14 research outputs found
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
Asymptotics of the Coleman-Gurtin model
This paper is concerned with the integrodifferential equation \partial_t
u-\Delta u -\int_0^\infty \kappa(s)\Delta u(t-s)\,\d s + \varphi(u)=f arising
in the Coleman-Gurtin's theory of heat conduction with hereditary memory, in
presence of a nonlinearity of critical growth. Rephrasing the
equation within the history space framework, we prove the existence of global
and exponential attractors of optimal regularity and finite fractal dimension
for the related solution semigroup, acting both on the basic weak-energy space
and on a more regular phase space.Comment: Accepted in Discrete and Continuous Dynamical Systems, Serie
A Statistical Framework for Domain Shape Estimation in Stokes Flows
We develop and implement a Bayesian approach for the estimation of the shape
of a two dimensional annular domain enclosing a Stokes flow from sparse and
noisy observations of the enclosed fluid. Our setup includes the case of direct
observations of the flow field as well as the measurement of concentrations of
a solute passively advected by and diffusing within the flow. Adopting a
statistical approach provides estimates of uncertainty in the shape due both to
the non-invertibility of the forward map and to error in the measurements. When
the shape represents a design problem of attempting to match desired target
outcomes, this "uncertainty" can be interpreted as identifying remaining
degrees of freedom available to the designer. We demonstrate the viability of
our framework on three concrete test problems. These problems illustrate the
promise of our framework for applications while providing a collection of test
cases for recently developed Markov Chain Monte Carlo (MCMC) algorithms
designed to resolve infinite dimensional statistical quantities
The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction
We study a class of semi-linear differential Volterra equations with
polynomial-type potentials that incorporates the effects of memory while being
subjected to random perturbations via an additive Gaussian noise. We show that
for a broad class of non-linear potentials and sufficiently regular noise the
system always admits invariant probability measures, defined on the extended
phase space, that possess higher regularity properties dictated by the
structure of the nonlinearities in the equation. Furthermore, we investigate
the singular limit as the memory kernel collapses to a Dirac function.
Specifically, provided sufficiently many directions in the phase space are
stochastically forced, we show that there is a unique stationary measure to
which the system converges, in a suitable Wasserstein distance, at exponential
rates independent of the decay of the memory kernel. We then prove the
convergence of the statistically steady states to the unique invariant
probability of the classical stochastic reaction-diffusion equation in the
desired singular limit. As a consequence, we establish the validity of the
small memory approximation for solutions on the infinite time horizon
On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2
The continuous-time Markov chain (CTMC) is the mathematical workhorse of
evolutionary biology. Learning CTMC model parameters using modern,
gradient-based methods requires the derivative of the matrix exponential
evaluated at the CTMC's infinitesimal generator (rate) matrix. Motivated by the
derivative's extreme computational complexity as a function of state space
cardinality, recent work demonstrates the surprising effectiveness of a naive,
first-order approximation for a host of problems in computational biology. In
response to this empirical success, we obtain rigorous deterministic and
probabilistic bounds for the error accrued by the naive approximation and
establish a "blessing of dimensionality" result that is universal for a large
class of rate matrices with random entries. Finally, we apply the first-order
approximation within surrogate-trajectory Hamiltonian Monte Carlo for the
analysis of the early spread of SARS-CoV-2 across 44 geographic regions that
comprise a state space of unprecedented dimensionality for unstructured
(flexible) CTMC models within evolutionary biology