7,371 research outputs found
Dynamic Programming for controlled Markov families: abstractly and over Martingale Measures
We describe an abstract control-theoretic framework in which the validity of
the dynamic programming principle can be established in continuous time by a
verification of a small number of structural properties. As an application we
treat several cases of interest, most notably the lower-hedging and
utility-maximization problems of financial mathematics both of which are
naturally posed over ``sets of martingale measures''
Bayesian Filtering for ODEs with Bounded Derivatives
Recently there has been increasing interest in probabilistic solvers for
ordinary differential equations (ODEs) that return full probability measures,
instead of point estimates, over the solution and can incorporate uncertainty
over the ODE at hand, e.g. if the vector field or the initial value is only
approximately known or evaluable. The ODE filter proposed in recent work models
the solution of the ODE by a Gauss-Markov process which serves as a prior in
the sense of Bayesian statistics. While previous work employed a Wiener process
prior on the (possibly multiple times) differentiated solution of the ODE and
established equivalence of the corresponding solver with classical numerical
methods, this paper raises the question whether other priors also yield
practically useful solvers. To this end, we discuss a range of possible priors
which enable fast filtering and propose a new prior--the Integrated Ornstein
Uhlenbeck Process (IOUP)--that complements the existing Integrated Wiener
process (IWP) filter by encoding the property that a derivative in time of the
solution is bounded in the sense that it tends to drift back to zero. We
provide experiments comparing IWP and IOUP filters which support the belief
that IWP approximates better divergent ODE's solutions whereas IOUP is a better
prior for trajectories with bounded derivatives.Comment: 14 pages, 9 figrue
Probability Measures for Numerical Solutions of Differential Equations
In this paper, we present a formal quantification of epistemic uncertainty
induced by numerical solutions of ordinary and partial differential equation
models. Numerical solutions of differential equations contain inherent
uncertainties due to the finite dimensional approximation of an unknown and
implicitly defined function. When statistically analysing models based on
differential equations describing physical, or other naturally occurring,
phenomena, it is therefore important to explicitly account for the uncertainty
introduced by the numerical method. This enables objective determination of its
importance relative to other uncertainties, such as those caused by data
contaminated with noise or model error induced by missing physical or
inadequate descriptors. To this end we show that a wide variety of existing
solvers can be randomised, inducing a probability measure over the solutions of
such differential equations. These measures exhibit contraction to a Dirac
measure around the true unknown solution, where the rates of convergence are
consistent with the underlying deterministic numerical method. Ordinary
differential equations and elliptic partial differential equations are used to
illustrate the approach to quantifying uncertainty in both the statistical
analysis of the forward and inverse problems
PHOENICS: A universal deep Bayesian optimizer
In this work we introduce PHOENICS, a probabilistic global optimization
algorithm combining ideas from Bayesian optimization with concepts from
Bayesian kernel density estimation. We propose an inexpensive acquisition
function balancing the explorative and exploitative behavior of the algorithm.
This acquisition function enables intuitive sampling strategies for an
efficient parallel search of global minima. The performance of PHOENICS is
assessed via an exhaustive benchmark study on a set of 15 discrete,
quasi-discrete and continuous multidimensional functions. Unlike optimization
methods based on Gaussian processes (GP) and random forests (RF), we show that
PHOENICS is less sensitive to the nature of the co-domain, and outperforms GP
and RF optimizations. We illustrate the performance of PHOENICS on the
Oregonator, a difficult case-study describing a complex chemical reaction
network. We demonstrate that only PHOENICS was able to reproduce qualitatively
and quantitatively the target dynamic behavior of this nonlinear reaction
dynamics. We recommend PHOENICS for rapid optimization of scalar, possibly
non-convex, black-box unknown objective functions
On numerical density approximations of solutions of SDEs with unbounded coefficients
We study a numerical method to compute probability density functions of
solutions of stochastic differential equations. The method is sometimes called
the numerical path integration method and has been shown to be fast and
accurate in application oriented fields. In this paper we provide a rigorous
analysis of the method that covers systems of equations with unbounded
coefficients. Working in a natural space for densities, , we obtain
stability, consistency, and new convergence results for the method, new
well-posedness and semigroup generation results for the related
Fokker-Planck-Kolmogorov equation, and a new and rigorous connection to the
corresponding probability density functions for both the approximate and the
exact problems. To prove the results we combine semigroup and PDE arguments in
a new way that should be of independent interest
Probability & incompressible Navier-Stokes equations: An overview of some recent developments
This is largely an attempt to provide probabilists some orientation to an
important class of non-linear partial differential equations in applied
mathematics, the incompressible Navier-Stokes equations. Particular focus is
given to the probabilistic framework introduced by LeJan and Sznitman [Probab.
Theory Related Fields 109 (1997) 343-366] and extended by Bhattacharya et al.
[Trans. Amer. Math. Soc. 355 (2003) 5003-5040; IMA Vol. Math. Appl., vol. 140,
2004, in press]. In particular this is an effort to provide some foundational
facts about these equations and an overview of some recent results with an
indication of some new directions for probabilistic consideration.Comment: Published at http://dx.doi.org/10.1214/154957805100000078 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Overdamped limit of generalized stochastic Hamiltonian systems for singular interaction potentials
First weak solutions of generalized stochastic Hamiltonian systems (gsHs) are
constructed via essential m-dissipativity of their generators on a suitable
core. For a scaled gsHs we prove convergence of the corresponding semigroups
and tightness of the weak solutions. This yields convergence in law of the
scaled gsHs to a distorted Brownian motion. In particular, the results confirm
the convergence of the Langevin dynamics in the overdamped regime to the
overdamped Langevin equation. The proofs work for a large class of (singular)
interaction potentials including, e.g., potentials of Lennard--Jones type
Quantitative model-checking of controlled discrete-time Markov processes
This paper focuses on optimizing probabilities of events of interest defined
over general controlled discrete-time Markov processes. It is shown that the
optimization over a wide class of -regular properties can be reduced to
the solution of one of two fundamental problems: reachability and repeated
reachability. We provide a comprehensive study of the former problem and an
initial characterisation of the (much more involved) latter problem. A case
study elucidates concepts and techniques
Nonnegative entire bounded solutions to some semilinear equations involving the fractional Laplacian
The main goal is to establish necessary and sufficient conditions under which
the fractional semilinear elliptic equation admits nonnegative nontrivial bounded solutions in the
whole space
Multiplicatively interacting point processes and applications to neural modeling
We introduce a nonlinear modification of the classical Hawkes process, which
allows inhibitory couplings between units without restrictions. The resulting
system of interacting point processes provides a useful mathematical model for
recurrent networks of spiking neurons with exponential transfer functions. The
expected rates of all neurons in the network are approximated by a first-order
differential system. We study the stability of the solutions of this equation,
and use the new formalism to implement a winner-takes-all network that operates
robustly for a wide range of parameters. Finally, we discuss relations with the
generalised linear model that is widely used for the analysis of spike trains.Comment: 22 pages, 7 figures. Submitted to J. Comp. Neurosci. Overall changes
according to suggestions of different reviewers. A conceptual error in a
derivation has been correcte
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