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We consider an elliptic optimal control problem where the objective functional contains evaluations of the state at a finite number of points. In particular, we use a fidelity term that encourages the state to take certain values at these points, which means our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the $L^2$ norm because we need the state space to embed into the space of continuous functions. In this paper we discretise the problem using two different piecewise linear finite element methods. For each discretisation we use two different approaches to prove a priori $L^2$ error estimates for the control. We discuss the differences between these methods and approaches and present numerical results that agree with our analytical results.

We explore an optimal partition problem on surfaces using a computational approach. The problem is to minimise the sum of the first Dirichlet Laplace--Beltrami operator eigenvalues over a given number of partitions of a surface. We consider a method based on eigenfunction segregation and perform calculations using modern high performance computing techniques. We first test the accuracy of the method in the case of three partitions on the sphere then explore the problem for higher numbers of partitions and on other surfaces.

In this paper we study time-periodic solutions to advection-diffusion equations of a scalar quantity $u$ on a periodically moving $n$-dimensional hypersurface $\Gamma(t) \subset \mathbb{R}^{n+1}$. We prove existence and uniqueness of solutions in suitable H\"older spaces.

We present an abstract framework for treating the theory of well-posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational formulations of PDEs on evolving spatial domains including moving hypersurfaces. We formulate an appropriate time derivative on evolving spaces called the material derivative and define a weak material derivative in analogy with the usual time derivative in fixed domain problems; our setting is abstract and not restricted to evolving domains or surfaces. Then we show well-posedness to a certain class of parabolic PDEs under some assumptions on the parabolic operator and the data., Comment: 38 pages. Minor typos corrected

We consider an arbitrary-Lagrangian-Eulerian evolving surface finite element method for the numerical approximation of advection and diffusion of a conserved scalar quantity on a moving surface. We describe the method, prove optimal order error bounds and present numerical simulations that agree with the theoretical results.

In this paper, we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements. The key idea is that the $n$-dimensional hypersurface, $\Gamma \subset \mathbb{R}^{n+1}$, is embedded in a polyhedral domain in $\mathbb R^{n+1}$ consisting of a union, $\mathcal{T}_h$, of $(n+1)$-simplices. The finite element approximating space is based on continuous piece-wise linear finite element functions on $\mathcal{T}_h$. Our first method is a sharp interface method, \emph{SIF}, which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation, $\Gamma_{h}$, of $\Gamma$. The full gradient is used rather than the projected tangential gradient and it is this which distinguishes \emph{SIF} from the method of [42]. The second method, \emph{NBM}, is a narrow band method in which the region of integration is a narrow band of width $O(h)$. \emph{NBM} is similar to the method of [13]. but again the full gradient is used in the discrete weak formulation. The a priori error analysis in this paper shows that the methods are of optimal order in the surface $L^{2}$ and $H^{1}$ norms and have the advantage that the normal derivative of the discrete solution is small and converges to zero. Our third method combines bulk finite elements, discrete sharp interfaces and narrow bands in order to give an unfitted finite element method for parabolic equations on evolving surfaces. We show that our method is conservative so that it preserves mass in the case of an advection diffusion conservation law. Numerical results are given which illustrate the rates of convergence.

Reverberation mapping (RM) is an important technique in studies of active galactic nuclei (AGN). The key idea of RM is to measure the time lag $\tau$ between variations in the continuum emission from the accretion disc and subsequent response of the broad line region (BLR). The measurement of $\tau$ is typically used to estimate the physical size of the BLR and is combined with other measurements to estimate the black hole mass $M_{\rm BH}$. A major difficulty with RM campaigns is the large amount of data needed to measure $\tau$. Recently, Fine et al (2012) introduced a new approach to RM where the BLR light curve is sparsely sampled, but this is counteracted by observing a large sample of AGN, rather than a single system. The results are combined to infer properties of the sample of AGN. In this letter we implement this method using a hierarchical Bayesian model and contrast this with the results from the previous stacked cross-correlation technique. We find that our inferences are more precise and allow for more straightforward interpretation than the stacked cross-correlation results., Comment: Accepted for publication in MNRAS Letters. 5 pages, 5 figures. Source code for this paper is available at https://github.com/eggplantbren/RMHB

We present a novel parameter identification algorithm for the estimation of parameters in models of cell motility using imaging data of migrating cells. Two alternative formulations of the objective functional that measures the difference between the computed and observed data are proposed and the parameter identification problem is formulated as a minimisation problem of nonlinear least squares type. A Levenberg-Marquardt based optimisation method is applied to the solution of the minimisation problem and the details of the implementation are discussed. A number of numerical experiments are presented which illustrate the robustness of the algorithm to parameter identification in the presence of large deformations and noisy data and parameter identification in three dimensional models of cell motility. An application to experimental data is also presented in which we seek to identify parameters in a model for the monopolar growth of fission yeast cells using experimental imaging data., Comment: 31 Pages, 13 Figures

