on regularization and active set methods with complexity for constrained optimization the main objective of this research is to introduce a practical method for smooth bound constrained optimization that possesses worst case evaluation complexity o varepsilon 3 2 for finding an varepsilon approximate first order stationary point when the hessian of the objective function is lipschitz continuous as other well established algorithms for optimization with box constraints the algorithm proceeds visiting the different faces of the domain aiming to reduce the norm of an internal projected gradient and abandoning active constraints when no additional progress is expected in the current face the introduced method emerges as a particular case of a method for minimization with linear constraints moreover the linearly constrained minimization algorithm is an instance of a minimization algorithm with general constraints whose implementation may be unaffordable when the constraints are complicated as a procedure for leaving faces a different method is employed that may be regarded as a

a note on the uniqueness of entropy solutions of nonlinear degenerate parabolic equations abstract following the lead of carrillo arch ration mech anal 147 1999 269 361 recently several authors have used kru kov s device of doubling the variables to prove uniqueness results for entropy solutions of nonlinear degenerate parabolic equations in all these results the second order differential operator is not allowed to depend explicitly on the spatial variable which certainly restricts the range of applications of entropy solution theory the purpose of this paper is to extend a version of carrillo s uniqueness result to a class of degenerate parabolic equations with spatially dependent second order differential operator the class is large enough to encompass several interesting nonlinear partial differential equations coming from the theory of porous media flow and the phenomenological theory of sedimentation consolidation processes

towards the resolution of the gibbs phenomena it is well known that the expansion of an analytic nonperiodic function on a finite interval in a fourier series leads to spurious oscillations at the interval boundaries this result is known as the gibbs phenomenon the present paper introduces a new method for the resolution of the gibbs phenomenon which follows on the reconstruction method of gottlieb and coworkers siam rev 39 1997 644 based on gegenbauer polynomials orthogonal with respect to weight function 1 x2 1 2 we refer to their approach as the direct method and to the new methodology as the inverse method both methods use the finite set of fourier coefficients of some given function as input data in the re expansion of the function in gegenbauer polynomials or in other orthogonal basis sets the finite partial sum of the new expansion provides a spectrally accurate approximation to the function in the direct method this requires that certain conditions are met concerning the parameter in the weight function the number of fourier coefficients n and the number of gegenbauer polynomials m we show that the new inverse method can give exact results for polynomials independent of and with m n the paper presents several numerical examples applied to a single domain or to subdomains of the main domain so as to illustrate the superiority of the inverse method in comparison with the direct method

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

a filtration model with hysteresis abstract we study a filtration model for which a hysteresis effect is accounted in the saturation versus pressure constitutive relation the problem is formulated in a quasi variational inequality we prove the existence of a time global solution

optimality condition and complexity analysis for linearly constrained optimization without differentiability on the boundary in this paper we consider the minimization of a continuous function that is potentially not differentiable or not twice differentiable on the boundary of the feasible region by exploiting an interior point technique we present first and second order optimality conditions for this problem that reduces to classical ones when the derivative on the boundary is available for this type of problems existing necessary conditions often rely on the notion of subdifferential or become non trivially weaker than the kkt condition in the twice differentiable counterpart problems in contrast this paper presents a new set of first and second order necessary conditions that are derived without the use of subdifferential and reduce to exactly the kkt condition when twice differentiability holds as a result these conditions are stronger than some existing ones considered for the discussed minimization problem when only non negativity constraints are present to solve for these optimality conditions in the special but important case of linearly constrained problems we present two novel interior point trust region algorithms and show that their worst case computational efficiency in achieving the potentially stronger optimality conditions match the best known complexity bounds since this work considers a more general problem than those in the literature our results also indicate that best known existing complexity bounds are actually held for a wider class of nonlinear programming problems this new development is significant since optimality conditions play a fundamental role in computational optimization and more and more nonlinear and nonconvex problems need to be solved in practice

a newton cg algorithm with complexity guarantees for smooth unconstrained optimization we consider minimization of a smooth nonconvex objective function using an iterative algorithm based on newton s method and the linear conjugate gradient algorithm with explicit detection and use of negative curvature directions for the hessian of the objective function the algorithm tracks newton conjugate gradient procedures developed in the 1980s closely but includes enhancements that allow worst case complexity results to be proved for convergence to points that satisfy approximate first order and second order optimality conditions the complexity results match the best known results in the literature for second order methods

