Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

Tumour susceptibility to innate and adaptive immunotherapy changes during tumour maturation

Immunotherapy of tumours using T cells expanded in vitro has met with mixed clinical success suggesting that a greater understanding of tumour/T-cell interaction is required. We used a HPV16E7 oncoprotein-based mouse tumour model to study this further. In this study, we demonstrate that a HPV16E7 tumour passes through at least three stages of immune susceptibility over time. At the earliest time point, infusion of intravenous immune cells fails to control tumour growth although the same cells given subcutaneously at the tumour site are effective. In a second stage, the tumour becomes resistant to subcutaneous infusion of cells but is now susceptible to both adjuvant activated and HPV16E7-specific immune cells transferred intravenously. In the last phase, the tumour is susceptible to intravenous transfer of HPV16E7-specific cells, but not adjuvant-activated immune cells. The requirement for IFN-gamma and perforin also changes with each stage of tumour development. Our data suggest that effective adoptive T-cell therapy of tumour will need to be matched with the stage of tumour development.

Adaptive phase measurements in linear optical quantum computation

Photon counting induces an effective non-linear optical phase shift in certain states derived by linear optics from single photons. Although this non-linearity is non-deterministic, it is sufficient in principle to allow scalable linear optics quantum computation (LOQC). The most obvious way to encode a qubit optically is as a superposition of the vacuum and a single photon in one mode-so-called 'single-rail' logic. Until now this approach was thought to be prohibitively expensive (in resources) compared to 'dual-rail' logic where a qubit is stored by a photon across two modes. Here we attack this problem with real-time feedback control, which can realize a quantum-limited phase measurement on a single mode, as has been recently demonstrated experimentally. We show that with this added measurement resource, the resource requirements for single-rail LOQC are not substantially different from those of dual-rail LOQC. In particular, with adaptive phase measurements an arbitrary qubit state a alpha/0 > + beta/1 > can be prepared deterministically.