Time-fractional diffusion equations with randomness, and efficient numerical estimations of expected values

Abstract In this work, we explore a time-fractional diffusion equation of order $$\alpha \in (0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with a stochastic diffusivity coefficient $${\kappa }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> . We focus on efficient estimation of the expected values of linear functionals acting on the solution of our model problem. To estimate the expected value computationally, the infinite expansions of $${\kappa }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>κ</mml:mi> </mml:math> need to be truncated. Then we approximate the high-dimensional integral over the random field using a high-order quasi-Monte Carlo method. This follows by approximating the deterministic solution over the space-time domain via a second-order accurate time-stepping scheme in combination with a spatial discretization by Galerkin finite elements. We show some regularity properties of the parametric solution and investigate the errors from estimating the expected value. We report on numerical experiments that complement the theoretical results.