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

a kinetic approach to comparison properties for degenerate parabolic hyperbolic equations with boundary conditions abstract we study the comparison principle for degenerate parabolic hyperbolic equations with initial and nonhomogeneous boundary conditions we prove a comparison theorem for any entropy sub and supersolution the l 1 contractivity and therefore uniqueness of entropy solutions has been obtained so far by some authors but it seems that any comparison theorem is not proven the method used there is the doubling variable technique due to kru kov our method is based upon the kinetic formulation and the kinetic techniques by developing the kinetic techniques for degenerate parabolic hyperbolic equations with boundary conditions we can obtain a comparison property which obviously extends the l 1 contractive property

the dirichlet problem for the total variation flow suppose that is an open bounded domain with a lipschitz boundary the purpose of this chapter is to study the dirichlet problem left begin gathered frac partial u partial t div left frac du left du right right in q left 0 infty right times omega hfill u left t x right phi left x right on s left 0 infty right times partial omega hfill u left 0 x right u 0 left x right in x in omega hfill end gathered right 5 1 where u0 l1 and l1 this evolution equation is related to the gradient descent method used to solve the problem begin gathered minimize int omega left du right int omega f udx int partial omega left u phi right d mathcal h n 1 hfill u in bv left omega right hfill end gathered 2 where f l1 l existence for this variational problem was proved in 118 theorem 1 4

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

a decoupled first second order steps technique for nonconvex nonlinear unconstrained optimization with improved complexity bounds in order to be provably convergent towards a second order stationary point optimization methods applied to nonconvex problems must necessarily exploit both first and second order information however as revealed by recent complexity analyses of some of these methods the overall effort to reach second order points is significantly larger when compared to the one of approaching first order ones on the other hand there are other algorithmic schemes initially designed with first order convergence in mind that do not appear to maintain the same first order performance when modified to take second order information into account in this paper we propose a technique that separately computes first and second order steps and that globally converges to second order stationary points it consists in better connecting the steps to be taken and the stationarity criteria potentially guaranteeing larger steps and decreases in the objective our approach is shown to lead to an improvement of the corresponding complexity bound with respect to the first order optimality tolerance while having a positive impact on the practical behavior although the applicability of our ideas is wider we focus the presentation on trust region methods with and without derivatives and motivate in both cases the interest of our strategy

a strongly degenerate quasilinear equation the parabolic case we prove the existence and uniqueness of entropy solutions of the neumann problem for the quasilinear parabolic equation ut a u du where a z f z and f is a convex function of with linear growth as satisfying other additional assumptions in particular this class includes the case where f z z 0 and is a convex function with linear growth as

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

on the uniqueness of the solution of an evolution free boundary problem in theory of lubrication abstract we study an evolution free boundary problem issued from hydrodynamic lubrication the cavitation phenomenon takes place and is described by the elrod adams model this model is suggested in preference to the classical variational inequality due to its ability to describe input and output flow the considered model is governed by the parabolic reynolds equation and the unknows are the pressure of the lubricant contained in the narrow gap between two circular cylinders and its percentage in an elementary volume the main result of this paper is the uniqueness of the solution for the parabolic free boundary problem the proof of this result is based on a comparison principle permitting compare two solutions of the problem when their initial values and their values on the boundary can be compared previously we state a continuity result and some monotonicity properties of the solution

