discontinuous galerkin methods for friedrichs systems i general theory this paper presents a unified analysis of discontinuous galerkin methods to approximate friedrichs systems an abstract set of conditions is identified at the continuous level to guarantee existence and uniqueness of the solution in a subspace of the graph of the differential operator then a general discontinuous galerkin method that weakly enforces boundary conditions and mildly penalizes interface jumps is proposed all the design constraints of the method are fully stated and an abstract error analysis in the spirit of strang s second lemma is presented finally the method is formulated locally using element fluxes and links with other formulations are discussed details are given for three examples namely advection reaction equations advection diffusion reaction equations and the maxwell equations in the so called elliptic regime

an aggregate subgradient method for nonsmooth convex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

relaxation methods for monotropic programs we propose a dual descent method for the problem of minimizing a convex possibly nondifferentiable separable cost subject to linear constraints the method has properties reminiscent of the gauss seidel method in numerical analysis and uses the e complementary slackness mechanism introduced in bertsekas hosein and tseng 1987 to ensure finite convergence to near optimality as special cases we obtain the methods in bertsekas hosein and tseng 1987 for network flow programs and the methods in tseng and bertsekas 1987 for linear programs

a preconditioning proximal newton method for nondifferentiable convex optimization we propose a proximal newton method for solving nondifferentiable convex optimization this method combines the generalized newton method with rockafellar s proximal point algorithm at each step the proximal point is found approximately and the regularization matrix is preconditioned to overcome inexactness of this approximation we show that such a preconditioning is possible within some accuracy and the second order differentiability properties of the moreau yosida regularization are invariant with respect to this preconditioning based upon these superlinear convergence is established under a semismoothness condition

convergence of line search methods for unconstrained optimization line search methods are traditional and successful methods for solving unconstrained optimization problems its convergence has attracted more attention in recent years in this paper we analyze the general results on convergence of line search methods with seven line search rules it is clarified that the search direction plays a main role in these methods and that step size guarantees the global convergence in some cases it is also proved that many line search methods have same convergence property these convergence results can enable us to design powerful effective and stable algorithms in practice finally a class of special line search methods is investigated

stationarity and superlinear convergence of an algorithm for univariate locally lipschitz constrained minimization this paper introduces an algorithm for minimizing a single variable locally lipschitz function subject to a like function being nonpositive the method combines polyhedral and quadratic approximation a new type of penalty technique and a safeguard in such a way as to give convergence to a stationary point the convergence is shown to be superlinear under somewhat stronger assumptions that allow both nonsmooth and nonconvex cases the algorithm can be an effective subroutine for solving line search subproblems called for by multivariable optimization algorithms

a generalized recursive convolution method for time domain propagation in porous media an efficient numerical method referred to as the auxiliary differential equation ade method is proposed to compute convolutions between relaxation functions and acoustic variables arising in sound propagation equations in porous media for this purpose the relaxation functions are approximated in the frequency domain by rational functions the time variation of the convolution is thus governed by first order differential equations which can be straightforwardly solved the accuracy of the method is first investigated and compared to that of recursive convolution methods it is shown that while recursive convolution methods are first or second order accurate in time the ade method does not introduce any additional error the ade method is then applied for outdoor sound propagation using the equations proposed by wilson et al in the ground 2007 appl acoust 68 173 200 a first one dimensional case is performed showing that only five poles are necessary to accurately approximate the relaxation functions for typical applications finally the ade method is used to compute sound propagation in a three dimensional geometry over an absorbing ground results obtained with wilson s equations are compared to those obtained with zwikker and kosten s equations and with an impedance surface for different flow resistivities

separating plane algorithms for convex optimization the equivalent formulation of a convex optimization problem is the computation of a value of a conjugate function at the origin the latter can be achieved by approximation of the epigraph of the conjugate function around the origin and gradual refinement of the approximation this yields a generic algorithm of convex optimization which transforms into some well known techniques when certain strategies of approximation are employed it also suggests new algorithmic approaches with promising computational experience and provides a uniform treatment of constrained and unconstrained optimization

