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This thesis consists of two parts. In the first, we develop two new strategies for spatial white noise and Gaussian-Mat\'ern field sampling that work within a non-nested multilevel (quasi) Monte Carlo (ML(Q)MC) hierarchy. In the second, we apply the techniques developed to quantify the level of uncertainty in a new stochastic model for tracer transport in the brain. The new sampling techniques are based on the stochastic partial differential equation (SPDE) approach, which recasts the sampling problem as the solution of an elliptic equation driven by spatial white noise. We present a new proof of an a priori error estimate for the finite element (FEM) solution of the white noise SPDE. The proof does not require the approximation of white noise in practice, and includes higher order elliptic operators and p-refinement. Within the SPDE approach, the efficient sampling of white noise realisations can be computationally expensive. In this thesis, we present two new sampling techniques that can be used to efficiently compute white noise samples in a FEM-MLMC and FEM-MLQMC setting. The key idea is to exploit the finite element matrix assembly procedure and factorise each local mass matrix independently, hence avoiding the factorisation of a large matrix. In a multilevel framework, the white noise samples must be coupled between subsequent levels. We show how our technique can be used to enforce this coupling even in the case of non-nested mesh hierarchies. In the MLQMC case, the QMC integrand variables must also be ordered in order of decaying importance to achieve fast convergence with respect to the number of samples. We express white noise as a Haar wavelet series whose hierarchical structure naturally exposes the leading order dimensions. We split this series in two terms which we sample via a hybrid standard Monte Carlo/QMC approach. We demonstrate the efficacy of our sampling methods with numerical experiments. In a multilevel setting, a good coupling is enforced and the telescoping sum is respected. In the MLQMC case, the asymptotic convergence rate is the same as standard Monte Carlo, but significant computational gains are obtained in practice thanks to a pre-asymptotic QMC-like regime. In the final part of the thesis, we employ a combination of the methods presented to solve a PDE with random coefficients describing tracer transport within the interstitial fluid of the brain. Numerical simulations support the claim that diffusion alone cannot explain the penetration of tracers within deep brain regions as observed in clinical experiments, even when uncertainties in the diffusivity have been accounted for. A convective velocity field may however increase tracer transport, provided that a directional structure is present in the interstitial fluid circulation.

The E and B EXperiment (EBEX) is a balloon-borne telescope designed to map the polarization of the cosmic microwave background (CMB) and emission from galactic dust at millimeter wavelengths from 150 to 410 GHz. The primary science objectives of EBEX are to: detect or constrain the primordial B-mode polarization of the CMB predicted by in ationary cosmology; measure the CMB B-mode signal induced by gravitational lensing; and characterize the polarized thermal emission from interstellar dust. EBEX will observe a 420 square degree patch of the sky at high galactic latitude with a telescope and camera that provide an 80 beam at three observing bands (150, 250, and 410 GHz) and a 6:2#14; diffraction limited field of view to two large-format bolometer array focal planes. Polarimetry is achieved via a continuously rotating half-wave plate (HWP), and the optical system is designed from the ground up for control of sidelobe response and polarization systematic errors. EBEX is intended to execute y or more Antarctic long duration balloon campaigns. In June 2009 EBEX completed a North American engineering flight launched from NASA's Columbia Scientific Ballooning Facility (CSBF) in Ft. Sumner, NM and operated in the stratosphere above 30 km altitude for #24; 10 hours. During flight EBEX must be largely autonomous as it conducts pointed, scheduled observations; tunes and operates 1432 TES bolometers via 28 embedded Digital frequency-domain multiplexing (DfMux) computers; logs over 3 GiB/hour of science and housekeeping data to onboard redundant disk storage arrays; manages and dispatches jobs over a fault-tolerant onboard Ethernet network; and feeds a complex real-time data processing infrastructure on the ground via satellite and line-of-sight (LOS) downlinks. In this thesis we review the EBEX instrument, present the optical design and the computational architecture for in-flight control and data handling, and the quick-look software stack. Finally we describe the 2009 North American test flight and present analysis of data collected at the end of that flight that characterizes scan-synchronous signals and the expected response to emission from thermal dust in our galaxy.