Research

## PhD topic

My PhD project's title is Adaptive numerical algorithms for PDE problems with random data and I'm supervised by Alex Bespalov.

The project concerns the numerical solution and the a posteriori error estimation of partial differential equations with random input data in the FEM setting.

Many engineering applications have to take account of all those parameters, such as initial, boundary, or source terms, which are affected by a large amount of uncertainty. In a typical PDE based setting, this makes difficult the full description of the problem considered and then the uncertainty is modelled by introducing some random parameters (variables) in the model.

The model problem we are interested in is the following one. Given a source term $f \in L^2(D)$, we consider the solution $u:D\times \Gamma\to \mathbb{R}$ to the following parametric steady-state diffusion problem \begin{align} -\nabla \cdot (a(\textbf{x},\textbf{y})\nabla u(\textbf{x},\textbf{y})) &= f(\textbf{x}) && \textbf{x} \in D, \, \textbf{y} \in \Gamma, \\ u(\textbf{x},\textbf{y}) &= 0 && \textbf{x} \in \partial D, \, \textbf{y} \in \Gamma. \end{align} where $D \subset \mathbb{R^2}$ is a bounded spatial domain, $\Gamma$ is a parameter domain defined as $\Gamma := \bigotimes_{m\geq 1} \Gamma_m$ with $\Gamma_m := [-1,1]$, $\textbf{y} = (y_1,y_2,\dots) \in \Gamma$ is a multivariate random parameter, and finally, $a(\textbf{x},\textbf{y})$ is a second-order random field represented as $a(\textbf{x},\textbf{y}) = a_0(\textbf{x}) + \sum_{m\geq 1} a_m(\textbf{x})\, y_m, \quad \textbf{x} \in D, \; \textbf{y} \in \Gamma,$ for a family of regular functions $a_m(\mathbf{x}) \in L^{\infty}(D)$, for all $m \in \mathbb{N}_0$. For example, we assume $y_m \in \Gamma_m$ to be the images of independent uniformly distributed mean-zero random variables on $\Gamma_m$, for all $m \in \mathbb{N}$.

The method we use to solve such a problem is the Stochastic Galerkin Finite Element Method (SGFEM). This method couples together both approximations in the spatial and the parameter domain and it generates solutions in tensor product spaces $X \otimes \mathcal{P}$, where $X$ is a finite element space and $\mathcal{P}$ is a space of multivariate polynomials in the parameters.

## Experience

During my PhD I've been teaching assistant for the following undergraduate modules at the University of Birmingham:
• Computer Lab Demonstrating – Spring 2016, Spring 2017
• Methods in Partial Differential Equations – Spring 2016
• 1st-year Support Class – Spring 2016, Autumn 2016
• Numerical Methods II – Autumn 2016, Autumn 2017
• Linear Programming – Autumn 2016, Autumn 2017
• Mathematical Modelling and Problem Solving – Autumn 2016
• Mathematics in Industry – Spring 2017
• Differential Equations – Spring 2017
• Numerical Linear Algebra with Applications – Spring 2018
• Numerical Methods and Programming – Spring 2018

## Events

• 5–6 December, 2016 – I attended the workshop Adaptive algorithms for computational PDEs at the University of Birmingham.
• 14–17 June, 2016 – I attended the conference MAFELAP: the Mathematics of Finite Element and Applications, Brunel University, London.
• 12–16 September, 2016 – I attended the summer school Uncertainty quantification, Weierstrass Institute (WIAS), Berlin.
• 6 March, 2017 – I gave a talk at the postgraduate seminars (LAMS) of the School of Mathematics of the University of Birmingham. Talk's title: Adaptive Finite Element algorithms for PDEs with random data.
• 27–30 June, 2017 – I gave a talk at The 27th Biennial Numerical Analysis Conference, University of Strathclyde, Glasgow. Talk's title: An adaptive stochastic Galerkin FEM for parametric PDEs with singular solutions; slides.
• 31 January 2018 – I took part in a student poster competition held at the School of Mathematics of the University of Birmingham; poster.
• 9 March 2018 – I gave a talk at the postgraduate seminars of the School of Mathematics of the University of Birmingham. Talk's title: Efficient adaptive algorithms for PDEs with random input data; slides.
• 19 June 2018 – I gave a talk at the SIAM UKIE National Student Chapter Conference held at the University of Bath. Talk's title: An efficient adaptive algorithm for elliptic problems with random input data; slides.

## Publications

• A. Bespalov, D. Praetorius, L. R., and M. Ruggeri, Goal-Oriented error estimation and adaptivity for elliptic PDEs with parametric or uncertain outputs, submitted for publication; arXiv preprint arXiv:1806.03928.
• A. Bespalov and L. R., Efficient adaptive algorithms for elliptic PDEs with random data, SIAM/ASA J. Uncertain. Quantif., 6(1)243-272, 2018; link.
• S. Cacace, E. Cristiani, and L. R., A level set based method for fixing overhangs in 3D printing, Appl. Math. Model. 44:446-455, 2017; link.
The paper has been inserted among the Featured Articles of the April issue 2017; link. Also, the research has been mentioned by Le Scienze, the Italian edition of Scientific American; link (in Italian).