Eero Saksman

An adaptive Metropolis algorithm

A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis-Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation.

DRAM: Efficient adaptive MCMC

We propose to combine two quite powerful ideas that have recently appeared in the Markov chain Monte Carlo literature: adaptive Metropolis samplers and delayed rejection. The ergodicity of the resulting non-Markovian sampler is proved, and the efficiency of the combination is demonstrated with various examples. We present situations where the combination outperforms the original methods: adaptation clearly enhances efficiency of the delayed rejection algorithm in cases where good proposal distributions are not available. Similarly, delayed rejection provides a systematic remedy when the adaptation process has a slow start.

Componentwise adaptation for high dimensional MCMC

SummaryWe introduce a new adaptive MCMC algorithm, based on the traditional single component Metropolis-Hastings algorithm and on our earlier adaptive Metropolis algorithm (AM). In the new algorithm the adaption is performed component by component. The chain is no more Markovian, but it remains ergodic. The algorithm is demonstrated to work well in varying test cases up to 1000 dimensions.

Beltrami operators in the plane

We determine optimal Lp-properties for the solutions of the general nonlinear elliptic system in the plane of the form fz = H(z, fz), h ∈ L(C), where H is a measurable function satisfying |H(z, w1)−H(z, w2)| ≤ k|w1−w2| and k is a constant k < 1. We will also establish the precise invertibility and spectral properties in Lp(C) for the operators I − Tμ, I − μT, and T − μ, where T is the Beurling transform. These operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions. In particular, we prove invertibility in Lp(C) whenever 1 + ‖μ‖∞ < p < 1 + 1/‖μ‖∞. We also prove related results with applications to the regularity of weakly quasiconformal mappings.

On singular integral and martingale transforms

Linear equivalences of norms of vector-valued singular integral operators and vector-valued martingale transforms are studied. In particular, it is shown that the UMD-constant of a Banach space X equals the norm of the real (or the imaginary) part of the Beurling-Ahlfors singular integral operator, acting on L p X (R 2 ) with p ∈ (1, ∞). Moreover, replacing equality by a linear equivalence, this is found to be a typical property of even multipliers. A corresponding result for odd multipliers and the Hilbert transform is given. As a corollary we obtain that the norm of the real part of the Beurling-Ahlfors operator equals p * — 1 with p * := max{p, (p/(p - 1))}, where the novelty is the lower bound.

Integral means and boundary limits of Dirichlet series

We study the boundary behavior of functions in the Hardy spaces HD^p for ordinary Dirichlet series. Our main result, answering a question of H. Hedenmalm, shows that the classical F. Carlson theorem on integral means does not extend to the imaginary axis for functions in HD^\infty, i.e., for ordinary Dirichlet series in H^\infty of the right half-plane. We discuss an important embedding problem for HD^p, the solution of which is only known when p is an even integer. Viewing HD^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatous theorem.

Burkholder integrals, Morrey’s problem and quasiconformal mappings

Inspired by Morreys Problem (on rank-one convex functionals) and the Burkholder integrals (of his martingale theory) we find that the Burkholder functionals

Basic properties of critical lognormal multiplicative chaos

B_p

Harnack's Inequality for p-Harmonic Functions via Stochastic Games

,

Logarithmic potentials, quasiconformal flows, and

p \ge 2