# Projects

## SGMethods: A Python implementation of sparse grid interpolation.

### Approximate high dimensional functions based on point samples only.

### Check the repository of the project (to be released soon!).

**Sparse grid Interpolation** (also called **stochastic collocation**) is an approximation method for **high-dimensional functions**, i.e. functions that depend on many scalar input variables.

The algorithm uses **point samples** (the “sparse grid”) to generate a polynomial interpolant of the original function that can be **evaluated cheaply and quickly**.

SGMathods can be used to interpolate any function, but the focus is on **parametric coefficient PDEs**, with useful functions and a number of examples already implemented. This is the kind of problem you need to solve when approximating **random coefficient PDEs**.

Here are some of the key features:

**Python****object-oriented**implementation, with**tests**and**documentation**;- Wide range of
**implemented interpolation nodes**and**interpolation methods**to choose from; **User-defined 1D interpolation operators**(e.g. piecewise linear functions or global Chebyshev polynoamials) can esaily defined and are**automatically “upgraded” to a sparse grid method**;- Algorithms designed to
**scale well with the number of scalar input variables**, even for very high dimensions; **Adaptive sparse grid selection based**on profit maximisation (coming soon);- A
**multilevel sparse grid interpolation**class implementing the method from [Teckentrup et al., SJUQ, 2015]). It**dramatically reduces the cost of approximating random coefficient PDEs**by cleverly combining sparse grid and finite element resolutions.

- A

## Sparse grid approximation of nonlinear stochastic PDEs. With application to the stochastic LLG equation (micromagnetics).

### Research paper written at TU Wien and UNSW, under review, with Prof. M. Feischl, Dr. X. An, Prof. J. Dick, Prof. T. Tran.

### Download the Preprint (PDF).

We show convergence rates for a sparse grid approximation of the distribution of solutions of the stochastic Landau-Lifshitz-Gilbert equation.

Beyond being a frequently studied equation in engineering and physics, the stochastic Landau-Lifshitz-Gilbert equation poses many interesting challenges that do not appear simultaneously in previous works on uncertainty quantification: The equation is strongly non-linear, time-dependent, and has a non-convex side constraint. Moreover, the parametrization of the stochastic noise features countably many unbounded parameters and low regularity compared to other elliptic and parabolic problems studied in uncertainty quantification.

We use a novel technique to establish uniform holomorphic regularity of the parameter-to-solution map based on a Gronwall-type estimate and the implicit function theorem. This method is very general and based on a set of abstract assumptions. Thus, it can be applied beyond the Landau-Lifshitz-Gilbert equation as well.

We demonstrate numerically the feasibility of approximating with sparse grid and show a clear advantage of a multi-level sparse grid schemes

## Convergence of adaptive sparse grid-finite elements for affine diffusion Poisson.

### Research paper written at TU Wien, published on CAMWA in 2021, with Prof. M. Feischl.

### Link to Journal Article.

### Download the Preprint (PDF).

We consider a Poisson partial differential equation with random diffusion coefficient. This model has applications e.g. in ground water flow modelling, in which the composition of the soil is not precisely know.

We discretize it with a combination of finite elements (to approximate the spatial dependence) and a sparse grid stochastic collocation (to approximate the random component). Both methods are adaptive, in the sense that the algorithm computes a sequence of solutions and automatically chooses how to refine them to improve accuracy optimally.

We give a thorough theoretical description of the method and its theoretical properties. We prove for the first time the convergence of such a stochastic collocation-finite elements algorithm.

## Trimmed isogeometric discretization of the Stokes problem.

### Master Project carried out at EPFL, 2019, with Dr. R. Vasquez, Dr. P. Antolin, and Prof. A. Buffa.

### Download the Report (PDF).

**IsoGeometric Analysis** (**IGA**) is a Galerkin-type numerical method for approximating solutions of PDE. While finite elements are based on piecewise polynomials, IGA uses **s plines** as its building block, both to

**discretize the computational domain**, as in

**CAD**software, and as

**basis functions**to represent the approximate solution.

** Trimming** is an important

**CAD operation**that allows one to manipulate g

*eometric primitives*by taking their

**union**,

**different**, or

**making a cut**along a curve.

In this project, we investigate the effect of trimming on the approximation of the **Stokes problem** with **Taylor-Hood** (TH) and **Raviart-Thomas** (RT) isogeometric elements. Both methods can **converge faster** than finite elements, and isogeometric RT generates an **exactly zero-divergence numerical solution**. We first extend the **Matlab** package *GeoPDEs *with the necessary functionalities to treat trimmed. Then, we perform several **numerical experiments** with different trimming configurations and TH or RT methods.

## Global optimization algorithms for semiconductor devices.

### Project carried out at Fluxim AG, Winterthur, 2018, with Dr. S. Altazin and the Fluxim team.

During my internship at Fluxim AG, I **researched and implemented global optimization algorithms** to expand the capabilities of the simulation software Setfos.

I used **Python** to research and test global optimization algorithms, such as **simulated annealing**, **genetic algorithms**, **simplex optimization**, and **bayesian global optimization**.

Algorithms were **tested against problems from semiconductor engineering**, for example, the optimization of a tandem solar cell.

Finally, I **implemented** a selection of the researched algorithms in a **C++ library**, which was then **integrated into the company software** Setfos. The library, which I called **GLobOpt**, features extensive **documentation** and **unit tests**.

## Optimization of the rate of convergence of diffusions.

### Project carried out at EPFL, Lausanne, 2018, with Dr. S. Krumsheid and Prof. F. Nobile.

**Download the report (PDF)**

We study a method (see [Lelièvre et. al. J. Stat. Phys, 2013]) to accelerate the convergence of *Stochastic Diffusion Processes* through a perturbation of their drift term that does not alter the stationary distribution. This method is based on the connection between diffusion processes and the spectral gap of the *Fokker-Planck operator*, i.e. its first nonzero eigenvalue.

We study the problem of approximating eigenvalues of differential operators. We also solve the problem of finding the best perturbation, i.e. the one that leads to the maximal acceleration of convergence, with global optimization methods.

Eigenvalue problems are solved with the finite elements programming language **FreeFem++** and global optimization problems are solved in **Matlab**.

This method could have important applications e.g. in Markov-Chain Monte Carlo (MCMC) sampling, in which a desired probability measure is sampled by treating it as the stationary distribution of a Markov Chain.

## Haemodynamics simulations of abdominal aortic aneurysm.

### Project carried out at EPFL, Lausanne, 2017, with Dr. C. Colciago and Prof. A. Quarteroni.

**Download the report (PDF)**.

Abdominal aortic aneurysms (AAA) is a pathology that cause 1 to 3% of the deaths among men aged between 65 and 85 in developed countries. One possible therapy is Endovascular Aneurysm Sealing (EVAS), which consists of filling the AAA sac with a polymer and making the blood flow through a stent.

We simulate the blood flow in a section of the abdominal aorta with **Finite Elements** and **BDF time stepping**. We study pre- and post-operatorial settings and compute quantities of interest that allow for a quantitative comparison (e.g. the Wall Shear Stress, the oscillatory shear index, etc.).

Computation meshes are generated with Gmsh, finite element simulations (instationary Navier-Stokes) are carried out with the **C++** library LifeV, and results are visualized in ParaView