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We consider the Bayesian approach to the inverse problem of recovering the shape of an object from measurements of its scattered acoustic field. Working in the time-harmonic setting, we focus on a Helmholtz transmission problem and then extend our results to an exterior Dirichlet problem. We assume the scatterer to be star-shaped and we use, as prior, a truncated expansion with uniform random variables for a radial parametrization of the scatterer's boundary. The main novelty of our work is that we establish the well-posedness of the Bayesian shape inverse problem in a wavenumber-explicit way, under some conditions on the material parameters excluding quasi-resonant regimes. Our estimates highlight how the stability of the posterior with respect to the data is affected by the wavenumber (or, in other words, the frequency), whose magnitude has to be understood not in absolute terms but in relationship to the spatial scale of the problem.

Infra-red (IR) thermography is an essential diagnostic tool for understanding the edge plasma behavior in fusion devices. In this work, we present a new in-house numerical tool, Functional Analysis of Heat Flux (FAHF), for IR thermographic inversion on Tokamak Energy's spherical tokamak (ST40). FAHF, written in Python, is designed for multi-2D thermographic inversions by solving the heat conduction equation within the divertor tiles using the finite difference method, and an explicit time stepping scheme. Utilising IR camera data with the highest available effective spatial resolution, FAHF calculates the plasma perpendicular heat flux density on the divertor tile surfaces -- a crucial quantity for edge plasma analysis. The tool's internal numerics is first verified through formal time and space convergence analyses, and further corroborated by an energy balance assessment. Although FAHF demonstrates significant sensitivity to user-selected spatial resolution, precise heat flux values are recoverable by ensuring a sufficiently high resolution. Implications for the optimal resolution of both the code and the diagnostic system are discussed. Finally, FAHF's model and geometry simplifications are confirmed to be accurate within 10%, based on comparison with COMSOL Multiphysics simulations. As such, FAHF is proven to be a precise and accurate tool for IR thermographic inversions in ST40.

Surfactants that are deposited at aqueous liquid films have the ability to generate surface tension gradients at the air-water interface, and thereby induce Marangoni flow. Combined with the production and depletion of surfactants at different locations of source and drains, out-of-equilibrium surface tension gradients can be sustained, resulting in Marangoni flow patterns that drive e.g., self-organization of amphiphile myelin assemblies. Here, a mathematical model based on the thin-film equations is proposed to simulate these flow patterns. The model equations are based on the surfactant source and drain concentrations, film-height and surfactant bulk concentration. We present a numerical scheme for approximating the model equations and discuss the numerically observed properties of the model., Comment: 8 Figures

Plasma facing components (PFCs) in the next generation of tokamak devices will operate in challenging environments, with heat loads predicted to exceed 10 MWm$^{-2}$. The magnitude of these heat loads is set by the width of the channel, the "scrape-off layer" (SOL), into which heat is exhausted, and can be characterised by an e-folding length scale for the decay of heat flux across the channel. It is expected this channel will narrow as tokamaks move towards reactor relevant conditions. Understanding the processes involved in setting the SOL heat flux width is imperative to be able to predict the heat loads PFCs must handle in future devices. Measurements of the SOL width are performed on the high-field spherical tokamak, ST40, using a newly commissioned infrared thermography system. With its high on-axis toroidal magnetic field ($\geq$1.5 T) ST40 is uniquely positioned to investigate the influence of toroidal field on the heat flux width in spherical tokamaks, whilst also extending measurements of the SOL width in spherical tokamaks to increased poloidal field ($\geq$0.3 T). Due to the divertor on ST40 having a low degree of axisymmetry, it is necessary for a set of radial measurements of the heat flux to be taken across the divertor, made possible using an automated toolchain that fully incorporates its 3D geometry. These radial profiles are combined with the magnetic topology of the plasma to infer the width of the SOL, with both single and double exponential profiles of heat flux observed. A reduction in the heat flux is observed toroidally across part of the divertor, with preliminary investigations indicating that partial shadowing occurs, resulting from the separation between magnetic field lines and trailing edges upstream of the observed region becoming comparable to the ion gyro-radius., Comment: 7 pages, 5 figures, Submitted to the Journal of Nuclear Materials and Energy Special Issue : Proceedings of 26th International Conference on Plasma Surface Interactions in Controlled Fusion Devices

