On a singular perturbation problem involving a ``circular-well' potential

We study the asymptotic behavior, as a small parameter goes to 0, of the minimizers for a variational problem which involves a ``circular-well' potential, i.e., a potential vanishing on a closed smooth curve in . We thus generalize previous results obtained for the special case of the Ginzburg-Landau potential.

     

On the complexity of stochastic integration

We study the complexity of approximating stochastic integrals with error for various classes of functions. For Ito integration, we show that the complexity is of order , even for classes of very smooth functions. The lower bound is obtained by showing that Ito integration is not easier than Lebesgue integration in the average case setting with the Wiener measure. The upper bound is obtained by the Milstein algorithm, which is almost optimal in the considered classes of functions. The Milstein algorithm uses the values of the Brownian motion and the integrand. It is bilinear in these values and is very easy to implement. For Stratonovich integration, we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of Ito integration.

     

Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth

In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter . The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in .

     

A variational principle for domino tilings

We formulate and prove a variational principle (in the sense of thermodynamics) for random domino tilings, or equivalently for the dimer model on a square grid. This principle states that a typical tiling of an arbitrary finite region can be described by a function that maximizes an entropy integral. We associate an entropy to every sort of local behavior domino tilings can exhibit, and prove that almost all tilings lie within (for an appropriate metric) of the unique entropy-maximizing solution. This gives a solution to the dimer problem with fully general boundary conditions, thereby resolving an issue first raised by Kasteleyn. Our methods also apply to dimer models on other grids and their associated tiling models, such as tilings of the plane by three orientations of unit lozenges.

     

The probability that a slightly perturbed numerical analysis problem is difficult

We prove a general theorem providing smoothed analysis estimates for conic condition numbers of problems of numerical analysis. Our probability estimates depend only on geometric invariants of the corresponding sets of ill-posed inputs. Several applications to linear and polynomial equation solving show that the estimates obtained in this way are easy to derive and quite accurate. The main theorem is based on a volume estimate of -tubular neighborhoods around a real algebraic subvariety of a sphere, intersected with a spherical disk of radius . Besides and , this bound depends only on the dimension of the sphere and on the degree of the defining equations.

     

The stability of the exponential equation

We generalize the well-known Baker's superstability result for exponential mappings with values in the field of complex numbers to the case of an arbitrary commutative complex semisimple Banach algebra. It was shown by Ger that the superstability phenomenon disappears if we formulate the stability question for exponential complex-valued functions in a more natural way. We improve his result by showing that the maximal possible distance of an -approximately exponential function to the set of all exponential functions tends to zero as tends to zero. In order to get this result we have to prove a stability theorem for real-valued functions additive modulo the set of all integers .

     

A random variational principle with application to weak Hadamard differentiability of convex integral functionals

We present a random version of the Borwein-Preiss smooth variational principle, stating that under suitable conditions, the set of minimizers of a perturbed function depending on a random variable, admits a measurable selection. Two applications are given. The first one shows that if is a superreflexive Banach space, then any convex continuous integral functional on from a certain class (in particular the usual norm), is weak Hadamard differentiable on a subset whose complement is -very porous. The second application is a random version of the Caristi fixed point theorem for multifunctions.

     

A nonconforming combination of the finite element and volume methods with an anisotropic mesh refinement for a singularly perturbed convection-diffusion equation

In this paper we formulate and analyze a discretization method for a 2D linear singularly perturbed convection-diffusion problem with a singular perturbation parameter . The method is based on a nonconforming combination of the conventional Galerkin piecewise linear triangular finite element method and an exponentially fitted finite volume method, and on a mixture of triangular and rectangular elements. It is shown that the method is stable with respect to a semi-discrete energy norm and the approximation error in the semi-discrete energy norm is bounded by with independent of the mesh parameter , the diffusion coefficient and the exact solution of the problem.

     

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.

Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor is bounded by for some exponent and some positive constant . The -exponent of tractability is defined as the smallest power of in these bounds. It is shown by using Monte Carlo quadrature that the -exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the -exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

     

The covering numbers and ``low -estimate" for quasi-convex bodies

This article gives estimates on the covering numbers and diameters of random proportional sections and projections of quasi-convex bodies in . These results were known for the convex case and played an essential role in the development of the theory. Because duality relations cannot be applied in the quasi-convex setting, new ingredients were introduced that give new understanding for the convex case as well.

     

Maximum norm error analysis of a 2d singularly perturbed semilinear reaction-diffusion problem

A semilinear reaction-diffusion equation with multiple solutions is considered in a smooth two-dimensional domain. Its diffusion parameter is arbitrarily small, which induces boundary layers. Constructing discrete sub- and super-solutions, we prove existence and investigate the accuracy of multiple discrete solutions on layer-adapted meshes of Bakhvalov and Shishkin types. It is shown that one gets second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the discrete maximum norm, uniformly in for . Here is the maximum side length of mesh elements, while the number of mesh nodes does not exceed . Numerical experiments are performed to support the theoretical results.