practical aspects of the moreau yosida regularization theoretical preliminaries when computing the infimal convolution of a convex function f with the squared norm the so called moreau yosida regularization of f is obtained among other things this function has a lipschitzian gradient we investigate some more of its properties relevant for optimization the most important part of our study concerns second order differentiability existence of a second order development of f implies that its regularization has a hessian for the converse we disclose the importance of the decomposition of bbb r n along cal u the subspace where f is smooth and cal v the subspace parallel to the subdifferential of f

restricted center problems under polyhedral gauges abstract given a finite set ex ex 1 ex 2 ex m in the plane and a forbidden region r we want to find a point x r such that max m 1 m w m m x ex m is minimized this is a variant of the well known continuous center problem where we measure the distance with polyhedral gauges the unit ball of a polyhedral gauge may be any convex polyhedron containing the origin in its interior this large class of distance functions allows very general practical settings such as asymmetry to be modeled each ex m is allowed to have its own gauge m and the forbidden region r enables us to include negative information which means defining the infeasible region explicitly instead of the feasible region in the model efficient algorithms and structural properties for this non convex optimization problem based on combinatorial and geometrical methods are presented a classification scheme for location problems is also introduced

a bracketing technique to ensure desirable convergence in univariate minimization this paper gives a general safeguarded bracketing technique for minimizing a function of a single variable in certain cases the technique guarantees convergence to a stationary point and when combined with sequential polynomial and or polyhedral fitting algorithms preserves rapid convergence each bracket has an interior point whose function value does not exceed those of the two bracket endpoints the safeguarding technique consists of replacing the fitting algorithm s iterate candidate by a close point whose distance from the three bracket points exceeds a positive multiple of the square of the bracket length it is shown that a given safeguarded quadratic fitting algorithm converges in a certain better than linear manner with respect to the bracket endpoints for a strongly convex twice continuously differentiable function

design of broadband time domain impedance boundary conditions using the oscillatory diffusive representation of acoustical models a methodology to design broadband time domain impedance boundary conditions tdibcs from the analysis of acoustical models is presented the derived tdibcs are recast exclusively as first order differential equations well suited for high order numerical simulations broadband approximations are yielded from an elementary linear least squares optimization that is for most models independent of the absorbing material geometry this methodology relies on a mathematical technique referred to as the oscillatory diffusive or poles and cuts representation and is applied to a wide range of acoustical models drawn from duct acoustics and outdoor sound propagation which covers perforates semi infinite ground layers as well as cavities filled with a porous medium it is shown that each of these impedance models leads to a different tdibc comparison with existing numerical models such as multi pole or extended helmholtz resonator provides insights into their suitability additionally the broadly applicable fractional polynomial impedance models are analyzed using fractional calculus

existence and multiplicity for solutions of neumann problem for semilinear elliptic equations the existence and multiplicity results are obtained for solutions of neumann problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

the existence of infinitely many solutions of a class of nonlinear elliptic equations with neumann boundary condition for both resonance and oscillation problems abstract in this paper we study and discuss nonlinear elliptic equations with neumann boundary condition for both resonance and oscillation problem and obtain new results on the exactly infinitely many positive and negative solutions which are all nonconstant the study of problem 1 1 are based on the variational methods and critical point theory we prove the conclusion by using sub sup solution method mountain pass theorem in order intervals leray schauder degree theory and more theory

the multidimensional wave equation with generalized acoustic boundary conditions i strong stability we study the wave equation on a domain of mathbb r d d geq 2 with dynamical boundary control applied on a part of the boundary and a dirichlet boundary condition on the remaining part we prove the asymptotic stability and nonuniform stability of the associated semigroup

transient and multiple frequency sound transmission through perforated plates at high amplitude the transmission of complex periodic and transient acoustic signals through orifice plates at high amplitude and in the absence of mean fluid flow is discussed a simple fluid dynamical model involving a time varying mass end correction is the basis of the theory the equation of motion for the air in the orifice is solved numerically in the time domain for a specific instance an analytical solution in the frequency domain is possible good agreement is noted between experimental and theoretical results in both time and frequency domains

a superlinearly convergent algorithm for one dimensional constrained minimization problems with convex functions this paper presents a globally and superlinearly convergent algorithm for solving one dimensional constrained minimization problems involving not necessarily smooth convex functions the constraint is handled by what can be interpreted as a new type of penalty method the algorithm does not require the objective function to be evaluated at infeasible points and it does not use constraint function values at feasible points the penalty parameter is automatically generated by the algorithm via linear approximation of the constraint function as in the unconstrained case developed by lemarechal and the author the algorithm uses a step that is the shorter of a quadratic approximation step and a polyhedral approximation step here the latter is actually a penalized polyhedral step whose computation is well conditioned if the constraint satisfies a nondegeneracy assumption