The scrape-off layer parallel heat flux decay lengths measured at ST40, a high field, low aspect ratio spherical tokamak, have been observed to bifurcate into two groups. The wide group matches closely with the scale of ion poloidal Larmour radius and follows existing H-mode scalings, while the narrow group falls up to 10 times below scalings, on the scale of ion total Larmour radius. The onset of the narrow scrape-off layer width is observed to be associated with suppressed magnetic fluctuations, suggesting reduced electromagnetic turbulence levels in the SOL., Comment: 6 pages, 4 figures

Using a novel modeling approach based on the so-called environmental stress level (ESL), we develop a mathematical model to describe systematically the collective influence of oxygen concentration and stiffness of the extracellular matrix on the response of tumor cells to a combined chemotherapeutic treatment. We perform Bayesian calibrations of the resulting model using particle filters, with in vitro experimental data for different hepatocellular carcinoma cell lines. The calibration results support the validity of our mathematical model. Furthermore, they shed light on individual as well as synergistic effects of hypoxia and tissue stiffness on tumor cell dynamics under chemotherapy., Comment: Contains supplementary material

This work studies how the choice of the representation for parametric, spatially distributed inputs to elliptic partial differential equations (PDEs) affects the efficiency of a polynomial surrogate, based on Taylor expansion, for the parameter-to-solution map. In particular, we show potential advantages of representations using functions with localized supports. As model problem, we consider the steady-state diffusion equation, where the diffusion coefficient and right-hand side depend smoothly but potentially in a \textsl{highly nonlinear} way on a parameter $y\in [-1,1]^{\mathbb{N}}$. Following previous work for affine parameter dependence and for the lognormal case, we use pointwise instead of norm-wise bounds to prove $\ell^p$-summability of the Taylor coefficients of the solution. As application, we consider surrogates for solutions to elliptic PDEs on parametric domains. Using a mapping to a nominal configuration, this case fits in the general framework, and higher convergence rates can be attained when modeling the parametric boundary via spatially localized functions. The theoretical results are supported by numerical experiments for the parametric domain problem, illustrating the efficiency of the proposed approach and providing further insight on numerical aspects. Although the methods and ideas are carried out for the steady-state diffusion equation, they extend easily to other elliptic and parabolic PDEs., Comment: 38 pages, 2 figures, submitted to ESIAM M2AN

Survival of living tumor cells underlies many influences such as nutrient saturation, oxygen level, drug concentrations or mechanical forces. Data-supported mathematical modeling can be a powerful tool to get a better understanding of cell behavior in different settings. However, under consideration of numerous environmental factors mathematical modeling can get challenging. We present an approach to model the separate influences of each environmental quantity on the cells in a collective manner by introducing the "environmental stress level". It is an immeasurable auxiliary variable, which quantifies to what extent viable cells would get in a stressed state, if exposed to certain conditions. A high stress level can inhibit cell growth, promote cell death and influence cell movement. As a proof of concept, we compare two systems of ordinary differential equations, which model tumor cell dynamics under various nutrient saturations respectively with and without considering an environmental stress level. Particle-based Bayesian inversion methods are used to quantify uncertainties and calibrate unknown model parameters with time resolved measurements of in vitro populations of liver cancer cells. The calibration results of both models are compared and the quality of fit is quantified. While predictions of both models show good agreement with the data, there is indication that the model considering the stress level yields a better fitting. The proposed modeling approach offers a flexible and extendable framework for considering systems with additional environmental factors affecting the cell dynamics.

Hypersonic flight regime is conventionally defined for Mach larger than 5; in these conditions, the flying object becomes enveloped in a plasma. This plasma is densest in thin surface layers, but in typical situations of interest it impacts electromagnetic wave propagation in an electrically large volume. We address this problem with a hybrid approach. We employ Equivalence Theorem to separate the inhomogeneous plasma region from the surrounding free space via an equivalent (Huygens) surface, and the Eikonal approximation to Maxwell equations in the large inhomogeneous region for obtaining equivalent currents on the separating surface. Then, we obtain the scattered field via (exact) free space radiation of these surface equivalent currents. The method is extensively tested against reference results and then applied to a real-life re-entry vehicle with full 3D plasma computed via Computational Fluid Dynamic (CFD) simulations. We address both scattering (RCS) from the entire vehicle and radiation from the on-board antennas. From our results, significant radio link path losses can be associated with plasma spatial variations (gradients) and collisional losses, to an extent that matches well the usually perceived blackout in crossing layers in cutoff. Furthermore, we find good agreement with existing literature concerning significant alterations of the radar response (RCS) due to the plasma envelope.

