We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called 'walk-on-spheres' algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the 'walk-on-spheres' comes within ε > 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic α-stable process with α ∈ (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.issued from x ∈ D, then, by the strong law of large numbers,For practical purposes, since it is impossible to take the limit, one truncates the series of estimates for large n and the central limit theorem gives O(1/n) upper bounds on the variance of the n-term sum, which serves as a numerical error estimate. Although forming the fundamental basis of most Monte Carlo methods for diffusive Dirichlettype problems, (1.3) is an inefficient numerical approach. Least of all, this is because the Monte Carlo simulation of u(x) is independent for each x ∈ D. Moreover, it is unclear how exactly to simulate the path of a Brownian motion on its first exit from D, that is to say, the quantity W τ D . This is because of the fractal properties of Brownian motion, making its path difficult to simulate. This introduces additional numerical errors over and above that of Monte Carlo simulation.A method proposed by (Muller, 1956), for the case that D is convex, sub-samples special points along the path of Brownian motion to the boundary of the domain D. The method does not require a complete simulation of its path and takes advantage of the distributional symmetry of Brownian motion. In order to describe the so-called 'walk-on-spheres', we need to first introduce some notation. We may thus set ρ 0 = x for x ∈ D and define r 1 to be the radius of the largest sphere inscribed in D that is centred at x. This sphere we will call S 1 = {y ∈ R d : |y − ρ 0 | = r 1 }. To avoid special cases, we henceforth assume that the surface area of S 1 ∩ ∂D is zero (this excludes, for example, the case that x = 0 and D is a sphere centred at the origin). Now set ρ 1 ∈ D to be a point uniformly distributed on S 1 and note that, given the assumption in the previous sentence, P x (ρ 1 ∈ ∂D) = 0. Construct the remainder of the sequence (ρ n , n ≥ 1) inductively. Given ρ n−1 , we define the radius, r n , of the largest sphere inscribed in D that is centred at ρ n−1 . Calling this sphere S n , we have that ...

We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting -elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.

The problem of parameter fitting for nonlinear oscillator models to noisy time series is addressed using a combination of Ensemble Kalman Filter and optimisation techniques. Encouraging preliminary results for acceptable sampling rates and noise levels are presented. Application to the understanding and control of tokamak nuclear reactor operation is discussed.

No abstract

No abstract