homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

algebras of higher dimension for displacement decompositions and computations with toeplitz plus hankel matrices abstract using the concept of displacement rank we suggest new formulas for the representation of a matrix in the form of a sum of products of matrices belonging to two particular matrix algebras having dimension about 2n and being noncommutative so far only n dimensional commutative matrix algebras have been used in this kind of applications we exploit the higher dimension of these algebras in order to reduce with respect to other decompositions the number of matrix products that have to be added for representing certain matrices interesting results are obtained in particular for toeplitz plus hankel like matrices a class that includes for example the inverses of toeplitz plus hankel matrices actually the new representation allows us to improve the complexity bounds for the product with preprocessing of these matrices by a vector

extended fast fixed order rls adaptive filters the existing derivations of conventional fast rls adaptive filters are intrinsically dependent on the shift structure in the input regression vectors this structure arises when a tapped delay line fir filter is used as a modeling filter we show unlike what original derivations may suggest that fast fixed order rls adaptive algorithms are not limited to fir filter structures we show that fast recursions in both explicit and array forms exist for more general data structures such as orthonormally based models one of the benefits of working with orthonormal bases is that fewer parameters can be used to model long impulse responses

how to prove that a preconditioner cannot be superlinear in the general case of multilevel toeplitz matrices we recently proved that any multilevel circulant preconditioner is not superlinear a cluster it may provide cannot be proper the proof was based on the concept of quasi equimodular matrices although this concept does not apply for example to the sine transform matrices in this paper with a new concept of partially equimodular matrices we cover all trigonometric matrix algebras widely used in the literature we propose a technique for proving the nonsuperlinearity of certain frequently used preconditioners for some representative sample multilevel matrices at the same time we show that these preconditioners are in a certain sense the best among the sublinear preconditioners with only a general cluster for multilevel toeplitz matrices

matrix decompositions using displacement rank and classes of commutative matrix algebras using the notion of displacement rank we look for a unifying approach to representations of a matrix a as sums of products of matrices belonging to commutative matrix algebras these representations are then considered in case a is the inverse of a toeplitz or a toeplitz plus hankel matrix some well known decomposition formulas for a gohberg semencul or kailath et al gader bini pan and gohberg olshevsky turn out to be special cases of the above representations new formulas for a in terms of algebras of symmetric matrices are studied and their computational aspects are discussed

another approach to the existence of value functions of stochastic differential games abstract the existence of value functions for general two player zero sum stochastic differential games has been obtained by fleming and souganidis in this paper we present a new approach to this problem we prove optimality inequalities of dynamic programming for viscosity sub and supersolutions of the associated bellman isaacs equations these inequalities are well known for deterministic differential games but are new for stochastic differential games it then easily follows that value functions are the unique viscosity solutions of the bellman isaacs equations and satisfy the principle of dynamic programming the results presented here are not the same as those of fleming and souganidis because we work with different reference spaces and the independence of value functions of the choice of reference spaces is not clear to us

fast scattered data approximation with neumann and other boundary conditions abstract an important problem in applications such as signal and image procesing is the approximation of a function f from a finite set of randomly scattered data f x j a common and powerful approach is to construct a trigonometric least squares approximation based on the set of exponentials e 2 i kx this leads to fast numerical algorithms but suffers from disturbing boundary effects due to the underlying periodicity assumption on the data which is rarely satisfied in practice to overcome this drawback we impose neumann boundary conditions on the data this implies the use of cosine polynomials cos kx as basis functions we show that using cosine polynomials leads to a least squares problem involving certain toeplitz plus hankel matrices and derive estimates on the condition number of these matrices unlike other toeplitz plus hankel matrices these matrices cannot be diagonalized by the discrete cosine transform dct but they still allow a fast matrix vector multiplication via dct which gives rise to fast conjugate gradient type algorithms we show how the results can be generalized to higher dimensions we also consider anti symmetric boundary conditions leading to sine polynomials as proper trigonometric basis finally we demonstrate the performance of the proposed methods by an application to a two dimensional geophysical scattered data problem