on the robust flexible and consistent implementation of time domain impedance boundary conditions for compressible flow simulations the accurate simulation of compressible flows requires the appropriate modeling of the reflection of acoustic waves at the boundaries in the present study we discuss time domain impedance boundary conditions tdibc the formulation proposed allows to impose a desired reflection coefficient at the inflow and outflow boundaries our formulation is an extension of the well known navier stokes characteristic boundary conditions the frequency dependent reflections at the boundaries are implemented with a state space model in the time domain we provide a comprehensive discussion on how such state space models can be constructed and interpreted this discussion shows that the state space description allows a robust and flexible implementation it allows to consider complex reflection coefficients and account for non constant cfd time steps in a straight forward manner furthermore we prove analytically and demonstrate numerically that the formulation proposed is consistent i e the formulation ensures that the flow simulation exhibits the reflection coefficient imposed accurately as long as the waves impinging on the boundary are plane and it prohibits drift of the mean flow variables finally the boundary conditions are tested successfully for laminar and turbulent flows

proximal quasi newton methods for nondifferentiable convex optimization the method is based on rockafellar s proximal point algorithm and a cutting plane technique at each step we use an approximate proximal point pa xk of xk to define a vk ekf pa xk with ek vk where is a constant the method monitors the reduction in the value of vk to identify when a line search on f should be used the quasi newton step is used to reduce the value of vk without the differentiability of f the method converges globally and the rate of convergence is q linear superlinear convergence is also discussed to extend the characterization result of dennis and more numerical results show the good performance of the method

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

an implementation of an algorithm for univariate minimization and an application to nested optimization this paper provides an efficient numerical algorithm for solving quite general single variable constrained minimization problems also given are a listing of the corresponding fortran subroutine pq1 an explanation of its general use and a description of its particular use in solving a practical resource allocation problem via a dual technique requiring nested optimization

a algorithm for convex minimization for convex minimization we introduce an algorithm based on space decomposition the method uses a bundle subroutine to generate a sequence of approximate proximal points when a primal dual track leading to a solution and zero subgradient pair exists these points approximate the primal track points and give the algorithm s or corrector steps the subroutine also approximates dual track points that are gradients needed for the method s newton predictor steps with the inclusion of a simple line search the resulting algorithm is proved to be globally convergent the convergence is superlinear if the primal dual track points and the objective s hessian are approximated well enough

multiple solutions and sign changing solutions of a class of nonlinear elliptic equations with neumann boundary condition in this paper we study and discuss nonlinear elliptic equations with neumann boundary condition for oscillation problem we obtain infinitely many positive and negative solutions of 1 1 which are all nonconstant and get at least two nonconstant solutions in every order interval under resonance case moreover we yield infinitely many sign changing solutions of 1 1 under some assumptions furthermore we give a precise description of critical groups of some kinds of critical points we draw the conclusions by using sub sup solution method mountain pass theorem in order intervals leray schauder degree theory and morse theory

multiplicity results for two kinds of equivariant systems abstract two equivariant problems of the form e u f u are considered where f is a real function which is invariant under the action of a group g and using morse theory for each problem an arbitrarily great number of orbits of solutions is founded choosing e suitably small the first problem is a o 2 equivariant system of two equations which can be seen as a complex ginzburg landau equation while the second one is a system of m equations which is equivariant for the action of a finite group of real orthogonal matrices m m

nonlinear asymptotic impedance model for a helmholtz resonator liner the usual nonlinear corrections for a helmholtz resonator type impedance do not seem to be based on a systematic asymptotic solution of the pertaining equations we aim to present a systematic derivation of a solution of the nonlinear helmholtz resonator equation in order to obtain analytically expressions for impedances close to resonance while including nonlinear effects the amplitude regime considered is such that when we stay away from the resonance condition the nonlinear terms are relatively small and the solution obtained is of the linear equation formed after neglecting the nonlinear terms close to the resonance frequency the nonlinear terms can no longer be neglected and algebraic equations are obtained that describe the corresponding nonlinear impedance sample results are presented including a few comparisons with measurements available in the literature the validity of the model is understood in the near resonance and nonresonance regimes

