A geometrical approach to monotone functions in \mathbb R^n

The paper is concerned with the fine properties of monotone functions on . We study the continuity and differentiability properties of these functions, the approximability properties, the structure of the distributional derivatives and of the weak Jacobians. Moreover, we exhibit an example of a monotone function u which is the gradient of a convex function and whose weak Jacobian Ju is supported on a purely unrectifiable set. Received October 9, 1996; in final form April 21, 1997     

Finite volume schemes for Hamilton–Jacobi equations

Summary. We introduce two classes of monotone finite volume schemes for Hamilton-Jacobi equations. The corresponding approximating functions are piecewise linear defined on a mesh consisting of triangles. The schemes are shown to converge to the viscosity solution of the Hamilton–Jacobi equation. Received February 25, 1998 / Published online: June 29, 1999     

A multi-resolution,non-parametric,Bayesian framework for identification of spatially-varying model parameters

This paper proposes a hierarchical, multi-resolution framework for the identification of model parameters and their spatially variability from noisy measurements of the response or output. Such parameters are frequently encountered in PDE-based models and correspond to quantities such as density or pressure fields, elasto-plastic moduli and internal variables in solid mechanics, conductivity fields in heat diffusion problems, permeability fields in fluid flow through porous media etc. The proposed model has all the advantages of traditional Bayesian formulations such as the ability to produce measures of confidence for the inferences made and providing not only predictive estimates but also quantitative measures of the predictive uncertainty. In contrast to existing approaches it utilizes a parsimonious, non-parametric formulation that favors sparse representations and whose complexity can be determined from the data. The proposed framework in non-intrusive and makes use of a sequence of forward solvers operating at various resolutions. As a result, inexpensive, coarse solvers are used to identify the most salient features of the unknown field(s) which are subsequently enriched by invoking solvers operating at finer resolutions. This leads to significant computational savings particularly in problems involving computationally demanding forward models but also improvements in accuracy. It is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling which is embarrassingly parallelizable and circumvents issues with slow mixing encountered in Markov Chain Monte Carlo schemes. The capabilities of the proposed methodology are illustrated in problems from nonlinear solid mechanics with special attention to cases where the data is contaminated with random noise and the scale of variability of the unknown field is smaller than the scale of the grid where observations are collected.     

Rough isometry and Dirichlet finite harmonic functions on Riemannian manifolds

We prove that the dimension of harmonic functions with finite Dirichlet integral is invariant under rough isometries between Riemannian manifolds satisfying the local conditions, expounded below. This result directly generalizes those of Kanai, of Grigor'yan, and of Holopainen. We also prove that the dimension of harmonic functions with finite Dirichlet integral is preserved under rough isometries between a Riemannian manifold satisfying the same local conditions and a graph of bounded degree; and between graphs of bounded degree. These results generalize those of Holopainen and Soardi, and of Soardi, respectively. Received: 23 July 1998 / Revised version: 10 February 1999     

On the elementary theory of rings of Witt vectors with a multiplicative set of representatives for the residue field

We indicate a complete set of elementary invariants for the ring of Witt vectors over a perfect field of prime characteristic, where this ring is equipped with its unique multiplicative set of representatives for the residue field. Received: 31 March 1998     

A global slice theorem for proper Hamiltonian actions

An analogue of a theorem of Abels [A] reducing the proper action of a non-compact Lie group to a compact transformation group is proved for the Hamiltonian setting. Received: 26 June 1997 / Revised version: 12 October 1998     

Characterization of the smoothness and curvature of a marginal function for a trust-region problem

Received May 28, 1996 / Revised version received May 1, 1998 Published online October 9, 1998     

Symmetries of a generic coaction

If B is -algebra of dimension $4\leq n<\infty$ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say , have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for in the case where B is a matrix algebra (2) if then the dual is not amenable (3) if then the fixed point subfactor has index n and principal graph . Received: 3 December 1998 / in final form: 8 January 1999