Projects

The POD-Tangent Plane Scheme: Reduced Basis approximation of the stochastic LLG equation

In collaboration with Fernando Henriquez and Michael Feischl (11.2024 — in progress).

 

Deep Operator Learning of the Stochastic Wave Equation

In collaboration with Michael Feischl and Ilaria Perugia (3.2025 — in progress).

 

Sparse grid approximation of stochastic PDEs: Adaptivity and approximation of the stochastic Landau–Lifshitz–Gilbert equation

My Doctoral thesis, written at TU Wien under the supervision of Prof. M. Feischl. Download it from the TU Wien Publications Archive.

This dissertation tackles the approximation of partial differential equations (PDEs) with random data, focusing on random coefficient PDEs and stochastic PDEs (SPDEs). It gathers results from the two publications from my PhD organically together with an introduction, a basic tutorial of my SGMethods Python implementation of sparse grid interpolation, and additional results. Here is an overview of the contents:

  1. Introduction
    1.1 Random coefficient PDEs
    1.2 Numerical approximation of parametric coefficient PDEs
    1.3 The stochastic Landau–Lifshitz–Gilbert equation (SLLG)
    1.4 Contributions of this thesis
  2. Convergence of an adaptive sparse grid–finite element scheme
  3. Sparse grid approximation of nonlinear SPDEs: The SLLG equation
  4. SGMethods: A Python implementation of sparse grid interpolation
  5. Additional and partial results

SGMethods: Python implementation of sparse grid interpolation.

Approximate high dimensional functions based on point samples. Access the GitHub Repository.

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 generates a polynomial interpolation of the original function that can be evaluated cheaply and quickly, compared to the original.

This approximation is based on point samples of the original functions over a sparse grid, i.e. a generalization of tensor-product interpolation nodes that can be tuned to reflect the complexity of the function in each variable.

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 type of problem is also relevant when approximating  random coefficient PDEs if the random field is represented by a parametric equivalent (e.g. a Karhunen-Loève expansion).

Here are some of the key features of SGMethods:

  • 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 polynomials) can easily be 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., JUQ, 2015]. It dramatically reduces the cost of approximating random coefficient PDEs by cleverly combining sparse grid and finite element resolutions.
Example of sparse grid (red) adapted to a function with different frequencies in different directions (blue to yellow backgroud color).
The multi-index set determines the sparse grid nodes allocation in each single direction and for directions' tuples.

Sparse grid approximation of nonlinear stochastic PDEs. The LLG equation.

Research paper written at TU Wien and UNSW with Prof. M. Feischl, Dr. X. An, Prof. J. Dick, Prof. T. Tran. Read the ArXiv Preprint or the SIAM JUQ final Publication.

We develop an effective sparse grid interpolation scheme for the stochastic Landau-Lifshitz-Gilbert equation, i.e. a dynamical model in micromagnetics (sub-micrometer magnetic materials) affected by random heat perturbations represented with by a Gaussian process.

As a first step, we introduce an innovative procedure to transform the SLLG equation (a Stochastic PDE) into a parametric coefficient PDE, where the parameter is a high-dimensional and unbounded vector.

Secondly, we develop a general regularity analysis technique for such parameter-to-solution maps, which allows quantifying the sensitivity of the solution on each scalar parameter.

Thirdly, we design a Sparse Grid Interpolation algorithm tuned to the sensitivity of the current problem and achieves dimension-independent approximation despite the high-dimensional parameter.

Finally, we spell out and test a Multi-Level Sparse Grid Interpolation algorithm, which dramatically reduces the computational cost of fully discrete approximations with a clever combination of sparse grid and finite elements computations.

Magnetic disk with mesh and magnetization (color-coded with magnitude of the component perpendicular to the disk).

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. Read the ArXiv Preprint or the CAMWA final Publication.

We consider a Poisson partial differential equation with random diffusion coefficient. This model has applications e.g. in groundwater flow modelling (the composition of the soil may not precisely be known) or solid/fluid mechanics (the material’s properties may be affected by measurement uncertainties). 

We consider a non-intrusive and adaptive scheme resulting from two approximations:

  • Parametric Approximation: We use Adaptive Sparse Grid Interpolation (ASG) to generate a surrogate model of the parameter-to-solution map based only on point samples. A rigorous error indicator allows allocating more nodes where the parametric dependence is complex.
  • Space Approximation: Each point sample, i.e. a realization of the random solution, is computed with the Adaptive Finite Element Method (AFEM) to discretize the space-dependence of the solution. The mesh is refined where the solution has sharp gradients based on a reliable a-posteriori error indicator.

We give a thorough theoretical analysis of the algorithm, which guarantees it reliability and convergence towards the exact solution.

Computational mesh for the random diffusion Poisson equation. The color encodes the elements' density, from blue (low) to yellow (high). The mesh is automatically refined where the function is more difficult to approximate.

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 master project 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 splines 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 geometric 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 project report (PDF)

We study the method described in [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 non-zero 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.

Simulation of the blood flow in abdominal aortic aneurysms

Project carried out at EPFL, Lausanne, 2017, with Dr. C. Colciago and Prof. A. Quarteroni. Download the project report (PDF)

Abdominal aortic aneurysms (AAA) are 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.

Velocity of the bood flow in abdominal aortic aneurysm
Schematics of the stent in place to correct the blood flow through the aneurysm
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