global analysis of two parameter elliptic eigenvalue problems on considere le probleme aux valeurs limites non lineaire lu f u 0 x u 0 x ou l est un operateur elliptique d ordre 2 et et sont des parametres on etudie les proprietes globales du continuum de solution de ces problemes quand et varient

natural continuous extensions of runge kutta methods the present paper develops a theory of natural continuous extensions nces for the discrete approximate solution of an ode given by a runge kutta process these nces are defined in such a way that the continuous solutions furnished by the one step collocation methods are included

fundamental problems with the model of uniform flow over acoustic linings abstract the walls of ducts containing flow are often acoustically lined in order to reduce sound many simple models of acoustic linings assume the lining to be linear and locally reacting examples considered here include the three parameter mass spring damper helmholtz resonator and enhanced helmholtz resonator models all of these models have been found to have stability issues with uniform mean grazing flow and there has been some confusion over the existence of hydrodynamic instability surface waves over such linings mathematically the standard proven briggs bers stability analysis is not applicable computationally the hydrodynamic modes are routinely ignored in the frequency domain and instabilities filtered out in the time domain this confusion also causes significant problems for mode matching green functions and scattering analyses in this paper it is shown that any situation not capable of being analysed using the briggs bers criterion is illposed and that this is the root cause of the confusion over hydrodynamic instabilities a large class of lining models including all those mentioned above are shown to be illposed with uniform mean flow an explanation is given of the effects of this illposedness in practice and it is argued that illposed models should not be used an alternative stability criterion to briggs bers has been occasionally used in the literature this alternative criterion involving analytic continuation to purely imaginary frequencies was recently christened the crighton leppington stability criterion however this stability criterion is incorrect and several counter examples are given here

a rapidly convergent five point algorithm for univariate minimization this paper presents an algorithm for minimizing a function of one variable which uses function but not derivative values at five points to generate each iterate it employs quadratic and polyhedral approximations together with a safeguard the basic method without the safeguard exhibits a type of better than linear convergence for certain piecewise twice continuously differentiable functions the safeguard guarantees convergence to a stationary point for very general functions and preserves the better than linear convergence of the basic method

a globally and superlinearly convergent algorithm for nonsmooth convex minimization it is well known that a possibly nondifferentiable convex minimization problem can be transformed into a differentiable convex minimization problem by way of the moreau yosida regularization this paper presents a globally convergent algorithm that is designed to solve the latter problem under additional semismoothness and regularity assumptions the proposed algorithm is shown to have a q superlinear rate of convergence

time domain numerical simulation of a flow impedance tube an efficient method has been developed for the application of the surface acoustic impedance condition in time domain solutions of aeroacoustic problems such as the broadband frequency simulation of a flow impedance tube the basis for this method is the standard impedance condition stated in the frequency domain as the particle displacement continuity equation the development of the time domain impedance condition follows the relations among the frequency z and discrete time domains and a rational function representation of the impedance in thez domain the resultant impedance condition consists of finite infinite impulse response type digital filter operations in the time domain which is very suitable to computational aeroacoustics algorithms this paper describes the present approach and discusses the time domain numerical simulations of the nasa langley flow impedance tube with a constant depth ceramic tubular liner both single and broadband frequency simulations are performed excellent agreement is shown with experimental data at various frequencies and flow conditions

a controllable canonical form implementation of time domain impedance boundary conditions for broadband aeroacoustic computation a new method which can be effectively and efficiently applied in the simulations of broadband noise problems is proposed for time domain impedance boundary condition implementations by using the so called controllable canonical form that is well known in linear system usually the impedance boundary condition can be defined in frequency domain as a rational polynomial function with poles in the negative half of the complex plane to guarantee stability otherwise causality might be violated in the corresponding time domain implementation to address this issue various methodologies have been proposed previously that usually lead to complicated polynomials whose numerical implementations are often indirect and intricate the proposed method with a controllable canonical form on the other hand directly transforms the frequency domain transfer function a quotient of rational polynomials to an equivalent state space model which consists of a series of first order ordinary differential equations that can be numerically implemented in a straightforward way the proposed method is demonstrated by using two benchmark problems a two dimensional gaussian pulse propagating in a uniform flow with a lined wall and the test cases from the nasa langley grazing incidence tube experiments good agreements demonstrate the potential of the proposed computational method

existence and continuation of solutions for a nonlinear neumann problem abstract in this article we study the existence continuation and bifurcation from infinity of nonconstant solutions for a nonlinear neumann problem we apply the leray schauder degree and the degree for s o 2 equivariant gradient operators defined by the second author in s rybicki s o 2 degree for orthogonal maps and its applications to bifurcation theory nonlinear anal tma 23 1 1994 83 102

