumulation of discrete harmonic maps at harmonic maps as discretization parameters tend to zero is established in two dimensions under certain assumptions on the underlying sequence of triangulations numerical simulations illustrate the performance of the algorithm for curved domains

relaxation methods for liquid crystal problems a point relaxation method for the liquid crystal problem is proposed and analyzed the liquid crystal problem is a minimization problem with a nonconvex local constraint it is proved that the energy of successive iterates is nonincreasing for the point relaxation method with relaxation parameter omega satisfying 0 omega 2 for a class of material constants it is also shown that the difference between successive iterates converges to zero and that limit points of the iteration sequence are minima with respect to perturbations which are supported at a point and which satisfy the constraint numerical results are given which demonstrate the improved rate of convergence for overrelaxation

finite element approximations of the ericksen leslie model for nematic liquid crystal flow the ericksen leslie model describes dynamics of low molar mass nematic liquid crystals where the spatiotemporal distribution of defects defining texture is represented by the director unit vector field bf d omega t rightarrow mathbb s 2 it consists of the navier stokes equations with an extra viscous stress tensor and a convective harmonic map heat flow equation to govern the dynamics of the director field two fully discrete finite element methods for a regularized system using the ginzburg landau regularization and for the limiting ericksen leslie model are proposed and well posedness and related discrete energy laws are established for the regularized model unconditional convergence of finite element solutions towards weak solutions of the continuum model as well as convergence towards measure valued solutions of the limiting ericksen leslie model are verified when the mesh parameters and the regularization parameter successively tend to zero computational experiments are also presented to show the importance of balancing numerical and regularization parameters to compare regularized and direct approaches and to show numerically the finite time formation annihilation and evolution of point defects

constraint preserving implicit finite element discretization of harmonic map flow into spheres discretization of the harmonic map flow into spheres often uses a penalization or projection strategy where the first suffers from the proper choice of an additional parameter and the latter from the lack of a discrete energy law and restrictive mesh constraints we propose an implicit scheme that preserves the sphere constraint at every node enjoys a discrete energy law and unconditionally converges to weak solutions of the harmonic map heat flow

a convergent and constraint preserving finite element method for the p harmonic flow into spheres an explicit fully discrete finite element method which satisfies the nonconvex side constraint at every node is developed for approximating the p harmonic flow for p in 1 infty convergence of the method is established under certain conditions on the domain and mesh computational examples are presented to demonstrate finite time blow ups and qualitative geometric changes of weak solutions of the p harmonic flow

stability and performance limits of adaptive primal dual networks this paper studies distributed primal dual strategies for adaptation and learning over networks from streaming data two first order methods are considered based on the arrow hurwicz ah and augmented lagrangian al techniques several revealing results are discovered in relation to the performance and stability of these strategies when employed over adaptive networks the conclusions establish that the advantages that these methods exhibit for deterministic optimization problems do not necessarily carry over to stochastic optimization problems it is found that they have narrower stability ranges and worse steady state mean square error performance than primal methods of the consensus and diffusion type it is also found that the ah technique can become unstable under a partial observation model while the other techniques are able to recover the unknown under this scenario a method to enhance the performance of al strategies is proposed by tying the selection of the step size to their regularization parameter it is shown that this method allows the al algorithm to approach the performance of consensus and diffusion strategies but that it remains less stable than these other strategies

the iterated kalman filter update as a gauss newton method it is shown that the iterated kalman filter ikf update is an application of the gauss newton method for approximating a maximum likelihood estimate an example is presented in which the iterated kalman filter update and maximum likelihood estimate show correct convergence behavior as the observation becomes more accurate whereas the extended kalman filter update does not

numerical methods for p harmonic flows and applications to image processing we propose in this paper an alternative approach for computing p harmonic maps and flows instead of solving a constrained minimization problem on sn 1 we solve an unconstrained minimization problem on the entire space of functions this is possible using the projection on the sphere of any arbitrary function then we show how this formulation can be used in practice for problems with both isotropic and anisotropic diffusion with applications to image processing using a new finite difference scheme

