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

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

Knowledge translation on dementia: a cluster randomized trial to compare a blended learning approach with a "classical" advanced training in GP quality circles

<p>Abstract</p> <p>Background</p> <p>Thus far important findings regarding the dementia syndrome have been implemented into patients' medical care only inadequately. A professional training accounting for both, general practitioners' (GP) needs and learning preferences as well as care-relevant aspects could be a major step towards improving medical care. In the WIDA-study, entitled "Knowledge translation on dementia in general practice" two different training concepts are developed, implemented and evaluated. Both concepts are building on an evidence-based, GP-related dementia guideline and communicate the guideline's essential insights.</p> <p>Methods/Design</p> <p>Both development and implementation emphasize a procedure that is well-accepted in practice and, thus, can achieve a high degree of external validity. This is particularly guaranteed through the preparation of training material and the fact that general practitioners' quality circles (QC) are addressed. The evaluation of the two training concepts is carried out by comparing two groups of GPs to which several quality circles have been randomly assigned. The primary outcome is the GPs' knowledge gain. Secondary outcomes are designed to indicate the training's potential effects on the GPs' practical actions. In the first training concept (study arm A) GPs participate in a structured case discussion prepared for by internet-based learning material ("blended-learning" approach). The second training concept (study arm B) relies on frontal medical training in the form of a slide presentation and follow-up discussion ("classical" approach).</p> <p>Discussion</p> <p>This paper presents the outline of a cluster-randomized trial which has been peer reviewed and support by a national funding organization – Federal Ministry of Education and Research (BMBF) – and is approved by an ethics commission. The data collection has started in August 2006 and the results will be published independently of the study's outcome.</p> <p>Trial Registration</p> <p>Current Controlled Trials [ISRCTN36550981]</p

Appraisal of a contour integral method for the Black-Scholes and Heston equations

A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parameters are derived theoretically and tested numerically. Test examples presented are the Black-Scholes PDE in one space dimension and the Heston PDE in two dimensions. In the latter case efficiency is compared to ADI splitting schemes for solving this problem. In the examples it is found that the contour integral method is superior for the range of medium to high accuracy requirements. Further improvements to the current implementation of the contour integral method are suggested.Comment: Paper has been publishe

Appraisal of a contour integral method for the Black-Scholes and Heston equations

A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parameters are derived theoretically and tested numerically. Test examples presented are the Black-Scholes PDE in one space dimension and the Heston PDE in two dimensions. In the latter case efficiency is compared to ADI splitting schemes for solving this problem. In the examples it is found that the contour integral method is superior for the range of medium to high accuracy requirements. Further improvements to the current implementation of the contour integral method are suggested.