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

on the limited memory bfgs method for large scale optimization we study the numerical performance of a limited memory quasi newton method for large scale optimization which we call the l bfgs method we compare its performance with that of the method developed by buckley and lenir 1985 which combines cycles of bfgs steps and conjugate direction steps our numerical tests indicate that the l bfgs method is faster than the method of buckley and lenir and is better able to use additional storage to accelerate convergence we show that the l bfgs method can be greatly accelerated by means of a simple scaling we then compare the l bfgs method with the partitioned quasi newton method of griewank and toint 1982a the results show that for some problems the partitioned quasi newton method is clearly superior to the l bfgs method however we find that for other problems the l bfgs method is very competitive due to its low iteration cost we also study the convergence properties of the l bfgs method and prove global convergence on uniformly convex problems

variable metric bundle methods from conceptual to implementable forms to minimize a convex function we combine moreau yosida regularizations quasi newton matrices and bundling mechanisms first we develop conceptual forms using reversal quasi newton formulae and we state their global and local convergence then to produce implementable versions we incorporate a bundle strategy together with a curve search no convergence results are given for the implementable versions however some numerical illustrations show their good behaviour even for large scale problems

a derivative based bracketing scheme for univariate minimization and the conjugate gradient method abstract a derivative based scheme for univariate minimization is developed this scheme has a quadratic convergence rate and requires an evaluation of the function and its derivative in each iteration an application involving the conjugate gradient method is presented

a stabilized finite element method using a discontinuous level set approach for the computation of bubble dynamics a novel numerical method for solving three dimensional two phase flow problems is presented this method combines a quadrature free discontinuous galerkin method for the level set equation with a pressure stabilized finite element method for the navier stokes equations the main challenge in the computation of such flows is the accurate evaluation of surface tension forces this involves the computation of the curvature of the fluid interface in the context of the discontinuous galerkin method we show that the use of a curvature computed by means of a direct derivation of the level set function leads to inaccurate and oscillatory results a more robust second order least squares computation of the curvature that filters out the high frequencies and produces converged results is presented this whole numerical technology allows to simulate a wide range of flow regimes with large density ratios to accurately capture the shape of the deforming interface of the bubble and to maintain good mass conservation

on superlinear convergence in univariate nonsmooth minimization this note demonstrates a new result on superlinear convergence in nonsmooth univariate minimization it also gives other concepts of rapid convergence for minimization of functions that may have discontinuous derivatives

multipeak solutions for the neumann problem of an elliptic system of fitzhugh nagumo type for systems of elliptic equations of fitzhugh nagumo type on bounded domains and with small diffusion in one equation we construct solutions with multiple sharp peaks close to each other and close to but not on the boundary this is a striking contrast to results for scalar equations for some symmetric domains we also construct similar multipeak solutions except that here the peaks are not close to each other

primal dual gradient structured functions second order results links to epi derivatives and partly smooth functions we give second order expansions for quite general nonsmooth functions from the cal v cal u space decomposition point of view the results depend on primal dual gradient structure which we relate to general concepts of second order epi derivatives and partly smooth functions expressions for the associated second order objects are given in terms of cal u subspace hessians

on a non linear differential equation modelling helmholtz resonator response abstract the non linear differential equation y dot y sgn y e y y sin t proposed by hersh and walker 1 as a model for helmholtz resonator response to a sound wave in the presence of grazing mean flow is studied by matched asymptotic expansion for e 1 the waveform contains a smooth outer part and a boundary layer of rapid change approximations to two terms are found in each region and approximate value for all fourier coefficients also found and compared very favourably with numerical curve fits for the first coefficients given in reference 1 the asymptotic approximations are found to be highly accurate over the whole waveform when compared with numerical solutions

