A Krylov subspace algorithm for evaluating the phi-functions appearing in exponential integrators

We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called phi-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the Matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.Comment: 20 pages, 3 colour figures, code available from http://www.maths.leeds.ac.uk/~jitse/software.html . v2: Various changes to improve presentation as suggested by the refere

Stability of central finite difference schemes for the Heston PDE

This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic spectral norm we prove practical, rigorous stability bounds. Our theoretical stability results are illustrated by ample numerical experiments

Visual attention in violent offenders: Susceptibility to distraction.

Impairments in executive functioning give rise to reduced control of behavior and impulses, and are therefore a risk factor for violence and criminal behavior. However, the contribution of specific underlying processes remains unclear. A crucial element of executive functioning, and essential for cognitive control and goaldirected behavior, is visual attention. To further elucidate the importance of attentional functioning in the general offender population, we employed an attentional capture task to measure visual attention. We expected offenders to have impaired visual attention, as revealed by increased attentional capture, compared to healthy controls. When comparing the performance of 62 offenders to 69 healthy community controls, we found our hypothesis to be partly confirmed. Offenders were more accurate overall, more accurate in the absence of distracting information, suggesting superior attention. In the presence of distracting information offenders were significantly less accurate compared to when no distracting information was present. Together, these findings indicate that violent offenders may have superior attention, yet worse control over attention. As such, violent offenders may have trouble adjusting to unexpected, irrelevant stimuli, which may relate to failures in selfregulation and inhibitory control

ADI splitting schemes for a fourth-order nonlinear partial differential equation from image processing

We present directional operator splitting schemes for the numerical solution of a fourth-order, nonlinear partial differential evolution equation which arises in image processing. This equation constitutes the H−1-gradient flow of the total variation and represents a prototype of higher-order equations of similar type which are popular in imaging for denoising, deblurring and inpainting problems. The efficient numerical solution of this equation is very challenging due to the stiffness of most numerical schemes. We show that the combination of directional splitting schemes with implicit time-stepping provides a stable and computationally cheap numerical realisation of the equation