the use of quadratic regularization with a cubic descent condition for unconstrained optimization cubic regularization and trust region methods with worst case first order complexity o varepsilon 3 2 and worst case second order complexity o varepsilon 3 have been developed in the last few years in this paper it is proved that the same complexities are achieved by means of a quadratic regularization method with a cubic sufficient descent condition instead of the more usual predicted reduction based descent asymptotic convergence and order of convergence results are also presented finally some numerical experiments comparing the new algorithm with a well established quadratic regularization method are shown

the porous medium equation in a two component domain abstract we consider a system of two porous medium equations defined on two different components of the real line which are connected by the nonlinear contact condition u x v x v u on the contact line s first we prove existence and uniqueness of a solution u v on a bounded domain furthermore we are interested in the behaviour of the interface of the porous medium equation when it crosses the contact line s between the two components to this end we solve the cauchy problem on unbounded components consider self similar solutions for special u m u and derive a formula for the shape of the interface in that case

solvability of boundary value problems for the general schrodinger equation via schrodinger type identity methods abstract in this article we consider a free boundary value problem for the general schrodinger equations with density dependent viscosity coefficients by applying a powerful technique based on the schrodinger type identity method under certain assumptions imposed on the initial data there exists a unique solution for this equation mentioned above finally we also give an application which reveals that the schrodinger type identity method is effective and simple

uniqueness of renormalized solutions of degenerate elliptic parabolic problems we consider a general class of degenerate elliptic paraboic problems associated with the equation b v t div a v dv f using kruzhkov s method of doubling variables both in space and time we prove uniqueness and a comparison principle in l 1 for renormalized solutions

on the uniqueness of the solution of the evolution dam problem the purpose here is to investigate some aspects of the evolution dam problem partial chi alpha u partial t delta u rm div chi e in q chi in h u in q u geq 0 in q u phi in sigma 2 partial u partial v chi ev leq 0 on sigma 2 cap phi 0 e in bf r n e 0 cdots 0 1 partial u partial v chi ev 0 on sigma 1 here h is the heaviside maximal graph h s 1 if s 0 h s 0 1 if s 0 alpha is the storativity constant alpha 0 if the fluid is incompressible and alpha 0 if the fluid is compressible also q omega times 0 t where omega is a bounded domain in bf r n omega represents a porous medium separating a finite number of reservoirs with lipschitz boundary g where g is divided into two parts g 1 which is the impervious part and g 2 which is the pervious part with the following assumptions g 2 is an open subset of g g 2 not emptyset g 2 cap g 1 emptyset g 2 cup g 1 g and e nu leq 0 a e on g 1 sigma i denotes g i times 0 t i 1 2 the pressure at sigma 2 is known and is represented by a nonnegative function phi in c 0 1 overline q the above problem can be completed by prescribing an initial value to chi alpha u the existence of a weak solution of the evolution dam problem is well known however many properties of the solutions of the initial value dam problem in particular comparison and uniqueness theorems continue to be open for a general domain omega the results of this paper give precise information about the comparison theorem 4 7 and the uniqueness corollary 4 9 statements the key point to prove comparison and uniqueness results is that the function h mapsto chi x x n h t h x in bf r n 1 is a e nonincreasing

generalization of the inverse polynomial reconstruction method in the resolution of the gibbs phenomenon the finite fourier representation of a function f x exhibits oscillations where the function or its derivatives are nonsmooth this is known as the gibbs phenomenon a robust and accurate reconstruction method that resolves the gibbs oscillations was proposed in a previous paper j comput appl math 161 2003 41 based on the inversion of the transformation matrix which represents the projection of a set of basis functions onto the fourier space if the function is a polynomial this inverse polynomial reconstruction method iprm is exact in this paper we develop the iprm by requiring that the proper error be orthogonal to the fourier or polynomial space the iprm is generalized to any set of basis functions the primitive basis polynomials nonclassical orthogonal polynomials and the gegenbauer polynomials are used to illustrate the wide validity of the iprm it is shown that the iprm yields a unique reconstruction irrespective of the basis set for any analytic function and yields spectral convergence the ill posedness of the transformation matrix due to the exponential growth of the condition number of the matrix is also discussed

