Spectral analysis of the Zig-Zag process on the torus

Spectral analysis of the Zig-Zag process on the torus

Wiarda, Sjoerd (TU Delft Electrical Engineering, Mathematics and Computer Science)

Bierkens, G.N.J.C. (mentor)
Veraar, M.C. (graduation committee)
Kraaij, R.C. (graduation committee)

Delft University of Technology

Applied Mathematics

2021-12-22

In this thesis, we analyse the spectrum of the generator of the one-dimensional Zig-Zag process defined on the torus $\mathbb{T}$. This is a piecewise deterministic Markov process (PDMP) used in Monte Carlo Markov chain methods (MCMC) for sampling from a probability distribution and calculating integrals \cite{Rejectionfree}, \cite{ZigZag}, \cite{Bouncy}. We show for Lipschitz potentials $U$ and bounded refreshment rates $\lambda_0 \in L^{\infty}(\mathbb{T})$ that the spectral gap $\kappa = \sup\{\operatorname{Re} \lambda : \lambda \in \sigma(\mathcal{L})\} \setminus \{0 \}$ of the associated $J$-self-adjoint generator $\mathcal{L}$ on $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ is positive. Moreover, we give two lower bounds for $\kappa$ by making use of one of the Schur complements associated with a block operator that is unitarily equivalent to $\mathcal{L}$. In addition we show that the spectrum of $L^2(\mathbb{T} ,\nu)$ and $C(\mathbb{T} \times \{+1,-1\})$ are the same and that the generator defined on both spaces generates a contraction semigroup. Under the assumption of unimodality of the potential $U$ and a zero refreshment rate, we show that a vertical "asymptotic line" exists to which all of the eigenvalues converge. Furthermore, we show that a spectral mapping theorem exists where, due to the spectral line, the spectrum of the semigroup can become uncountable or countable depending on the time parameter of the semigroup $P(t)$ generated by $\mathcal{L}$. Lastly, we show that a discretisation of the spectrum generates a semigroup that converges uniformly on each bounded time interval to the semigroup of the Zig-Zag process and we use these discretisations to numerically analyse the behaviour of general potentials and refreshment rates.

spectral gap
J-self-adjoint operator
spectral mapping theorem
Zig-Zag process
piecewise deterministic Markov process
MCMC
spectral theory
Schur complement

http://resolver.tudelft.nl/uuid:1a0930e4-a739-4666-b38e-cf8f46509178

Student theses

master thesis

© 2021 Sjoerd Wiarda