     

An extension of Elton's theorem to complex Banach spaces

Let 0$"> be sufficiently small. Then, for , there exists such that if are vectors in the unit ball of a complex Banach space which satisfy

(where are independent complex Steinhaus random variables), then there exists a set , with , such that

for all (). The dependence on of the threshold proportion is sharp.

     

Stability of small amplitude boundary layers for mixed hyperbolic-parabolic systems

We consider an initial boundary value problem for a symmetrizable mixed hyperbolic-parabolic system of conservation laws with a small viscosity , When the boundary is noncharacteristic for both the viscous and the inviscid system, and the boundary condition dissipative, we show that converges to a solution of the inviscid system before the formation of shocks if the amplitude of the boundary layer is sufficiently small. This generalizes previous results obtained for invertible and the linear study of Serre and Zumbrun obtained for a pure Dirichlet's boundary condition.

     

A proof of W. T. Gowers' theorem

W. T. Gowers' theorem asserts that for every Lipschitz function and 0$">, there exists an infinite-dimensional subspace of such that the oscillation of on is at most . The proof of this theorem has been reduced by W. T. Gowers to the proof of a new Ramsey type theorem. Our aim is to present a proof of the last result.

     

Prox-regular functions in variational analysis

The class of prox-regular functions covers all l.s.c., proper, convex functions, lower- functions and strongly amenable functions, hence a large core of functions of interest in variational analysis and optimization. The subgradient mappings associated with prox-regular functions have unusually rich properties, which are brought to light here through the study of the associated Moreau envelope functions and proximal mappings. Connections are made between second-order epi-derivatives of the functions and proto-derivatives of their subdifferentials. Conditions are identified under which the Moreau envelope functions are convex or strongly convex, even if the given functions are not.

     

A ``nonlinear' proof of Pitt's compactness theorem

Using Stegall's variational principle, we present a simple proof of Pitt's theorem that bounded linear operators from into are compact for .

     

Strong tractability of multivariate integration using quasi-Monte Carlo algorithms

We study quasi-Monte Carlo algorithms based on low discrepancy sequences for multivariate integration. We consider the problem of how the minimal number of function evaluations needed to reduce the worst-case error from its initial error by a factor of depends on and the dimension . Strong tractability means that it does not depend on and is bounded by a polynomial in . The least possible value of the power of is called the -exponent of strong tractability. Sloan and Wozniakowski established a necessary and sufficient condition of strong tractability in weighted Sobolev spaces, and showed that the -exponent of strong tractability is between 1 and 2. However, their proof is not constructive.

In this paper we prove in a constructive way that multivariate integration in some weighted Sobolev spaces is strongly tractable with -exponent equal to 1, which is the best possible value under a stronger assumption than Sloan and Wozniakowski's assumption. We show that quasi-Monte Carlo algorithms using Niederreiter's -sequences and Sobol sequences achieve the optimal convergence order for any 0$"> independent of the dimension with a worst case deterministic guarantee (where is the number of function evaluations). This implies that strong tractability with the best -exponent can be achieved in appropriate weighted Sobolev spaces by using Niederreiter's -sequences and Sobol sequences.

     

Estimates on the mean growth of functions in convex domains of finite type

Let be a bounded convex domain of finite type in with smooth boundary. In this paper, we prove the following inequality:

where , and . This is a generalization of some classical result of Hardy-Littlewood for the case of the unit disc. Using this inequality, we can embed the space into a weighted Bergman space in a convex domain of finite type.

     

Sectional bodies associated with a convex body

We define the sectional bodies associated to a convex body in and two related measures of symmetry. These definitions extend those of Grünbaum (1963). As Grünbaum conjectured, we prove that the simplices are the most dissymmetrical convex bodies with respect to these measures. In the case when the convex body has a sufficiently smooth boundary, we investigate some limit behaviours of the volume of the sectional bodies.

     

Generalized \mathcal{C}-concave conditions and their applications

We first introduce two general $\mathcal{C}$ -concave conditions, and show the implications between $\mathcal{C}$ -concave, diagonally $\mathcal{C}$ -concave, diagonally $\mathcal{C}$ -quasiconcave, and ??-diagonally $\mathcal{C}$ -quasiconcave conditions which generalize both concavity and quasiconcavity simultaneously without assuming the linear structure. Using the ??-diagonal $\mathcal{C}$ -quasiconcavity, we prove two non-compact minimax inequalities in a topological space which generalize Fan??s minimax inequality and its generalizations in several aspects. As applications, we will prove a general minimax theorem and basic geometric formulations of the minimax inequality in a topological space.