on the dam problem this paper is concerned with the qualitative behavior of solutions of the dam problem for flow of an incompressible fluid through a two dimensional porous medium the cross section of the flow region is allowed to have a complicated boundary the flow domain is assumed to have a lipschitz boundary and minimal assumptions are made on the parts where various boundary conditions are to be satisfied the main contribution of the paper is to determine the corresponding properties of the solution after giving a concise proof of the known result that a solution exists the authors introduce the so called s 3 connected solutions s 3 is the impervious part of the boundary and prove several interesting results about them solutions which do not have these properties are such that there are pools of fluid not in contact with the outside reservoir and they prove that any solution can be written as a sum of an s 3 connected solution and pools also it is shown that s 3 connected solutions are unique for some boundaries it can be shown that only s 3 connected solutions exist so that uniqueness holds without qualification

entropy solutions for nonlinear degenerate problems we consider a class of elliptic hyperbolic degenerate equations g u delta b u divg phi u f with dirichlet homogeneous boundary conditions and a class of elliptic parabolic hyperbolic degenerate equations g u t delta b u divg phi u f with homogeneous dirichlet conditions and initial conditions existence of entropy solutions for both problems is proved for nondecreasing continuous functions g and b vanishing at zero and for a continuous vectorial function satisfying rather general conditions comparison and uniqueness of entropy solutions are proved for g and b continuous and nondecreasing and for continuous

the equivalence of weak solutions and entropy solutions of nonlinear degenerate second order equations abstract we prove the equivalence of weak solutions and entropy solutions of an elliptic parabolic hyperbolic degenerate equation g t t b u div u f with homogeneous dirichlet conditions and initial conditions as a result of the equivalence we obtain the l1 contraction principle and uniqueness of weak solutions of elliptic parabolic degenerate equations

orthogonal polynomials constructive theory and applications we consider polynomials orthogonal with respect to some measure on the real line a basic problem in the constructive theory of such polynomials is the determination of their three term recurrence relation given the measure in question depending on what is known about this measure there are different ways to proceed if as is typical in applications in the physical sciences one knows the measure only through moment information the appropriate procedure is an algorithm that goes back to chebyshev the algorithm in effect implements the nonlinear map from the given moments or modified moments to the desired recursion coefficients the numerical effectiveness of this procedure is determined in an essential way by the numerical condition of this map it is known that the map is ill conditioned in the case of ordinary moments for modified moments it may or may not be well conditioned the matter depending on certain properties of the given measure and the additional measure defining modified moments a theorem to this effect will be given and illustrated on some typical examples if more is known about the given measure for example if it is absolutely continuous and can be evaluated pointwise then a procedure can be employed which is based on an observation of stieltjes stieltjes remarked that the desired recursion coefficients can be successively built up by alternating between known inner product formulae for these coefficients and the initial sections of the recurrence relation already obtained an effective implementation of this idea requires a suitable discretization of the inner product this requirement while possibly a weakness of the method also accounts for its beauty since it leaves room for imagination and ingenuity used skillfully the discretized stieltjes procedure is among the most widely applicable and effective methods for generating orthogonal polynomials some examples will be given to illustrate its use finally we show how our newly acquired capability of generating nonstandard orthogonal polynomials can be used to solve some special problems in approximation theory and in the summation of slowly convergent series a novel set of polynomials orthogonal on the semicircle will also be mentioned briefly in connection with cauchy principal value integrals

spectral methods for hyperbolic problems we review the current state of fourier and chebyshev collocation methods for the solution of hyperbolic problems with an eye to basic questions of accuracy and stability of the numerical approximations throughout the discussion we emphasize recent developments in the area such as spectral penalty methods the use of filters the resolution of the gibbs phenomenon and issues related to the solution of nonlinear conservations laws such as conservation and convergence we also include a brief discussion on the formulation of multi domain methods for hyperbolic problems and conclude with a few examples of the application of pseudospectral collocation methods for solving nontrivial systems of conservation laws