This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as = p n ; where n is the stepsize and provides the convergence on smooth initial data. The case of t...

Many models involve the Coulomb’s law in order to describe dynamical properties of friction phenomena. In order to generalize this Coulomb’s law and to deal with its correct mathematical expression, we study a nonlinear equation where we take into account a maximal monotone graph. In the particular case of Coulomb’s law, existence and uniqueness are proved. But in the general case, only existen...

When multivalent salts are added to dilute DNA solutions, the DNA chains condense into bundles of a well-defined size. We explore two physical problems motivated by this phenomenon: the origin of the attractive interactions between the highly negativelycharged DNA chains that cause them to concentrate into bundles, and the mechanism that prevents the bundles from growing indefinitely large.

Approximately 40 million people live within a 4-mile radius of waste sites that the Agency for Toxic Substances and Disease Registry (ATSDR) has assessed to date. Human populations living in the vicinity of such sites are often subjected to complex chemical exposures that may contribute to the total body burden of oxogenous chemicals. Apart from the contaminants found at waste sites, exposure m...

The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorit...

Out of the five members of the Ebolavirus family, four cause life-threatening disease, whereas the fifth, Reston virus (RESTV), is nonpathogenic in humans. The reasons for this discrepancy remain unclear. In this review, we analyze the currently available information to provide a state-of-the-art summary of the factors that determine the human pathogenicity of Ebolaviruses. RESTV causes sporadi...

A ‘correct’ interpretation of the computational complexity of an ill-posed problem is formulated as a cost/effectiveness balance for the use of available data to obtain adequate solutions for an application. This composition with an application, is seen as the real problem, leading to the conclusion that some apparently ill-posed problems are, in context, really well-posed with a reasonable ass...

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au = f is solvable, and ‖fδ − f ‖ δ, then the following results are provided: Problem Fδ(u) := ‖Au− fδ‖2 + α‖u‖2 has a unique global minimizer uα,δ for any fδ , uα,δ = A∗(AA∗ + αI)−1fδ . There is a function α = α(δ), limδ→0 α(δ)= 0 such that limδ→0 ‖uα(δ),δ − y...

| Projection methods based on wavelet functions combine optimal convergence rates with algorithmic eeciency. The proofs in this paper utilize the approximation properties of wavelets and results from the general theory of regularization methods. Moreover, adaptive strategies can be incorporated still leading to optimal convergence rates for the resulting algorithms. The so-called wavelet-vaguel...