There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Fractional Differential Equations

to be done in this area. Face recognition is a problem that scarcely clamoured for attention before the computer age but, having surfaced, has involved a wide range of techniques and has attracted the attention of some fine minds (David Mumford was a Fields Medallist in 1974). This singular application of mathematical modelling to a messy applied problem of obvious utility and importance but with no unique solution is a pretty one to share with students: perhaps, returning to the source of our opening quotation, we may invert Duncan's earlier observation, 'There is an art to find the mind's construction in the face!'.

Linear and nonlinear waves, by G. B. Whitham. Pp.636. £50. 1999. ISBN 0 471 35942 4 (Wiley).

This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

A Posteriori Error Estimation in Finite Element Analysis

The computational work and storage of numerically solving the time fractional PDEs are generally huge for the traditional direct methods since they require total memory and work, where NT and NS represent the total number of time steps and grid points in space, respectively. To overcome this difficulty, we present an efficient algorithm for the evaluation of the Caputo fractional derivative of order α∈(0,1). The algorithm is based on an efficient sum-of-exponentials (SOE) approximation for the kernel t –1–α on the interval [Δt, T] with a uniform absolute error e. We give the theoretical analysis to show that the number of exponentials N exp needed is of order for T≫1 or for TH1 for fixed accuracy e. The resulting algorithm requires only storage and work when numerically solving the time fractional PDEs. Furthermore, we also give the stability and error analysis of the new scheme, and present several numerical examples to demonstrate the performance of our scheme.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/AF5FDC74FD7A010ED0ACD291BB1A92B1/S1815240617000147a.pdf/div-class-title-fast-evaluation-of-the-caputo-fractional-derivative-and-its-applications-to-fractional-diffusion-equations-div.pdf

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

This is the first in a series of papers in which a new gradient recovery method is introduced and analyzed. It is proved that the method is superconvergent for translation invariant finite element spaces of any order. The method maintains the simplicity, efficiency, and superconvergence properties of the Zienkiewicz--Zhu patch recovery method. In addition, for uniform triangular meshes, the method is superconvergent for the linear element under the chevron pattern, and ultraconvergent at element edge centers for the quadratic element under the regular pattern. Applications of this new gradient recovery technique will be discussed in forthcoming papers.

A New Finite Element Gradient Recovery Method: Superconvergence Property

Difference and finite element methods are described, analyzed, and tested for numerical solution of linear parabolic and elliptic SPDEs driven by white noise. Weak and integral formulations of the stochastic partial differential equations are approximated, respectively, by finite element and difference methods. The white noise processes are approximated by piecewise constant random processes to facilitate convergence proofs for the finite element method. Error analyses of the two numerical methods yield estimates of convergence rates. Computational experiments indicate that the two numerical methods have similar accuracy but the finite element method is computationally more efficient than the difference method

Finite element and difference approximation of some linear stochastic partial differential equations

Some recovery type error estimators for linear finite elements are analyzed under O(h 1+α ) (α > 0) regular grids. Superconvergence of order O(h 1+ρ ) (0 < p < α) is established for recovered gradients by three different methods. As a consequence, a posteriori error estimators based on those recovery methods are asymptotically exact.

/pdf/analysis-of-recovery-type-a-posteriori-error-estimators-for-1uxdwde2kl.pdf

Analysis of recovery type a posteriori error estimators for mildly structured grids

Superconvergence of order $O(h^{1+\rho})$, for some $\rho > 0$, is established for the gradient recovered with the polynomial preserving recovery (PPR) when the mesh is mildly structured. Consequently, the PPR-recovered gradient can be used in building an asymptotically exact a posteriori error estimator.

/pdf/a-posteriori-error-estimates-based-on-the-polynomial-2mbfmowcea.pdf

Zhimin Zhang

Papers

Fast Evaluation of the Caputo Fractional Derivative and its Applications to Fractional Diffusion Equations

A New Finite Element Gradient Recovery Method: Superconvergence Property

Finite element and difference approximation of some linear stochastic partial differential equations

Analysis of recovery type a posteriori error estimators for mildly structured grids

A Posteriori Error Estimates Based on the Polynomial Preserving Recovery