Algebra Seminar: Gilbert -- Quasi-Monte Carlo for uncertainty quantification of tumour growth and treatment modelled by a semilinear parabolic PDE

Alec Gilbert (UNSW) will give a talk on Wed 6 May at 12pm in Carslaw 451.  We will then
invite him to lunch 1-2pm, all welcome, students get free lunch.  

Title: Quasi-Monte Carlo for uncertainty quantification of tumour growth and treatment
modelled by a semilinear parabolic PDE 

Abstract: This talk will discuss applying
quasi-Monte Carlo (QMC) methods to a class of semilinear parabolic reaction-diffusion
PDEs used to model tumour growth.  The underlying model captures several
phenomenological aspects of tumour growth: infiltration of tumour into surrounding
healthy tissue, proliferation of the existing tumour, and patient response to therapies,
each of which are subject to uncertainty and lead to uncertain model parameters.  In
order to quantify efficiently how this uncertainty propagates through the model to
quantities of interest of the solution we use tailored QMC rules.  We provide a full
error analysis of the QMC method, including well-posedness of the parametric PDE and
stochastic regularity analysis in the case of affine uniform random coefficients.  A key
difficulty in the analysis is that the nonlinear reaction term is of the form u(1 - u)
and so is not monotone.  I will discuss the overall strategy of how we overcome this,
including a key intermediate result that initial conditions bounded by [0, 1] lead to
solutions that stay in [0, 1] as time evolves.  To conclude the talk I will present
numerical experiments that validate the theoretical results and demonstrate the improved
performance of QMC compared to Monte Carlo.