viscosity method for homogenization of parabolic nonlinear equations in perforated domains abstract in this paper we develop a viscosity method for homogenization of nonlinear parabolic equations constrained by highly oscillating obstacles or dirichlet data in perforated domains the dirichlet data on the perforated domain can be considered as a constraint or an obstacle homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the porous medium equation in perforated domains for the simplicity we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls if the decay rate of the capacity of columns or the capacity of punctured ball is too high or too small the limit of u will converge to trivial solutions the critical decay rates of having nontrivial solution are obtained with the construction of barriers we also show the limit of u satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense

homogenization for fully nonlinear parabolic equations abstract this paper concerns the homogenization of fully nonlinear parabolic equations of the form t u e h t x t e 2 x e d 2 u e 0 in 0 t r n where the hamiltonian h t x x is periodic both in and in our aim is to establish sufficient conditions for the convergence as e 0 of u e to a solution u to the effective equation t u h t x d 2 u 0 in 0 t r n where the effective hamiltonian h is obtained by a parabolic equation called cell problem we shall prove that h inherits several properties of h we also consider the case that u e 0 x h x x e on r n we point out a sufficient condition for having u 0 x h x on r n with an effective initial datum h given by the asymptotic behaviour of the solution to the recession problem a parabolic cauchy problem related to 1 1

multiscale convergence and reiterated homogenisation this paper generalises the notion of two scale convergence to the case of multiple separated scales of periodic oscillations it allows us to introduce a multi scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients this new method is applied to a model problem with a finite or infinite number of microscopic scales namely the homogenisation of the heat equation in a composite material finally it is generalised to handle the homogenisation of the neumann problem in a perforated domain

homogenization of linear and semilinear second order parabolic pdes with periodic coefficients a probabilistic approach we study the limit of the solution of linear and semilinear second order pdes of parabolic type with rapidly oscillating periodic coefficients singular drift and singular coefficient of the zeroth order term our method of proof is fully probabilistic and builds upon the arguments in earlier work in the linear case we use the feynman kac formula to represent the solution of the parabolic pde and in the semilinear case we use an associated backward stochastic differential equation

directional homogenization of elliptic equations in non divergence form abstract the paper concerns regularities of solutions of elliptic equations in non divergence form with directional homogenization the main interest is to investigate the differences of regularities of datums that are needed and of solutions that are obtained between directions with homogenization and without homogenization we establish uniform interior l p 1 p estimates for d 2 u e and uniform interior c 0 1 estimates for d d x u e where x are the non homogenization directions our results are optimal

on uniqueness and existence of viscosity solutions of fully nonlinear second order elliptic pde s we prove several comparison and existence theorems for viscosity solutions of fully nonlinear degenerate elliptic equations one of them extends some recent uniqueness results by jensen some establish the uniqueness of solutions for second order isaacs equations and hence include the uniqueness results for bellman equations by p l lions our comparison results apply even for discontinuous solutions and so perron s method readily yields the existence of continuous solutions

an initial and boundary value problem modeling of fish like swimming in this paper we consider an initial and boundary value problem that models the self propelled motion of solids in a bidimensional viscous incompressible fluid the self propelling mechanism consisting of appropriate deformations of the solids is a simplified model of the propulsion mechanism of fish like swimmers the governing equations consist of the navier stokes equations for the fluid coupled to newton s laws for the solids since we consider the case in which the fluid solid system fills a bounded domain we have to tackle a free boundary value problem the main theoretical result in the paper asserts the global existence and uniqueness up to possible contacts of strong solutions of this problem the second novel result of this work is the provision of a numerical method for the fluid solid system this method provides a simulation of the simultaneous displacement of several swimmers and is tested on several examples

rank one plus a null lagrangian is an inherited property of two dimensional compliance tensors under homogenisation assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank one tensor plus a null lagrangian there is only one symmetric one in 2d the purpose of this paper is to prove that the effective compliance tensor has the same representation rank one plus the null lagrangian this statement generalizes the well known result of hill 14 15 that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus it also generalizes the surprising discovery in 1 that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined the present paper sheds light on this effect by placing it in a more general framework and using some elliptic pde theory rather than the translation method our results allow us to calculate the polycrystalline g closures of the special class of crystals under consideration our analysis is contrasted with a two dimensional model problem for shape memory polycrystals we show that the two problems can be thought of as elastic percolation problems one elliptic one hyperbolic

