85 research outputs found
Long Time Propagation of Stochasticity by Dynamical Polynomial Chaos Expansions
Stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs) play an important role in many areas of engineering and applied sciences such as atmospheric sciences, mechanical and aerospace engineering, geosciences, and finance. Equilibrium statistics and long-time solutions of these equations are pertinent to many applications. Typically, these models contain several uncertain parameters which need to be propagated in order to facilitate uncertainty quantification and prediction. Correspondingly, in this thesis, we propose a generalization of the Polynomial Chaos (PC) framework for long-time solutions of SDEs and SPDEs driven by Brownian motion forcing.
Polynomial chaos expansions (PCEs) allow us to propagate uncertainties in the coefficients of these equations to the statistics of their solutions. Their main advantages are: (i) they replace stochastic equations by systems of deterministic equations; and (ii) they provide fast convergence. Their main challenge is that the computational cost becomes prohibitive when the dimension of the parameters modeling the stochasticity is even moderately large. In particular, for equations with Brownian motion forcing, the long-time simulation by PC-based methods is notoriously difficult as the dimension of stochastic variables increases with time.
With the goal in mind to deliver computationally efficient numerical algorithms for stochastic equations in the long time, our main strategy is to leverage the intrinsic sparsity in the dynamics by identifying the influential random parameters and construct spectral approximations to the solutions in terms of those relevant variables. Once this strategy is employed dynamically in time, using online constructions, approximations can retain their sparsity and accuracy; even for long times. To this end, exploiting Markov property of Brownian motion, we present a restart procedure that allows PCEs to expand the solutions at future times in terms of orthogonal polynomials of the measure describing the solution at a given time and the future Brownian motion. In case of SPDEs, the Karhunen-Loeve expansion (KLE) is applied at each restart to select the influential variables and keep the dimensionality minimal. Using frequent restarts and low degree polynomials, the algorithms are able to capture long-time solutions accurately. We will also introduce, using the same principles, a similar algorithm based on a stochastic collocation method for the solutions of SDEs.
We apply the methods to the numerical simulation of linear and nonlinear SDEs, and stochastic Burgers and Navier-Stokes equations with white noise forcing. Our methods also allow us to incorporate time-independent random coefficients such as a random viscosity. We propose several numerical simulations, and show that the algorithms compare favorably with standard Monte Carlo methods in terms of accuracy and computational times. To demonstrate the efficiency of the algorithms for long-time simulations, we compute invariant measures of the solutions when they exist
Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Points
We present a machine learning (ML)-assisted framework bridging manifold
learning, neural networks, Gaussian processes, and Equation-Free multiscale
modeling, for (a) detecting tipping points in the emergent behavior of complex
systems, and (b) characterizing probabilities of rare events (here,
catastrophic shifts) near them. Our illustrative example is an event-driven,
stochastic agent-based model (ABM) describing the mimetic behavior of traders
in a simple financial market. Given high-dimensional spatiotemporal data --
generated by the stochastic ABM -- we construct reduced-order models for the
emergent dynamics at different scales: (a) mesoscopic Integro-Partial
Differential Equations (IPDEs); and (b) mean-field-type Stochastic Differential
Equations (SDEs) embedded in a low-dimensional latent space, targeted to the
neighborhood of the tipping point. We contrast the uses of the different models
and the effort involved in learning them.Comment: 29 pages, 8 figures, 6 table
Semi-Parametric Drift and Diffusion Estimation for Multiscale Diffusions
We consider the problem of statistical inference for the effective dynamics
of multiscale diffusion processes with (at least) two widely separated
characteristic time scales. More precisely, we seek to determine parameters in
the effective equation describing the dynamics on the longer diffusive time
scale, i.e. in a homogenization framework. We examine the case where both the
drift and the diffusion coefficients in the effective dynamics are
space-dependent and depend on multiple unknown parameters. It is known that
classical estimators, such as Maximum Likelihood and Quadratic Variation of the
Path Estimators, fail to obtain reasonable estimates for parameters in the
effective dynamics when based on observations of the underlying multiscale
diffusion. We propose a novel algorithm for estimating both the drift and
diffusion coefficients in the effective dynamics based on a semi-parametric
framework. We demonstrate by means of extensive numerical simulations of a
number of selected examples that the algorithm performs well when applied to
data from a multiscale diffusion. These examples also illustrate that the
algorithm can be used effectively to obtain accurate and unbiased estimates.Comment: 32 pages, 10 figure
Mathematics of Quantitative Finance
The workshop on Mathematics of Quantitative Finance, organised at the Mathematisches Forschungsinstitut Oberwolfach from 26 February to 4 March 2017, focused on cutting edge areas of mathematical finance, with an emphasis on the applicability of the new techniques and models presented by the participants
Comparison of stochastic parameterizations in the framework of a coupled ocean-atmosphere model
A new framework is proposed for the evaluation of stochastic subgrid-scale
parameterizations in the context of MAOOAM, a coupled ocean-atmosphere model of
intermediate complexity. Two physically-based parameterizations are
investigated, the first one based on the singular perturbation of Markov
operator, also known as homogenization. The second one is a recently proposed
parameterization based on the Ruelle's response theory. The two
parameterization are implemented in a rigorous way, assuming however that the
unresolved scale relevant statistics are Gaussian. They are extensively tested
for a low-order version known to exhibit low-frequency variability, and some
preliminary results are obtained for an intermediate-order version. Several
different configurations of the resolved-unresolved scale separations are then
considered. Both parameterizations show remarkable performances in correcting
the impact of model errors, being even able to change the modality of the
probability distributions. Their respective limitations are also discussed.Comment: 44 pages, 12 figures, 4 table
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