We consider the point evaluation of the solution to interface problems with geometric uncertainties, where the uncertainty in the obstacle is described by a high-dimensional parameter $\boldsymbol{y}\in[-1,1]^d$, $d\in\mathbb{N}$. We focus in particular on an elliptic interface problem and a Helmholtz transmission problem. Point values of the solution in the physical domain depend in general non-smoothly on the high-dimensional parameter, posing a challenge when one is interested in building surrogates. Indeed, high-order methods show poor convergence rates, while methods which are able to track discontinuities usually suffer from the so-called curse of dimensionality. For this reason, in this work we propose to build surrogates for point evaluation using deep neural networks. We provide a theoretical justification for why we expect neural networks to provide good surrogates. Furthermore, we present extensive numerical experiments showing their good performance in practice. We observe in particular that neural networks do not suffer from the curse of dimensionality, and we study the dependence of the error on the number of point evaluations (that is, the number of discontinuities in the parameter space), as well as on several modeling parameters, such as the contrast between the two materials and, for the Helmholtz transmission problem, the wavenumber.

In this work, we present and analyse a system of coupled partial differential equations, which models tumour growth under the influence of subdiffusion, mechanical effects, nutrient supply, and chemotherapy. The subdiffusion of the system is modelled by a time fractional derivative in the equation governing the volume fraction of the tumour cells. The mass densities of the nutrients and the chemotherapeutic agents are modelled by reaction diffusion equations. We prove the existence and uniqueness of a weak solution to the model via the Faedo--Galerkin method and the application of appropriate compactness theorems. Lastly, we propose a fully discretised system and illustrate the effects of the fractional derivative and the influence of the fractional parameter in numerical examples.

We consider the generation of samples of a mean-zero Gaussian random field with Mat\'ern covariance function. Every sample requires the solution of a differential equation with Gaussian white noise forcing, formulated on a bounded computational domain. This introduces unwanted boundary effects since the stochastic partial differential equation is originally posed on the whole $\mathbb{R}^d$, without boundary conditions. We use a window technique, whereby one embeds the computational domain into a larger domain, and postulates convenient boundary conditions on the extended domain. To mitigate the pollution from the artificial boundary it has been suggested in numerical studies to choose a window size that is at least as large as the correlation length of the Mat\'ern field. We provide a rigorous analysis for the error in the covariance introduced by the window technique, for homogeneous Dirichlet, homogeneous Neumann, and periodic boundary conditions. We show that the error decays exponentially in the window size, independently of the type of boundary condition. We conduct numerical experiments in 1D and 2D space, confirming our theoretical results.

Methods for generating sequences of surrogates approximating fine scale models of two-phase random heterogeneous media are presented that are designed to adaptively control the modeling error in key quantities of interest (QoIs). For specificity, the base models considered involve stochastic partial differential equations characterizing, for example, steady-state heat conduction in random heterogeneous materials and stochastic elastostatics problems in linear elasticity. The adaptive process involves generating a sequence of surrogate models defined on a partition of the solution domain into regular subdomains and then, based on estimates of the error in the QoIs, assigning homogenized effective material properties to some subdomains and full random fine scale properties to others, to control the error so as to meet a preset tolerance. New model-based Multilevel Monte Carlo (mbMLMC) methods are presented that exploit the adaptive sequencing and are designed to reduce variances and thereby accelerate convergence of Monte Carlo sampling. Estimates of cost and mean squared error of the method are presented. The results of several numerical experiments are discussed that confirm that substantial saving in computer costs can be realized through the use of controlled surrogate models and the associated mbMLMC algorithms.

In the framework of uncertainty quantification, we consider a quantity of interest which depends non-smoothly on the high-dimensional parameter representing the uncertainty. We show that, in this situation, the multilevel Monte Carlo algorithm is a valid option to compute moments of the quantity of interest (here we focus on the expectation), as it allows to bypass the precise location of discontinuities in the parameter space. We illustrate how such lack of smoothness occurs for the point evaluation of the solution to a (Helmholtz) transmission problem with uncertain interface, if the point can be crossed by the interface for some realizations. For this case, we provide a space regularity analysis for the solution, in order to state converge results in the L1-norm for the finite element discretization. The latter are then used to determine the optimal distribution of samples among the Monte Carlo levels. Particular emphasis is given on the robustness of our estimates with respect to the dimension of the parameter space.