representations of toeplitz plus hankel matrices using trigonometric transformations with application to fast matrix vector multiplication abstract representations of real toeplitz and toeplitz plus hankel matrices are presented that involve real trigonometric transformations dct dst dht and diagonal matrices these representations can be used for fast matrix vector multiplication in particular it is shown that the multiplication of an n n toeplitz plus hankel matrix by a vector requires only 4 transformations of length n plus o n operations

point vortex dynamics as zero radius limit of the motion of a rigid body in an irrotational fluid the vortex point system is usually considered as an idealized model where the vorticity of an ideal incompressible two dimensional fluid is concentrated in a finite number of moving points in the case of a single vortex in an otherwise irrotational ideal fluid occupying a bounded and simply connected two dimensional domain the motion is given by the so called kirchhoff routh velocity which depends only on the domain the main result of this paper establishes that this dynamics can also be obtained as the limit of the motion of a rigid body immersed in such a fluid when the body shrinks to a pointwise massless particle with fixed circulation the rigid body is assumed to be only accelerated by the force exerted by the fluid pressure on its boundary the fluid velocity and pressure being given by the incompressible euler equations with vanishing vorticity the circulation of the fluid velocity around the particle is conserved as time proceeds according to kelvin s theorem and gives the strength of the limit point vortex in order to understand the solid dynamics one has to evaluate the pressure field on the boundary of the solid that is where the singularity is concentrated at the limit moreover the newton equations driving the particle s dynamics involve a singular perturbation problem in time whereas the fluid state may be seen as solving an auxiliary problem the fluid velocity can indeed be recovered by an elliptic type problem where time appears only as a parameter since the analysis involves singular perturbation problem both in space and in time and that nothing excludes a priori some sharp energy transfer from the fluid toward the rigid immersed particle it is crucial to recast the solid dynamics under normal forms this form makes appear the gyroscopic structure of these singularities and allows one to obtain uniform estimates on the dynamics thanks to some energy type quantities modulated by the limit dynamics and therefore to describe the transition of the dynamics in the limit in order to get such normal forms we first establish that the newton equations can be seen as a geodesic equation with a metric associated with the well known added inertia phenomenon under an applied force similar to the lorentz force which can be seen as an extension of the celebrated kutta joukowski lift force then in the limit surprising relations between leading and subprincipal orders of various terms and the modulation variables show up and allow us to establish a normal form with a more gyroscopic structure as a byproduct of our analysis we also prove that in the different regime where the body shrinks with a fixed mass the limit equation is a second order differential equation involving a kutta joukowski type lift force which extends the result of glass o lacave c sueur f on the motion of a small body immersed in a two dimensional incompressible perfect fluid bull soc math france volume 142 fascicule 3 489 536 2014 to the case where the domain occupied by the solid fluid system is bounded

trigonometric transform splitting methods for real symmetric toeplitz systems abstract in this paper we study efficient iterative methods for real symmetric toeplitz systems based on the trigonometric transformation splitting tts of the real symmetric toeplitz matrix a theoretical analyses show that if the generating function f of the n n toeplitz matrix a is a real positive even function then the tts iterative methods converge to the unique solution of the linear system of equations for sufficient large n moreover we derive an upper bound of the contraction factor of the tts iteration which is dependent solely on the spectra of the two tts matrices involved different from the cscs iterative method in ng 2003 in which all operations counts concern complex operations when the dfts are employed even if the toeplitz matrix a is real and symmetric our method only involves real arithmetics when the dcts and dsts are used the numerical experiments show that our method works better than cscs iterative method and much better than the positive definite and skew symmetric splitting pss iterative method in bai et al 2005 and the symmetric gauss seidel sgs iterative method

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

singular perturbations of nonlinear degenerate parabolic pdes a general convergence result the main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic pdes partial differential equations with highly oscillating initial data it substantially generalizes some results obtained previously in 2 under the only assumptions that the hamiltonian is ergodic and stabilizing in a suitable sense the solutions are proved to converge in a relaxed sense to the solution of a limit cauchy problem with appropriate effective hamiltonian and initial data in its formulation our convergence result is analogous to the stability property of barles and perthame it should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques a detailed exposition of ergodicity and stabilization is given with many examples applications to homogenization and averaging are also discussed

