Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ⁿ⁺¹× ⁿ⁺¹, and then develop a periodisation procedure for the calculus of symbols on the cylinder ×. We show Gårding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.     

Computational complexity of the integration problem for anisotropic classes

We determine the exact order of -complexity of the numerical integration problem for the anisotropic class W^r(I^d) and H^r(I^d) with respect to the worst case randomized methods and the average case deterministic methods. We prove this result by developing a decomposition technique of Borel measure on unit cube of d-dimensional Euclidean space. Moreover by the imbedding relationship between function classes we extend our results to the classes of functions W_p(I^d) and H_p(I^d). By the way we highlight some typical results and stress the importance of some open problems related to the complexity of numerical integration. Project supported by the fund of Personnel Division of Nankai University and the Program of One Hundred Distinguished Chinese Scientists of the Chinese Academy of Sciences.     

Some Properties of the Anisotropic Riesz-Bessel Potential

The Hardy-Littlewood-Bessel maximal functions (B-maximal functions), Morrey-Bessel and BMO-Bessel spaces were introduced and studied in [6]. In the present paper, we study the anisotropic Riesz-Bessel potential (B-potential) in the Morrey-Bessel and BMO-Bessel spaces. We obtain a theorem analogous to the Sobolev theorem, for the anisotropic Riesz-Bessel potential in Morrey-Bessel spaces. We introduce a metric characteristic _p, in the space of locally integrable functions and establish estimates connecting the characteristics of the image and preimage of the corresponding integral transform. These estimates are of independent interest. Moreover, they are used for the investigation of integral operators in different scales of Banach function spaces, in particular, in weighted L _p -spaces. The results seem to be new even in the isotropic case.     

Atomic and molecular decompositions of anisotropic Besov spaces

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic -transforms of Frazier and Jawerth [19, 21] to non-isotropic settings.Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the -transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].The author was partially supported by the NSF grant DMS-0441817.     

A Note on the Robust 1-Center Problem on Trees

We consider the robust 1-center problem on trees with uncertainty in vertex weights and edge lengths. The weights of the vertices and the lengths of the edges can take any value in prespecified intervals with unknown distribution. We show that this problem can be solved in O(n ³ logn) time thus improving on Averbakh and Berman's algorithm with time complexity O(n ⁶). For the case when the vertices of the tree have weights equal to 1 we show that the robust 1-center problem can be solved in O(nlogn) time, again improving on Averbakh and Berman's time complexity of O(n ² logn).     

Anisotropic Fourier multipliers and singular integrals for vector-valued functions

We show that an anisotropic Mihlin-type condition on the symbol guarantees the boundedness of the associated Fourier multiplier operator on L ^p (R ⁿ ,X) for 1 < p < ∞ and an arbitrary UMD space X. In many cases, this result can be used as a substitute for the Marcinkiewicz–Lizorkin multiplier theorem, which is invalid in general UMD spaces. An application to anisotropic singular integrals is given. Mathematics Subject Classification (2000) 42B15, 42B20, 46E40     

Conservation laws with continuous flux functions

In the present paper we investigate uniqueness problem for generalized entropic solution of stationnary and evolution conservation laws under anisotropic conditions on vector-flux functions in the classesL andL L ¹.This research was performed while visiting the Equipe de Mathématiques in Besançon.     

An inexact primal–dual path following algorithm for convex quadratic SDP

We propose primal–dual path-following Mehrotra-type predictor–corrector methods for solving convex quadratic semidefinite programming (QSDP) problems of the form: , where is a self-adjoint positive semidefinite linear operator on , b∈R ^m , and is a linear map from to R ^m . At each interior-point iteration, the search direction is computed from a dense symmetric indefinite linear system (called the augmented equation) of dimension m + n(n + 1)/2. Such linear systems are typically very large and can only be solved by iterative methods. We propose three classes of preconditioners for the augmented equation, and show that the corresponding preconditioned matrices have favorable asymptotic eigenvalue distributions for fast convergence under suitable nondegeneracy assumptions. Numerical experiments on a variety of QSDPs with n up to 1600 are performed and the computational results show that our methods are efficient and robust. Research supported in part by Academic Research Grant R146-000-076-112.     

Estimates for stable hypersurfaces of prescribed F-mean curvature

We consider immersed hypersurfaces :Mⁿ→ℝⁿ⁺¹ with prescribed anisotropic mean curvature . Such hypersurfaces can be characterized as critical points of parametric functionals of the type with an elliptic Lagrangian F depending on normal directions and a smooth vectorfield Q satisfying . We establish curvature estimates for stable hypersurfaces of dimension n≤5, provided F is C³-close to the area integrand.     

A Locking-Free Nonconforming Finite Element Method for Planar Linear Elasticity

We propose a locking-free nonconforming finite element method based on quadrilaterals to solve for the displacement variable in the pure displacement boundary value problem of planar linear elasticity. The method proposed in this paper is optimal and robust in the sense that the convergence estimates in the energy and L ²-norms are independent of the Lamé parameter .     

Spline approximation methods for multidimensional periodic pseudodifferential equations

We investigate several numerical methods for solving the pseudodifferential equationAu=f on the n-dimensional torusT ⁿ . We examine collocation methods as well as Galerkin-Petrov methods using various periodical spline functions. The considered spline spaces are subordinated to a uniform rectangular or triangular grid. For given approximation method and invertible pseudodifferential operatorA we compute a numerical symbol ^C , resp. ^G , depending onA and on the approximation method. It turns out that the stability of the numerical method is equivalent to the ellipticity of the corresponding numerical symbol. The case of variable symbols is tackled by a local principle. Optimal error estimates are established.The second author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant namber Ko 634/32-1.     

