# Alexander Gilbert - Quasi-Monte Carlo methods for the uncertainty quantification of eigenvalue problems

Dates: | 20 March 2020 |
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Times: | 14:00 - 15:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Speaker: | Alexander Gilbert |

Dr. Alexander Gilbert from Institute for Applied Mathematics at Heidelberg University will be speaking at this seminar. Abstract: Eigenvalue problems are useful for modelling many important physical phenomena, ranging from photonic crystal structures to quantum mechanics to the neutron diffusion criticality problem. In many of these applications the model parameters are unknown, in which case one aims to quantify the uncertainty in the eigenproblem model. With this as motivation, we will study an elliptic eigenvalue problem with coefficients that depend on infinitely many stochastic parameters. The stochasticity in the coefficients causes the eigenvalues and eigenfunctions to also be stochastic, and so our goal will be to compute the expectation of the minimal eigenvalue. In practice, to approximate this expectation one must: 1) truncate the stochastic dimension; 2) discretise the eigenvalue problem in space (e.g., by finite elements); and 3) apply a quadrature rule to estimate the expected value.In this talk, we will present a multilevel quasi-Monte Carlo method for approximating the expectation of the minimal eigenvalue, which is based on a hierarchy of finite element meshes and truncation dimensions. To improve the sampling efficiency over Monte Carlo we will use a quasi-Monte Carlo rule to generate the sampling points. Quasi-Monte Carlo rules are deterministic (or quasi-random) quadrature rules that are well-suited to high-dimensional integration and can converge at a rate of 1/N, which is faster than the rate for Monte Carlo. Also, to make each eigenproblem solve on a given level more efficient, we utilise the two-grid scheme from & Zhou 1999 to obtain the eigenvalue on the fine mesh from the coarse eigenvalue (and eigenfunction) with a single linear solve.

### Speaker

Alexander Gilbert

Organisation: Institute for Applied Mathematics, Heidelberg University