the perturbed test function method for viscosity solutions of nonlinear pde the method of viscosity solutions for nonlinear partial differential equations pdes justifies passages to limits by in effect using the maximum principle to convert to the corresponding limit problem for smooth test functions we describe in this paper a perturbed test function device which entails various modifications of the test functions by lower order correctors applications include homogenisation for quasilinear elliptic pdes and approximation of quasilinear parabolic pdes by systems of hamilton jacobi equations

higher order convergence rates in theory of homogenization ii oscillatory initial data abstract we establish higher order convergence rates in periodic homogenization of fully nonlinear uniformly parabolic cauchy problems accompanied with rapidly oscillating initial data such result is new even for linear problems here we construct higher order initial layer and interior correctors which describe the oscillatory behavior near the initial and interior time zone of the domain to construct higher order correctors we develop a regularity theory in macroscopic scales and prove an exponential decay estimate for initial layer correctors the higher order expansion requires an iteration process successively correcting the initial layer then the interior this leads to a more complicated asymptotic expansion as compared to the non oscillating data case and this complexity is present even in the linear case a notable observation for fully nonlinear operators is that even if the given operator is space time periodic the interior correctors become aperiodic in the time variable as we proceed with the iteration process moreover each interior corrector of higher order is paired with a space time periodic version and the difference between the two decays exponentially fast with time

an embedding approach to frequency domain and subband adaptive filtering frequency domain and subband implementations improve the computational efficiency and the convergence rate of adaptive schemes the well known multidelay adaptive filter mdf belongs to this class of block adaptive structures and is a dft based algorithm we develop adaptive structures that are based on the trigonometric transforms discrete cosine transform dct and discrete sine transform dst and on the discrete hartley transform dht as a result these structures involve only real arithmetic and are attractive alternatives in cases where the traditional dft based scheme exhibits poor performance the filters are derived by first presenting a derivation for the classical dft based filter that allows us to pursue these extensions immediately the approach used in this paper also provides further insights into subband adaptive filtering

korovkin theorems and linear positive gram matrix algebra approximations of toeplitz matrices abstract in this paper we are concerned with the approximation of toeplitz matrices generated by continuous 2 periodic functions f i r with i for this purpose we define a class of matrix algebras related to suitable choices of gram functions where we look for good preconditioners in particular we construct these preconditioners through linear operators and linear positive operators lpos approximating in some sense the function f then by making use of some matrix versions 39 42 41 of the korovkin and weierstrass theorems we analyze the convergence features of old and new preconditioners finally among the given results are adapted in order to deal with l1 generating functions and the related preconditioners are compared with the ones devised by using band toeplitz matrices 6 36

on cell problems for hamilton jacobi equations with non coercive hamiltonians and their application to homogenization problems abstract we study a cell problem arising in homogenization for a hamilton jacobi equation whose hamiltonian is not coercive we introduce a generalized notion of effective hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective hamiltonian under some sufficient conditions the result is applied to the associated homogenization problem we also show that homogenization for non coercive equations fails in general

homogenization of elliptic equations with principal part not in divergence form and hamiltonian with quadratic growth in this paper we consider the following problem here the coefficients aij and bi are smooth periodic with respect to the second variable and the matrix aij ij is uniformly elliptic the hamiltonian h is locally lipschitz continuous with respect to u and du and has quadratic growth with respect to du the hamilton jacobi beliman equations of some stochastic control problems are of this type our aim is to pass to the limit in 0 as tends to zero we assume the coefficients bi to be centered with respect to the invariant measure of the problem see the main assumption 3 13 then we derive l h and w p0 2 estimates for the solutions of 0 we also prove the following corrector s result this allows us to pass to the limit in 0 and to obtain this problem is of the same type as the initial one when 0 is the hamilton jacobi bellman equation of a stochastic control problem then 00 is also a hamilton jacobi bellman equation but one corresponding to a modified set of controls