helmholtz resonator impedance model part 1 nonlinear behavior a one dimensional preliminary design impedance prediction model is derived for helmholtz resonators constructed with circular orie ces the model addresses only the effects of resonator geometry and incident sound pressure amplitude the effects of grazing e ow are not included an experimental program was conducted to calibrate unknown empirical parameters derived in the model the model was shown to predict reasonably accurately impedance data published in the open literature for resonators constructed with single and multiple orie ces over a wide range of sound pressure amplitudes and frequencies the model provides a rapid means of calculating resonator impedance as a function of resonator geometry incident sound pressure amplitude and frequency

global and superlinear convergence of an algorithm for one dimensional minimization of convex functions this paper studies an algorithm for minimizing a convex function based upon a combination of polyhedral and quadratic approximation the method was given earlier but without a good specification for updating the algorithm s curvature matrix here for the case of onedimensional minimization we provide a specification that insures convergence even in cases where the curvature scalar tends to zero or infinity under mild additional assumptions we show that the convergence is superlinear

fluid structure algorithms based on steklov poincare operators in this paper we review some classical algorithms for fluid structure interaction problems and we propose an alternative viewpoint mutuated from the domain decomposition theory this approach yields preconditioned richardson iterations on the steklov poincare nonlinear equation at the fluid structure interface

existence and continuation of periodic solutions of autonomous newtonian systems abstract in this article we study the existence and the continuation of periodic solutions of autonomous newtonian systems to prove the results we apply the infinite dimensional version of the degree for so 2 equivariant gradient operators defined by the third author in nonlinear anal theory methods appl 23 1 1994 83 102 and developed in topol meth nonlinear anal 9 2 1997 383 417 using the results due to rabier symmetries topological degree and a theorem of z q wang j math 24 3 1994 1087 1115 and wang symmetries and calculation of the degree chinese ann math 10 1989 520 536 we show that the leray schauder degree is not applicable in the proofs of our theorems because it vanishes

runge kutta methods for parabolic equations and convolution quadrature we study the approximation properties of runge kutta time discretizations of linear and semilinear parabolic equations including incompressible navier stokes equations we derive asymptotically sharp error bounds and relate the temporal order of convergence which is generally noninteger to spatial regularity and the type of boundary conditions the analysis relies on an interpretation of runge kutta methods as convolution quadratures in a different context these can be used as efficient computational methods for the approximation of convolution integrals and integral equations they use the laplace transform of the confolution kernal via a discrete operational calculus 23 refs 2 tabs

wave equation stabilization by delays equal to even multiples of the wave propagation time for a string equation with time delays in the output feedback loop we study stability and show that the system is a riesz spectral system and prove that the spectrum determined growth condition holds for all delays when the delay is equal to the even multiples of the wave propagation time we develop the necessary and sufficient conditions for the feedback gain and time delay which guarantee the exponential stability of the closed loop system in particular we show that as the delay of even multiples is increasing to infinity the stability bound on the feedback gain decays to zero we also show that whenever the delay is an odd multiple of the wave propagation time the closed loop system is unstable the lack of robustness to a small perturbation in time delay is specifically discussed for the delay equal to two a numerical simulation for the case of the delay equal to two is presented to illustrate the convergence finally an alternative stability analysis is conducted within the framework of well posed infinite dimensional systems

existence theorems for equations in normed spaces and boundary value problems for nonlinear vector ordinary differential equations on demontre l existence des solutions des equations dans des espaces normes quand la partie non lineaire de l equation satisfait des conditions de croissance et d asymptote

high order accurate thin layer approximations for time domain electromagnetics part i general metal backed coatings we present a family of high order accurate thin layer approximations for time domain electromagnetics the thin layer approximations are valid for metal backed coatings of general isotropic materials and certain classes of anisotropic materials on smooth curvilinear backgrounds both dielectric and magnetic materials can be considered these models are non trivial and we discuss their properties and implementation in the context of discontinuous galerkin methods which are particularly well suited for these models the range of validity accuracy and stability of the resulting schemes is demonstrated through one and two dimensional examples