Lateral diffusion of molecules on surfaces plays a very important role in various biological processes, including lipid transport across the cell membrane, synaptic transmission and other phenomena such as exo- and endocytosis, signal transduction, chemotaxis and cell growth. In many cases, the surfaces can possess spatial inhomogeneities and/or be rapidly changing shape. Using a generalisation of the model for a thermally excited Helfrich elastic membrane, we consider the problem of lateral diffusion on quasi-planar surfaces, possessing both spatial and temporal fluctuations. Using results from homogenisation theory, we show that, under the assumption of scale separation between the characteristic length and time scales of the membrane fluctuations and the characteristic scale of the diffusing particle, the lateral diffusion process can be well approximated by a Brownian motion on the plane with constant diffusion tensor $D$ which depends in a highly nonlinear way on the detailed properties of the surface. The effective diffusion tensor will depend on the relative scales of the spatial and temporal fluctuations and, for different scaling regimes, we prove the existence of a macroscopic limit in each case., Comment: 56 pages, 9 figures, submitted to J. Nonlin. Sci

We consider the inverse problem of recovering a binary function from blurred and noisy data. Such problems arise in many applications, for example image processing and optimal control of PDEs. Our formulation is based on the Mumford-Shah model, but with a phase field approximation to the perimeter regularisation. We use a double obstacle potential as well as a smooth double well potential. We introduce an iterative method for solving the problem, develop a suitable discretisation of this iterative method, and prove some convergence results. Numerical simulations are presented which illustrate the usefulness of the approach and the relative merits of the phase field models.

We use the evolving surface finite element method to solve a Cahn- Hilliard equation on an evolving surface with prescribed velocity. We start by deriving the equation using a conservation law and appropriate transport for- mulae and provide the necessary functional analytic setting. The finite element method relies on evolving an initial triangulation by moving the nodes accord- ing to the prescribed velocity. We go on to show a rigorous well-posedness result for the continuous equations by showing convergence, along a subse- quence, of the finite element scheme. We conclude the paper by deriving error estimates and present various numerical examples.

We formulate stochastic partial differential equations on Riemannian manifolds, moving surfaces, general evolving Riemannian manifolds (with appropriate assumptions) and Riemannian manifolds with random metrics, in the variational setting of the analysis to stochastic partial differential equations. Considering mainly linear stochastic partial differential equations, we establish various existence and uniqueness theorems., Comment: 42 pages, 1 figure

We develop rigorous, analytic techniques to study the behaviour of biased random walks on combs. This enables us to calculate exactly the spectral dimension of random comb ensembles for any bias scenario in the teeth or spine. Two specific examples of random comb ensembles are discussed; the random comb with nonzero probability of an infinitely long tooth at each vertex on the spine and the random comb with a power law distribution of tooth lengths. We also analyze transport properties along the spine for these probability measures., Comment: minor changes in technical definitions

[Abridged] Norris et al. (1981) found that there was a distinct lack of cyanogen-strong (CN-strong) stars in their sample of AGB stars, as compared to their sample of RGB stars (which had roughly equal numbers of CN-normal and CNstrong stars). Similar features have been discovered in the AGB populations of other clusters. Unfortunately all of these studies suffer from low AGB star counts so the conclusions are not necessarily robust -- larger, statistically signifcant, sample sizes are needed. We outline the results of a literature search for relevant CN observations and describe our observing proposal to test the suggestion that there are substantial abundance differences between the AGB and RGB in galactic globular clusters. The literature search revealed that the AGB star counts for all studies (which are not, in general, studies about AGB stars in particular) are low, usually being < or = 10. The search also revealed that the picture may not be consistent between clusters. Although most clusters appear to have CN-weak AGBs, at least two seem to have CN-strong AGBs (M5 & 47 Tuc). To further complicate the picture, clusters often appear to have a combination of both CN-strong and CN-weak stars on their AGBs -- although one population tends to dominate. Again, all these assertions are however based on small sample sizes.We aim to increase the sample sizes by an order of magnitude using existing high quality photometry in which the AGB and RGB can be reliably separated. For the observations we will use a wide-field, low- to mid-resolution multi-object spectroscope. With the new information we hope to ascertain whether significant abundance differences really exist., Comment: 5 pages. To appear in the conference proceedings of the "Eighth Torino Workshop on Nucleosynthesis in AGB Stars" (Universidad de Granada, Spain, 2006)

We study chemical equilibration of quarks and gluons in central nuclear collisions at RHIC and LHC energies. The initial quark and gluon densities are taken from earlier studies as well as from recent perturbative QCD estimates and are then evolved via rate equations coupled to longitudinally boost-invariant fluid dynamics. We find that, for RHIC initial conditions, the lifetime of quark-gluon matter is too short in order for the quark and gluon number densities to chemically equilibrate prior to hadronization. In contrast, at LHC energies chemical equilibration is complete before the system hadronizes. Entropy production due to chemical equilibration can be as large as 30%., Comment: 30 pages (latex2e), 13 postscript figures, corrected one figure, further analysis performed, to be published in NPA