30 research outputs found

    On the surprising effectiveness of a simple matrix exponential derivative approximation, with application to global SARS-CoV-2

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    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

    The short memory limit for long time statistics in a stochastic Coleman-Gurtin model of heat conduction

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    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 [0,)[0,\infty)

    A Statistical Framework for Domain Shape Estimation in Stokes Flows

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    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

    Embracing Uncertainty in "Small Data" Problems: Estimating Earthquakes from Historical Anecdotes

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    We apply the Bayesian inversion process to make principled estimates of the magnitude and location of a pre-instrumental earthquake in Eastern Indonesia in the mid 19th century, by combining anecdotal historical accounts of the resultant tsunami with our modern understanding of the geology of the region. Quantifying the seismic record prior to modern instrumentation is critical to a more thorough understanding of the current risks in Eastern Indonesia. In particular, the occurrence of such a major earthquake in the 1850s provides evidence that this region is susceptible to future seismic hazards on the same order of magnitude. More importantly, the approach taken here gives evidence that even "small data" that is limited in scope and extremely uncertain can still be used to yield information on past seismic events, which is key to an increased understanding of the current seismic state. Moreover, sensitivity bounds indicate that the results obtained here are robust despite the inherent uncertainty in the observations