a quadratic approximation method for minimizing a class of quasidifferentiable functions an unconstrained nonlinear programming problem with nondifferentiabilities is considered the nondifferentiabilities arise from terms of the form max f 1 x f n x which may enter nonlinearly in the objective function local convex polyhedral upper approximations to the objective function are introduced these approximations are used in an iterative method for solving the problem the algorithm proceeds by solving quadratic programming subproblems to generate search directions approximate line searches ensure global convergence of the method to stationary points the algorithm is conceptually simple and easy to implement it generalizes efficient variable metric methods for minimax calculations

a quasi second order proximal bundle algorithm this paper introduces an algorithm for convex minimization which includes quasi newton updates within a proximal point algorithm that depends on a preconditioned bundle subalgorithm the method uses the hessian of a certain outer function which depends on the jacobian of a proximal point mapping which in turn depends on the preconditioner matrix and on a lagrangian hessian relative to a certain tangent space convergence is proved under boundedness assumptions on the preconditioner sequence

multiple solutions of neumann problem for elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations at resonance or nonresonance with unbounded nonlinearity by the reduction method and the critical point theory in particular a three critical point theorem proposed by brezis nirenberg

energy analysis and discretization of nonlinear impedance boundary conditions for the time domain linearized euler equations time domain impedance boundary conditions tdibcs can be enforced using the impeda nce the admittance or the scattering operator this article demonstrates the computational advantage of the last even for nonlinear tdibcs with the linearized euler equations this is achieved by a systematic semi discrete energy analysis of the weak enforcement of a generic nonlinear tdibc in a discontinuous galerkin finite element method in particular the analysis highlights that the sole definition of a discrete model is not enough to fully define a tdibc to support the analysis an elementary physical nonlinear scattering operator is derived and its computational properties are investigated in an impedance tube then the derivation of time delayed broadband tdibcs from physical reflection coefficient models is carried out for single degree of freedom acoustical liners a high order discretization of the derived time local formulation which consists in composing a set of ordinary differential equations with a transport equation is applied to two flow ducts

diffusive approximation of a time fractional burger s equation in nonlinear acoustics a fractional time derivative is introduced into burger s equation to model losses of nonlinear waves this term amounts to a time convolution product which greatly penalizes the numerical modeling a diffusive representation of the fractional derivative is adopted here replacing this nonlocal operator by a continuum of memory variables that satisfy local in time ordinary differential equations then a quadrature formula yields a system of local partial differential equations well suited to numerical integration the determination of the quadrature coefficients is crucial to ensure both the well posedness of the system and the computational efficiency of the diffusive approximation for this purpose optimization with constraint is shown to be a very efficient strategy strang splitting is used to solve successively the hyperbolic part by a shock capturing scheme and the diffusive part exactly numerical experiments are proposed to assess the efficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation

a descent algorithm for nonsmooth convex optimization this paper presents a new descent algorithm for minimizing a convex function which is not necessarily differentiable the algorithm can be implemented and may be considered a modification of the e subgradient algorithm and lemarechal s descent algorithm also our algorithm is seen to be closely related to the proximal point algorithm applied to convex minimization problems a convergence theorem for the algorithm is established under the assumption that the objective function is bounded from below limited computational experience with the algorithm is also reported

descent methods for composite nondifferentiable optimization problems we present a framework for the development of globally defined descent algorithms for the minimization of non differentiable objective functionsf h o f withh convex within our structure the global convergence properties of the cauchy modified newton gauss newton and variable metric methods are easily established along with that of several new approaches examples illustrating the calculational techniques are provided

on mathcalvu theory for functions with primal dual gradient structure we consider a general class of convex functions having what we call primal dual gradient structure it includes finitely determined max functions and maximum eigenvalue functions as well as other infinitely defined max functions for a function in this class we discuss a space decomposition that allows us to identify a subspace on which the function appears to be smooth moreover using the special structure of such a function we compute smooth trajectories along which certain second order expansions can be obtained we also give an explicit expression for the hessian of a related lagrangian

an active set method for solving linearly constrained nonsmooth optimization problems an algorithm for solving linearly constrained optimization problems is proposed the search direction is computed by a bundle principle and the constraints are treated through an active set strategy difficulties that arise when the objective function is nonsmooth require a clever choice of a constraint to relax a certain nondegeneracy assumption is necessary to obtain convergence

multiple existence of solutions for a semilinear elliptic problem with neumann boundary condition the multiplicity of solutions for semilinear elliptic equations with exponential growth nonlinearities is treated the approach to the problem is a variational method

