Hollow fibre membrane bioreactors provide a fast and efficient method for engineering functional tissue for use in medical treatments. Flow is utilised to overcome mass transport limitations by perfusing a nutrient-rich culture medium through the fibre lumen, which can then transport along the fibre lumen or across the porous membrane wall. Cells seeded at the outer membrane wall consume the nutrient and subsequently produce waste metabolites, which are transported away through an external extra-capillary space (ECS) along with excess nutrient. We present and investigate a two-dimensional axisymmetric model for fluid flow and solute transport through a single-fibre bioreactor configuration, with cells seeded to the external fibre wall. Fluid flow is modelled by steady lubrication and Darcy equations, which are coupled to the solute transport problem modelled by a system of advection–diffusion equations, supplemented with a reaction term to model the cell layer. Our model analysis reveals how spatially varying wall permeability distributions can be utilised to provide uniform nutrient delivery to a spatially uniform, homogeneous cell population. We also reveal how maximising the transmural pressure drop across the membrane wall is the dominant mechanism for waste removal rather than traditional experimental methods of flushing the ECS.

Inference in spatial and spatio-temporal models can be challenging for a variety of reasons. For example, non-Gaussianity often leads to analytically intractable integrals; we may be in a ‘big’ data setting, whereby the number of observations renders traditional methods too computationally expensive; we may wish to make inferences over spatial supports that are different to those of our measurements; or, we may wish to use a statistical model whose likelihood function is either unavailable or computationally intractable. In this thesis, I develop several techniques that help to alleviate these challenges.

Atmospheric aerosols influence the Earth’s climate, primarily by affecting cloud formation and scattering visible radiation. However, aerosol-related physical processes in climate simulations are highly uncertain. Constraining these processes could help improve model-based climate predictions. We propose a scalable statistical framework for constraining the parameters of expensive climate models by comparing model outputs with observations. Using the C3.AI Suite, a cloud computing platform, we use a perturbed parameter ensemble of the UKESM1 climate model to efficiently train a surrogate model. A method for estimating a data-driven model discrepancy term is described. The strict bounds method is applied to quantify parametric uncertainty in a principled way. We demonstrate the scalability of this framework with 2 weeks’ worth of simulated aerosol optical depth data over the South Atlantic and Central African region, written from the model every 3 hr and matched in time to twice-daily MODIS satellite observations. When constraining the model using real satellite observations, we establish constraints on combinations of two model parameters using much higher time-resolution outputs from the climate model than previous studies. This result suggests that within the limits imposed by an imperfect climate model, potentially very powerful constraints may be achieved when our framework is scaled to the analysis of more observations and for longer time periods.

Flow fields are determined from image sequences obtained in an experiment in which benthic macrofauna, Arenicola marina, causes water flow and the images depict the distribution of a tracer that is carried with the flow. The experimental setup is such that flow is largely two-dimensional, with a localised region where the Arenicola resides, from which flow originates. Here, we propose a novel parametric framework that quantifies such flow that is dominant along the image plane. We adopt a Bayesian framework so that we can impart certain physical constraints on parameters into the estimation process via prior distribution. The primary aim is to approximate the mean of the posterior distribution to present the parameter estimate via Markov Chain Monte Carlo. We demonstrate that the results obtained from the proposed method provide more realistic flows (in terms of divergence magnitude) than those computed from classical approaches such as the multi-resolution Horn–Schunk method. This highlights the usefulness of our approach if motion is largely constrained to the image plane with localised fluid sources.

A digital twin (DT) relies on a detailed, virtual representation of a physical product. Since uncertainties and deviations can lead to significant changes in the functionality and quality of products, they should be considered in the DT. However, valuable product properties are often hidden and thus difficult to integrate into a DT. In this work, a Bayesian inverse approach based on surrogate models is applied to infer hidden composite laminate ply angles from strain measurements. The approach is able to find the true values even for ill-posed problems and shows good results up to 6 plies.

In this paper, we consider non-self-adjoint Dirac operators on a finite interval with complex-valued potentials and quasi-periodic boundary conditions. Necessary and sufficient conditions for a set of complex numbers to be the spectrum of the indicated problem are established.

Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.

We extend previous weak well-posedness results obtained in Frigeri et al. (2017, Solvability, Regularity, and Optimal Control of Boundary Value Problems for PDEs, Vol. 22, Springer, Cham, pp. 217–254) concerning a non-local variant of a diffuse interface tumour model proposed by Hawkins-Daarud et al. (2012, Int. J. Numer. Method Biomed. Engng.28, 3–24). The model consists of a non-local Cahn–Hilliard equation with degenerate mobility and singular potential for the phase field variable, coupled to a reaction–diffusion equation for the concentration of a nutrient. We prove the existence of strong solutions to the model and establish some high-order continuous dependence estimates, even in the presence of concentration-dependent mobilities for the nutrient variable in two spatial dimensions. Then, we apply the new regularity results to study an inverse problem identifying the initial tumour distribution from measurements at the terminal time. Formulating the Tikhonov regularised inverse problem as a constrained minimisation problem, we establish the existence of minimisers and derive first-order necessary optimality conditions.

This paper presents two modifications of compressive sensing (CS)-based approach applied to the near-field diagnosis of active phased arrays. CS-based antenna array diagnosis allows a significant reduction of measurement time, which is crucial for the characterization of electrically large active antenna arrays, e.g. used in synthetic aperture radar. However, practical implementation of this method is limited by two factors: first, it is sensitive to thermal instabilities of the array under test, and second, excitation reconstruction accuracy strongly depends on the accuracy of the elements of the measurement matrix. First proposed modification allows taking into account of thermal instability of the array by using an iterative ℓ1-minimization procedure. The second modification increases the accuracy of reconstruction using several simple additional measurements.

In this paper, a theoretical study for the design of multi-source transmitters suitable for perpendicular dynamic wireless power transfer is presented. Unlike conventional systems, the concept presented here overcomes the traditional limitation on the receiver's orientation by providing an optimal distribution of the transmitted energy obtained by using different sources. For this purpose, a theoretical study of different transmitters has been achieved by solving the inverse problem. Comparison with conventional single-source transmitters carrying the same total current as the multi-source transmitters, shows a significant enhancement of the power gain when a Genetic Algorithm is used. The obtained theoretical results show power gain levels over 7.5 dB for different path lengths at different heights. At the end, a solution for a path of an infinite length is presented.

We study the numerical identification of an unknown portion of the boundary on which either the Dirichlet or the Neumann condition is provided from the knowledge of Cauchy data on the remaining, accessible and known part of the boundary of a two-dimensional domain, for problems governed by Helmholtz-type equations. This inverse geometric problem is solved using the plane waves method (PWM) in conjunction with the Tikhonov regularization method. The value for the regularization parameter is chosen according to Hansen's L-curve criterion. The stability, convergence, accuracy and efficiency of the proposed method are investigated by considering several examples.

An inverse energy method for the identification of the structural force in high frequency ranges from radiated noise measurements is presented in this paper. The radiation of acoustic energy of the structure coupled to an acoustic cavity is treated using an energetic method called the simplified energy method. The main novelty of this paper consists in using the same energy method to solve inverse structural problem. It consists of localization and quantification of the vibration source through the knowledge of acoustic energy density. Numerical test cases with different measurement points are used for validation purpose. The numerical results show that the proposed method has an excellent performance in detecting the structural force with a few acoustical measurements.

An inverse problem of determining unknown source parameter in a parabolic equation is considered. The variational iteration method (VIM) is presented to solve inverse problems. The solution gives good approximations by VIM. A numerical example shows that the VIM works effectively for an inverse problem.