Abstract:
Kolmogorov $n$-widths and low-rank approximations are studied for families of elliptic diffusion PDEs parametrized by the diffusion coefficients. The decay of the $n$-widths can be controlled by that of the error achieved by best $n$-term approximations using polynomials in the parametric variable. However, we prove that in certain relevant instances where the diffusion coefficients are piecewise constant over a partition of the physical domain, the $n$-widths exhibit significantly faster decay. This, in turn, yields a theoretical justification of the fast convergence of reduced basis or POD methods when treating such parametric PDEs. Our results are confirmed by numerical experiments, which also reveal the influence of the partition geometry on the decay of the $n$-widths.

Abstract:
We derive error estimates for multinomial approximations of American options in a multidimensional jump--diffusion Merton's model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approximations theorems for i.i.d. random vectors which were obtained [14]. For the multidimensional Black--Scholes model our results can be extended also to a general path dependent payoffs which satisfy Lipschitz type conditions. For the case of multinomial approximations of American options for the Black--Scholes model our estimates are a significant improvement of those which were obtained in [8] (for game options in a more general setup)

Abstract:
Theoretical studies of localization, anomalous diffusion and ergodicity breaking require solving the electronic structure of disordered systems. We use free probability to approximate the ensemble- averaged density of states without exact diagonalization. We present an error analysis that quantifies the accuracy using a generalized moment expansion, allowing us to distinguish between different approximations. We identify an approximation that is accurate to the eighth moment across all noise strengths, and contrast this with the perturbation theory and isotropic entanglement theory.

Abstract:
Finite difference approximations to multi-asset American put option price are considered. The assets are modelled as a multi-dimensional diffusion process with variable drift and volatility. Approximation error of order one quarter with respect to the time discretisation parameter and one half with respect to the space discretisation parameter is proved by reformulating the corresponding optimal stopping problem as a solution of a degenerate Hamilton-Jacobi-Bellman equation. Furthermore, the error arising from restricting the discrete problem to a finite grid by reducing the original problem to a bounded domain is estimated.

Abstract:
In this correspondence our aim is to use some tight lower and upper bounds for the differential quaternary phase shift keying transmission bit error rate in order to deduce accurate approximations for the bit error rate by improving the known results in the literature. The computation of our new approximate expressions are significantly simpler than that of the exact expression.

Abstract:
Different approximations to describe nucleation kinetics have been analyzed. Some new approximations for solution of diffusion problem are proposed. Error of the approximation with constant radius of an embryo is clarified. The theory is illustrated by numerical calculations.

Abstract:
We justify and give error estimates for binomial approximations of game (Israeli) options in the Black--Scholes market with Lipschitz continuous path dependent payoffs which are new also for usual American style options. We show also that rational (optimal) exercise times and hedging self-financing portfolios of binomial approximations yield for game options in the Black--Scholes market ``nearly'' rational exercise times and ``nearly'' hedging self-financing portfolios with small average shortfalls and initial capitals close to fair prices of the options. The estimates rely on strong invariance principle type approximations via the Skorokhod embedding.

Abstract:
We introduce the notion of \delta-viscosity solutions for fully nonlinear uniformly parabolic PDE on bounded domains. We prove that \delta-viscosity solutions are uniformly close to the actual viscosity solution. As a consequence we obtain an error estimate for implicit monotone finite difference approximations of uniformly parabolic PDE.

Abstract:
We evaluate the self-diffusion and transport diffusion of interacting particles in a discrete geometry consisting of a linear chain of cavities, with interactions within a cavity described by a free-energy function. Exact analytical expressions are obtained in the absence of correlations, showing that the self-diffusion can exceed the transport diffusion if the free-energy function is concave. The effect of correlations is elucidated by comparison with numerical results. Quantitative agreement is obtained with recent experimental data for diffusion in a nanoporous zeolitic imidazolate framework material, ZIF-8.