free steering relaxation methods for problems with strictly convex costs and linear constraints we consider dual coordinate ascent methods for minimizing a strictly convex possibly nondifferentiable function subject to linear constraints such methods are useful in large scale applications e g entropy maximization quadratic programming network flow because they are simply can exploit sparsity and in certain cases are highly parallelizable we establish their global convergence under weak conditions and a free steering order of relaxation previous comparable results were restricted to special problems with separable costs and equality constraints our convergence framework unifies to a certain extent the approaches of bregman censor and lent de pierro and iusem and luo and tseng and complements that of bertsekas and tseng

an aggregate subgradient method for nonsmooth and nonconvex minimization a class of implementable algorithms is described for minimizing any convex not necessarily differentiable functionf of several variables the methods require only the calculation off and one subgradient off at designated points they generalize lemarechal s bundle method more specifically instead of using all previously computed subgradients in search direction finding subproblems that are quadratic programming problems the methods use an aggregate subgradient which is recursively updated as the algorithms proceed each algorithm yields a minimizing sequence of points and iff has any minimizers then this sequence converges to a solution of the problem particular members of this algorithm class terminate whenf is piecewise linear the methods are easy to implement and have flexible storage requirements and computational effort per iteration that can be controlled by a user

optimal runge kutta schemes for discontinuous galerkin space discretizations applied to wave propagation problems we study the performance of methods of lines combining discontinuous galerkin spatial discretizations and explicit runge kutta time integrators with the aim of deriving optimal runge kutta schemes for wave propagation applications we review relevant runge kutta methods from literature and consider schemes of order q from 3 to 4 and number of stages up to q 4 for optimization from a user point of view the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method two scenarios are defined in the first one the element size is totally free and a 8 stage fourth order runge kutta scheme is found to minimize a cost measure depending on both accuracy and stability in the second one the elements are assumed to be constrained to such a small size by geometrical features of the computational domain that accuracy is disregarded we then derive one 7 stage third order scheme and one 8 stage fourth order scheme that maximize the stability limit the performance of the three new schemes is thoroughly analyzed and the benefits are illustrated with two examples for each of these runge kutta methods we provide the coefficients for a 2n storage implementation along with the information needed by the user to employ them optimally

modeling shape analysis and computation of the equilibrium pore shape near a pem pem intersection abstract in this paper we study the equilibrium shape of an interface that represents the lateral boundary of a pore channel embedded in an elastomer the model consists of a system of pdes comprising a linear elasticity equation for displacements within the elastomer and a nonlinear poisson equation for the electric potential within the channel filled with protons and water to determine the equilibrium interface a variational approach is employed we analyze i the existence and uniqueness of the electrical potential ii the shape derivatives of state variables and iii the shape differentiability of the corresponding energy and the corresponding euler lagrange equation the latter leads to a modified young laplace equation on the interface this modified equation is compared with the classical young laplace equation by computing several equilibrium shapes using a fixed point algorithm

multiple solutions of a class of neumann problem for semilinear elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations with unbounded nonlinearity by the implicit function theorem and the morse theory

bfgs with update skipping and varying memory we give conditions under which limited memory quasi newton methods with exact line searches will terminate in n steps when minimizing n dimensional quadratic functions we show that although all broyden family methods terminate in n steps in their full memory versions only bfgs does so with limited memory additionally we show that full memory broyden family methods with exact line searches terminate in at most n p steps when p matrix updates are skipped we introduce new limited memory bfgs variants and test them on nonquadratic minimization problems

broadband impedance boundary conditions for the simulation of sound propagation in the time domain an accurate and practical surface impedance boundary condition in the time domain has been developed for application to broadband frequency simulation in aeroacoustic problems to show the capability of this method two kinds of numerical simulations are performed and compared with the analytical experimental results one is acoustic wave reflection by a monopole source over an impedance surface and the other is acoustic wave propagation in a duct with a finite impedance wall both single frequency and broadband frequency simulations are performed within the framework of linearized euler equations a high order dispersion relation preserving finite difference method and a low dissipation low dispersion runge kutta method are used for spatial discretization and time integration respectively the results show excellent agreement with the analytical experimental results at various frequencies the method accurately predicts both the amplitude and the phase of acoustic pressure and ensures the well posedness of the broadband time domain impedance boundary condition