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

homogenization for stochastic hamilton jacobi equations homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales we prove homogenization for certain stochastic hamilton jacobi partial differential equations the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale separation limit in some cases we also establish a central limit theorem

viscosity solutions methods for singular perturbations in deterministic and stochastic control viscosity solutions methods are used to pass to the limit in some penalization problems for first order and second order degenerate parabolic hamilton jacobi bellman equations this characterizes the limit of the value functions of singularly perturbed optimal control problems for deterministic systems and for controlled degenerate diffusions the results apply to cases where the usual order reduction method does not give the correct limit and to systems with fast state variables depending nonlinearly on the control some connections with ergodic control and periodic homogenization are discussed

stability of bsdes with random terminal time and homogenization of semilinear elliptic pdes in this paper we extend the probabilistic method for homogenization of semi linear parabolic pdes developed by buckdahn hu and peng to the case of elliptic pdes first we give a stability result for bsdes with random terminal time which are related to elliptic pdes as shown in peng stochastics stochastics rep 37 1991 61 74 in the one dimensional case we also partially relax the monotonicity assumption on the coefficient then we use these stability results for bsdes with random terminal time to study homogenization of systems of semilinear elliptic pdes

liouville properties and critical value of fully nonlinear elliptic operators abstract we prove some liouville properties for sub and supersolutions of fully nonlinear degenerate elliptic equations in the whole space our assumptions allow the coefficients of the first order terms to be large at infinity provided they have an appropriate sign as in ornstein uhlenbeck operators we give two applications the first is a stabilization property for large times of solutions to fully nonlinear parabolic equations the second is the solvability of an ergodic hamilton jacobi bellman equation that identifies a unique critical value of the operator

probabilistic approach to singular perturbations of semilinear and quasilinear parabolic pdes in 5 buckdahn et al study some homogenization properties of systems of semilinear parabolic partial di erential equations pdes in short with periodic structures using a probabilistic approach involving stability of backward stochastic di erential equations bsdes this method is also developed in 3 to take into account systems of semilinear elliptic pdes which are related to bsdes with random terminal time moreover buckdahn and hu in 4 deal with homogenization of quasilinear pdes showing o some kind of stability property of forward backward stochastic di erential equations fbsdes in short in this paper we adopt the same method in order to study singular perturbations of semilinear pdes as well as those of quasilinear pdes the strong stability properties of bsdes allow us to study the asymptotic behavior of the solution of a system of singularly perturbed semilinear pdes by using the nonlinear feynman kac formula developed by pardoux and peng 11 singular perturbations of quasi linear pdes are treated in the same way in view of their connections to fbsdes revealed by ma et al 13

control at a distance of the motion of a rigid body immersed in a two dimensional viscous incompressible fluid abstract we consider the motion of a rigid body immersed in a two dimensional viscous incompressible fluid with navier slip with friction conditions at the solid boundary the fluid solid system occupies the whole plane we prove the small time global exact controllability of the position and velocity of the solid when the control takes the form of a distributed force supported in a compact subset with nonvoid interior of the fluid domain away from the body the strategy relies on the introduction of a small parameter we consider fast and strong amplitude controls for which the navier stokes rigid body system behaves like a perturbation of the euler rigid body system by the means of a multi scale asymptotic expansion we construct a controlled solution to the navier stokes rigid body system thanks to some controlled solutions to euler rigid body type systems and by using that the influence of the boundary layer on the solid motion turns out to be sufficiently small

dft representations of toeplitz plus hankel bezoutians with application to fast matrixvector multiplication abstract representations for inverses of toeplitz plus hankel matrices and more general t h bezoutians with discrete fourier transformations are presented which can be used for fast matrix vector multiplication with the help of some of the formulas multiplication by a toeplitz plus hankel matrix inverse can be carried out with six dft s plus eight dft s for preprocessing which improves the so far best known result of seven dft s plus 10 dft s for preprocessing in e bozzo linear algebra appl 230 1995 127 150

generic controllability of 3d swimmers in a perfect fluid we address the problem of controlling a dynamical system governing the motion of a 3d weighted shape changing body swimming in a perfect fluid the rigid displacement of the swimmer results from the exchange of momentum between prescribed shape changes and the flow with the total impulse of the fluid swimmer system being null at every time we prove the following tracking results i synchronized swimming up to an arbitrarily small change of its density any swimmer can approximately follow any given trajectory while in addition undergoing approximately any given shape changes in this statement the control consists in arbitrarily small superimposed deformations ii freestyle swimming up to an arbitrarily small change of its density any swimmer can approximately track any given trajectory by suitably combining at most five basic movements that can be generically chosen no macro shape changes are prescribed in this statement

multiscale problems and homogenization for second order hamilton jacobi equations we prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic pdes as a special case we cover the iterated homogenization for such equations with oscillating initial data explicit examples among others are the two scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n scale homogenization of uniformly parabolic fully nonlinear pdes

