## Data Assimilation and Reduced Modeling for High Dimensional Problems

##### CIRM, Luminy, France

##### July 19-August 27, 2021

#### Model order reduction by spectral gap optimization

**Supervisors:**T. Lelièvre, L. Pillaud-Vivien et G. Stoltz (EPFL, Ecole des Ponts ParisTech)

**Students:**Open to 2 students

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Project Description:
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Sujet 2: Model order reduction by spectral gap optimization [T. Lelièvre, L. Pillaud-Vivien et G. Stoltz ]
Obtaining a good collective variable is a crucial problem in computational statistical physics, in particular to apply
free energy biasing methods (the free energy being associated with the chosen collective variable). In this context, a
good collective variable is such that the free energy biased dynamics reached equilibrium much quicker than the original
unbiased dynamics. In a recent work [1], an algorithm has been proposed to estimate the spectral gap of the overdamped
Langevin dynamics for a given target probability measure, using samples of this measure. This can be used as a measure
of the rate of convergence to equilibrium, and thus leads to optimization algorithm. The objective of this project is to
investigate if this idea can be used on concrete examples from computational statistical physics or Bayesian inference.

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References:
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[1] L. Pillaud-Vivien, F. Bach, T. Lelièvre, A. Rudi et G. Stoltz, Statistical Estimation of the Poincaré constant and
Application to Sampling Multimodal Distributions, https://hal.archives-ouvertes.fr/hal-02327453v2