nonstationary monotone iterative methods for nonlinear partial differential equations a simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations with the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system picard gauss seidel and jacobi monotone iterative methods are presented and analyzed for the adaptive solutions the adaptive meshes are generated by the 1 irregular mesh refinement scheme which together with the m matrix of the finite element stiffness matrix lead to existence uniqueness comparison theorems with simple upper and lower solutions as initial iterates some numerical examples including a test problem with known analytical solution are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties numerical results of simulations on a mosfet with the gate length down to 34 nm are also given

solving variational problems and partial differential equations that map between manifolds via the closest point method abstract maps from a source manifold m to a target manifold n appear in liquid crystals color image enhancement texture mapping brain mapping and many other areas a numerical framework to solve variational problems and partial differential equations pdes that map between manifolds is introduced within this paper our approach the closest point method for manifold mapping reduces the problem of solving a constrained pde between manifolds m and n to the simpler problems of solving a pde on m and projecting to the closest points on n in our approach an embedding pde is formulated in the embedding space using closest point representations of m and n this enables the use of standard cartesian numerics for general manifolds that are open or closed with or without orientation and of any codimension an algorithm is presented for the important example of harmonic maps and generalized to a broader class of pdes which includes p harmonic maps improved efficiency and robustness are observed in convergence studies relative to the level set embedding methods harmonic and p harmonic maps are computed for a variety of numerical examples in these examples we denoise texture maps diffuse random maps between general manifolds and enhance color images

the ericksen model of liquid crystals with colloidal and electric effects we present a robust discretization of the ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields the total energy consists of the ericksen energy a weak anchoring or penalized dirichlet energy to model colloids and an electrical energy for a given electric field we describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi gradient flow algorithm which has a strictly monotone energy decreasing property numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non trivial defect patterns such as the saturn ring defect we conclude with a rigorous proof of the convergence of our discrete energy to the continuous energy

maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements the problem of estimating the frequencies phases and amplitudes of sinusoidal signals is considered a simplified maximum likelihood gauss newton algorithm which provides asymptotically efficient estimates of these parameters is proposed initial estimates for this algorithm are obtained by a variation of the overdetermined yule walker method and periodogram based procedure use of the maximum likelihood gauss newton algorithm is not however limited to this particular initialization method some other possibilities to get suitable initial estimates are briefly discussed an analytical and numerical study of the shape of the likelihood function associated with the sinusoids in noise process reveals its multimodal structure and clearly sets the importance of the initialization procedure some numerical examples are presented to illustrate the performance of the proposed estimation procedure comparison to the performance corresponding to the cramer rao lower bound is also presented using a simple expression for the asymptotic cramer rao bound covariance matrix derived in the paper

a modified accelerated monotone iterative method for finite difference reaction diffusion convection equations this paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction diffusion convection equations with nonlinear boundary conditions a modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time dependent problem and its corresponding steady state problem this method leads to a simple and yet efficient linear iterative algorithm it yields two sequences of iterations that converge monotonically from above and below respectively to a unique solution of the system the monotone property of the iterations gives concurrently improving upper and lower bounds for the solution it is shown that the rate of convergence for the sum of the two sequences is quadratic under an additional requirement quadratic convergence is attained for one of these two sequences in contrast with the existing accelerated monotone iterative methods our new method avoids computing local maxima in the construction of these sequences an application using a model problem gives numerical results that illustrate the effectiveness of the proposed method

constructive existence and uniqueness for some nonlinear two point boundary value problems abstract constructive existence and uniqueness theorems are presented for the problem y x y y 0 y 0 y 1 y 1 applications to several problems are also given including one in which the boundary values are y 0 y 0 y 1 y1

a renormalized newton method for liquid crystal director modeling we consider the nonlinear systems of equations that result from discretizations of a prototype variational model for the equilibrium director field characterizing the orientational properties of a liquid crystal material in the presence of pointwise unit vector constraints and coupled electric fields the numerical solution of such equations by lagrange newton methods leads to linear systems with a double saddle point form for which we have previously proposed a preconditioned nullspace method as an effective solver a ramage and e c gartland jr siam j sci comput 35 2013 pp b226 b247 here we propose and analyze a modified outer iteration renormalized newton method in which the orientation variables are normalized onto the constraint manifold at each iterative step this scheme takes advantage of the special structure of these problems and we prove that it is locally quadratically convergent the renormalized newton method bears some resemblance to the truncated newton method of