exit time problems in optimal control and vanishing viscosity method the authors study the connections between deterministic exit time control problems and possibly discontinuous viscosity solutions of a first order hamilton jacobi hj equation up to the boundary this equation admits a maximum and a minimum solution that are the value functions associated to stopping time problems on the boundary when these solutions are equal they can be obtained through the vanishing viscosity method finally when the hj equation has a continuous solution it is proved to be the value function for the first exit time of the domain it is also the vanishing viscosity limit arising in particular in some large deviations problems

compactness methods in the theory of homogenization ii equations in non divergence form we prove c0 c1 and c1 1 a priori estimates for solutions of boundary value problems for elliptic operators with periodic coefficients of the form j 1ai j x 2 xi xj the constants in these estimates are independent of the small parameter and hence our results imply strengthened versions of the classical averaging theorems these results generalize to a wide class of linear operators in non divergence form including equations with lower order terms and nonuniformly oscillating coefficients as well as to certain nonlinear problems which we discuss in the last section

wiener hopf and spectral factorization of real polynomials by newton s method abstract we present a new method for factoring a real polynomial into the product of two polynomials which have their zeros inside and outside the unit circle respectively the approach is based on solving a nonlinear system by newton s method the jacobi matrix is a polynomial of a companion matrix and thus a bezoutian times the inverse of a triangular toeplitz matrix after getting deeper understanding of the displacement structure of the bezoutian each newton step can be reduced to a few applications of discrete fourier or cosine transforms and the lu decomposition of a conceived linear system with a cauchy like matrix these structural features are employed to design a fast algorithm which shows excellent numerical behavior for example in the case of the extremely ill conditioned test polynomial 2 z n k 0 n 1 z k 2 k 1 n z k of the degree 2 n 20000 we obtain the factorization after 22 newton steps with an error of 3 4 10 8 in the euclidean norm the execution time on a laptop being a couple of minutes the local quadratic convergence of our newton method is proved and explicit convergence balls are given on the basis of estimates for the lipschitz constant which occurs in kantorovich s theorem we also design and test two superfast versions of our method which for polynomials of degree 20000 typically need about half a minute to produce an initial coefficient vector and then perform the newton iteration within 1 s

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

transformation techniques for toeplitz and toeplitz plus hankel matrices part i transformations transformations of the form a c1ac2 are investigated that transform toeplitz and toeplitz plus hankel matrices into generalized cauchy matrices c1 and c2 are matrices related to the discrete fourier transformation or to various real trigonometric transformations combining these results with pivoting techniques in part ii algorithms for toeplitz and toeplitz plus hankel systems will be presented that are more stable than classical algorithms

a generalization of trigonometric transform splitting methods for spatial fractional diffusion equations abstract the spatial fractional diffusion equation can be discretized by employing the implicit finite difference scheme with the shifted grunwald formula and the given discretized linear systems have a diagonal plus toeplitz structure in this paper based on trigonometric transformation splitting tts we study efficient iterative method called gtts has two parameters as a focus we give the simple and effective optimal forms of these two parameters respectively we carry out some numerical experiments to illustrate the effectiveness and accuracy of this new algorithm

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

continuous dependence estimates and homogenization of quasi monotone systems of fully nonlinear second order parabolic equations abstract the aim of this paper is to extend the continuous dependence estimates proved by jakobsen and karlsen 2002 3 to quasi monotone systems of fully nonlinear second order parabolic equations as a by product of these estimates we get an holder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation in the second part of the paper we employ similar techniques to study the periodic homogenization of quasi monotone systems of fully nonlinear second order uniformly parabolic equations finally some examples are discussed

optimal trigonometric preconditioners for nonsymmetric toeplitz systems abstract this paper is concerned with the solution of systems of linear equations tnxn bn where tn n n denotes a sequence of nonsingular nonsymmetric toeplitz matrices arising from a generating function of the wiener class we present a technique for the fast construction of optimal trigonometric preconditioners mn mn t ntn of the corresponding normal equation which can be extended to toeplitz least squares problems in a straightforward way moreover we prove that the spectrum of the preconditioned matrix mn1t ntn is clustered at 1 such that the pcg method applied to the normal equation converges superlinearly numerical tests confirm the theoretical expectations