Intervals of periodicity and absolute stability of explicit nyström methods fory″=f(x,y)

We examine intervals of periodicity and absolute stability of explicit Nyström methods fory=f(x,y) by applying these methods to the test equationy=– ² y,>0. We consider in detail general families of fourth-order explicit Nyström methods; necessary and sufficient conditions are given to characterize methods which possess non-vanishing intervals of periodicity and absolute stability. We establish closed-form expressions giving intervals of periodicity and/or absolute stability, in case these exist, for any fourth-order method. We then show that the methodM ₄ (1/6, 5/6) has the largest interval of periodicity out of all fourth-order methods; we also obtain the fourth-order method with the largest interval of absolute stability. The corresponding results for second and third-order explicit Nyström methods are also included.     

Crack Singularities for General Elliptic Systems

We consider general homogeneous Agmon‐Douglis‐Nirenberg elliptic systems with constant coefficients complemented by the same set of boundary conditions on both sides of a crack in a two‐dimensional domain. We prove that the singular functions expressed in polar coordinates (r, θ) near the crack tip all have the form r^{k + 1/2}φ(θ) with k ≥ 0 integer, with the possible exception of a finite number of singularities of the form r^k log r φ(θ). We also prove results about singularities in the case when the boundary conditions on the two sides of the crack are not the same, and in particular in mixed Dirichlet‐Neumann boundary value problems for strongly coercive systems: in the latter case, we prove that the exponents of singularity have the form with real η and integer k. This is valid for general anisotropic elasticity too.     

Dual coordinate step methods for linear network flow problems

We review a class of recently-proposed linear-cost network flow methods which are amenable to distributed implementation. All the methods in the class use the notion of-complementary slackness, and most do not explicitly manipulate any global objects such as paths, trees, or cuts. Interestingly, these methods have stimulated a large number of newserial computational complexity results. We develop the basic theory of these methods and present two specific methods, the-relaxation algorithm for the minimum-cost flow problem, and theauction algorithm for the assignment problem. We show how to implement these methods with serial complexities of O(N ³ logNC) and O(NA logNC), respectively. We also discuss practical implementation issues and computational experience to date. Finally, we show how to implement-relaxation in a completely asynchronous, chaotic environment in which some processors compute faster than others, some processors communicate faster than others, and there can be arbitrarily large communication delays.Supported by Grant NSF-ECS-8217668 and by the Army Research Office under grant DAAL03-86-K-0171. Thanks are due to David Castañon, Paul Tseng, and Jim Orlin for their helpful comments.     

Whitney type inequalities for local anisotropic polynomial approximation

We prove a multivariate Whitney type theorem for the local anisotropic polynomial approximation in L_p(Q) with 1≤p≤∞. Here Q is a d-parallelepiped in R^d with sides parallel to the coordinate axes. We consider the error of best approximation of a function f by algebraic polynomials of fixed degree at most r_i−1 in variable , and relate it to a so-called total mixed modulus of smoothness appropriate to characterizing the convergence rate of the approximation error. This theorem is derived from a Johnen type theorem on equivalence between a certain K-functional and the total mixed modulus of smoothness which is proved in the present paper.     

Robust M-estimation of a dispersion matrix with a structure

An iterative algorithm for the robust M-estimation of the dispersion matrix of the form + ² I _p has been given. This algorithm converges after some steps and reduces the effect of outliers on the covariance matrix. The consistency and asymptotic normality of the estimator are established.     

Minimax-robust prediction of discrete time series

Summary We discuss a robust approach for predicting a weakly stationary discrete time series whose spectral density f is not exactly known. We assume that we know that f ), where is a convex set of spectral densities fulfilling some not too stringent conditions. We proof the existence of a most indeterministic density f ⁰ in , and we show that the classical optimal linear predictor for a time series with spectral density f ⁰ is mini-max-robust in the sense that it minimizes the maximal possible prediction error.We investigate some special models , and, in doing so, we illustrate a generally applicable method for determining the robust predictor. In particular, we discuss model sets which are defined by conditions on a finite part of the autocovariance sequence of the corresponding time series. These examples are of particular interest as the most indeterministic density is an autoregressive one, i.e. the robust predictor is finite. We discuss connections between this type of model set and maximum entropy and generalized maximum entropy spectral estimates.     

Some results on surface measures in calculus of variations

We study some relations between the concepts of perimeter, Hausdorff measure, and Minkowsky content, when R ^N is endowed with a convex Finsler metric depending in a continuous way on the position. We show some connections with the theory of -convergence and with the anisotropic motion of a smooth hypersurface by mean curvature.This work was partially supported by NSF Grant DMS-9008999, and by MURST (Progetto Nazionale «Equazioni di Evoluzione e Applicazioni Fisico-Matematiche» and «Analisi Numerica e Matematica Computazionale») and CNR (IAN and Contracts 92.00833.01, 93.00564.01) of Italy.     

On accelerating PL continuation algorithms by predictor—corrector methods

We consider path following methods designed to trace the zeroes of a continuous or differentiable mapF:R ⁿ⁺¹ R ⁿ . These methods are applicable e.g. in the numerical study of nonlinear eigenvalue and bifurcation problems. Traditionally a simplicial algorithm is based on a fixed triangulationT ofR ⁿ⁺¹ and a corresponding piecewise linear approximationF _T :R ⁿ⁺¹ R ⁿ .4 A fixed triangulation algorithm then traces the zeroes ofF _T via a complementary pivoting procedure. We present two kinds of hybrid algorithms that have the structure of a predictor—corrector method using simplicial methods to carry out the corrector steps. Numerical experience is reported showing the improvement in efficiency as compared to the fixed triangulation algorithm.