a numerical method for the nonlinear neumann problem an iterative scheme for the nonlinear neumann problem over a smooth bounded domain in r2 is studied an equivalent system of coupled problems is obtained by the method of alternative problems a combination of the capacitance matrix method and one dimensional nonlinear solver is then applied to this system yielding an efficient numerical algorithm

convergence and applications of a gossip based gauss newton algorithm the gauss newton algorithm is a popular and efficient centralized method for solving non linear least squares nlls problems in this paper a multi agent distributed version of this algorithm is proposed to solve general nlls problems in a network named gossip based gauss newton ggn algorithm furthermore we analyze and present sufficient conditions for its convergence and show numerically that the ggn algorithm achieves performance comparable to the centralized algorithm with graceful degradation in case of network failures more importantly the ggn algorithm provides significant performance gains compared to other distributed first order methods

finite element analysis of the landau de gennes minimization problem for liquid crystals this paper describes the landau de gennes free energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational properties of liquid crystal materials analytical and numerical issues are addressed conditions guaranteeing well posedness existence regularity of the problem are given as is a nonlinear finite element convergence analysis

accelerated monotone iterations for numerical solutions of nonlinear elliptic boundary value problems abstract an accelerated monotone iterative scheme for numerical solutions of a class of nonlinear elliptic boundary value problems is presented the mathematical analysis is devoted to a system of discretized equations of the elliptic boundary value problem by the finite difference or finite element method it is shown that the sequence of iterations from a linear iteration process converges monotonically and quadratically to a unique solution in a sector between a pair of upper and lower solutions this result is then used to show the quadratic convergence of the iterations to a maximal solution and a minimal solution when the nonlinear discrete system possesses multiple solutions an application is given to a tabular reactor model from chemical engineering for numerical solutions and the number of iterations are compared with that by the regular monotone iterative scheme

a saddle point approach to the computation of harmonic maps in this paper we consider numerical approximations of a constraint minimization problem where the object function is a quadratic dirichlet functional for vector fields and the interior constraint is given by a convex function the solutions of this problem are usually referred to as harmonic maps the solution is characterized by a nonlinear saddle point problem and the corresponding linearized problem is well posed near strict local minima the main contribution of the present paper is to establish a corresponding result for a proper finite element discretization in the case of two space dimensions iterative schemes of newton type for the discrete nonlinear saddle point problems are investigated and mesh independent preconditioners for the iterative methods are proposed

monotone iterative methods for nonlinear equations involving a noninvertible linear part we study monotone iterative methods for the numerical solution of a class of nonlinear elliptic boundary value problems by applying a theorem of ortega and rheinboldt this generalizes the work of earlier authors to the case of nonlinear perturbations of linear problems with a nontrivial kernel

low dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem abstract a nonlinear boundary value problem exhibiting a nonlocal bifurcation in its solution set is considered it is shown that for some values of the problem parameters there is only the zero solution while for other values of these parameters two nontrivial solutions also appear these nontrivial solutions do not lie in small neighborhoods of the zero function so that the bifurcation analysis is not local to the origin the method of reduction to alternative finite dimensional problems is applied to give a numerical description of the bifurcation and in particular to describe a surface of bifurcating solutions in addition the solution set of the 1 dimensional galerkin approximation is also shown to have the same qualitative nature as the solution set of the full problem

on the existence of smooth breathers for nonlinear wave equations previous work has shown that there is a finite dimensional manifold of generalized solutions of the nonlinear wave equation utt u g u subject to t periodicity in t plus radial symmetry and decay to zero at infinity in x the generalized solutions may lack regularity at the origin if 0 and u depends on t these solutions are called breathers numerical evidence for the existence of smooth solutions which are more regular at the origin is presented