skip to main content


Title: Quantification of model uncertainty on path-space via goal-oriented relative entropy
Quantifying the impact of parametric and model-form uncertainty on the predictions of stochastic models is a key challenge in many applications. Previous work has shown that the relative entropy rate is an effective tool for deriving path-space uncertainty quantification (UQ) bounds on ergodic averages. In this work we identify appropriate information-theoretic objects for a wider range of quantities of interest on path-space, such as hitting times and exponentially discounted observables, and develop the corresponding UQ bounds. In addition, our method yields tighter UQ bounds, even in cases where previous relative-entropy-based methods also apply, e.g. , for ergodic averages. We illustrate these results with examples from option pricing, non-reversible diffusion processes, stochastic control, semi-Markov queueing models, and expectations and distributions of hitting times.  more » « less
Award ID(s):
2008970
NSF-PAR ID:
10356229
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
ESAIM: Mathematical Modelling and Numerical Analysis
Volume:
55
Issue:
1
ISSN:
0764-583X
Page Range / eLocation ID:
131 to 169
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract

    The Mass Spectrometer and Incoherent Scatter radar (MSIS) model family has been developed and improved since the early 1970's. The most recent version of MSIS is the Naval Research Laboratory (NRL) MSIS 2.0 empirical atmospheric model. NRLMSIS 2.0 provides species density, mass density, and temperature estimates as function of location and space weather conditions. MSIS models have long been a popular choice of thermosphere model in the research and operations community alike, but—like many models—does not provide uncertainty estimates. In this work, we develop an exospheric temperature model based in machine learning that can be used with NRLMSIS 2.0 to calibrate it relative to high‐fidelity satellite density estimates directly through the exospheric temperature parameter. Instead of providing point estimates, our model (called MSIS‐UQ) outputs a distribution which is assessed using a metric called the calibration error score. We show that MSIS‐UQ debiases NRLMSIS 2.0 resulting in reduced differences between model and satellite density of 25% and is 11% closer to satellite density than the Space Force's High Accuracy Satellite Drag Model. We also show the model's uncertainty estimation capabilities by generating altitude profiles for species density, mass density, and temperature. This explicitly demonstrates how exospheric temperature probabilities affect density and temperature profiles within NRLMSIS 2.0. Another study displays improved post‐storm overcooling capabilities relative to NRLMSIS 2.0 alone, enhancing the phenomena that it can capture.

     
    more » « less
  2. Abstract

    Climate models are generally calibrated manually by comparing selected climate statistics, such as the global top‐of‐atmosphere energy balance, to observations. The manual tuning only targets a limited subset of observational data and parameters. Bayesian calibration can estimate climate model parameters and their uncertainty using a larger fraction of the available data and automatically exploring the parameter space more broadly. In Bayesian learning, it is natural to exploit the seasonal cycle, which has large amplitude compared with anthropogenic climate change in many climate statistics. In this study, we develop methods for the calibration and uncertainty quantification (UQ) of model parameters exploiting the seasonal cycle, and we demonstrate a proof‐of‐concept with an idealized general circulation model (GCM). UQ is performed using the calibrate‐emulate‐sample approach, which combines stochastic optimization and machine learning emulation to speed up Bayesian learning. The methods are demonstrated in a perfect‐model setting through the calibration and UQ of a convective parameterization in an idealized GCM with a seasonal cycle. Calibration and UQ based on seasonally averaged climate statistics, compared to annually averaged, reduces the calibration error by up to an order of magnitude and narrows the spread of the non‐Gaussian posterior distributions by factors between two and five, depending on the variables used for UQ. The reduction in the spread of the parameter posterior distribution leads to a reduction in the uncertainty of climate model predictions.

     
    more » « less
  3. null (Ed.)
    The authors develop a theory characterizing optimal stopping times for discrete-time ergodic Markov processes with discounted rewards. The theory differs from prior work by its view of per-stage and terminal reward functions as elements of a certain Hilbert space. In addition to a streamlined analysis establishing existence and uniqueness of a solution to Bellman's equation, this approach provides an elegant framework for the study of approximate solutions. In particular, the authors propose a stochastic approximation algorithm that tunes weights of a linear combination of basis functions in order to approximate a value function. They prove that this algorithm converges (almost surely) and that the limit of convergence has some desirable properties. The utility of the approximation method is illustrated via a computational case study involving the pricing of a path dependent financial derivative security that gives rise to an optimal stopping problem with a 100-dimensional state space 
    more » « less
  4. Ta-Shma, Amnon (Ed.)
    In a recent work, Gryaznov, Pudlák and Talebanfard (CCC '22) introduced a linear variant of read-once branching programs, with motivations from circuit and proof complexity. Such a read-once linear branching program is a branching program where each node is allowed to make 𝔽₂-linear queries, and is read-once in the sense that the queries on each path is linearly independent. As their main result, they constructed an explicit function with average-case complexity 2^{n/3-o(n)} against a slightly restricted model, which they call strongly read-once linear branching programs. The main tool in their lower bound result is a new type of extractor, called directional affine extractors, that they introduced. Our main result is an explicit function with 2^{n-o(n)} average-case complexity against the strongly read-once linear branching program model, which is almost optimal. This result is based on a new connection from this problem to sumset extractors, which is a randomness extractor model introduced by Chattopadhyay and Li (STOC '16) as a generalization of many other well-studied models including two-source extractors, affine extractors and small-space extractors. With this new connection, our lower bound naturally follows from a recent construction of sumset extractors by Chattopadhyay and Liao (STOC '22). In addition, we show that directional affine extractors imply sumset extractors in a restricted setting. We observe that such restricted sumset sources are enough to derive lower bounds, and obtain an arguably more modular proof of the lower bound by Gryaznov, Pudlák and Talebanfard. We also initiate a study of pseudorandomness against linear branching programs. Our main result here is a hitting set generator construction against regular linear branching programs with constant width. We derive this result based on a connection to Kakeya sets over finite fields. 
    more » « less
  5. Deterministic and stochastic approaches to handle uncertainties may incur very different complexities in computation and memory, in addition to different uncertainty models. For linear systems with delay and rate constrained communications between the observer and controller, previous work shows that the deterministic approach l_infty control has low complexity but only handles bounded disturbance. In this paper, we take a stochastic approach and propose an LQ controller that can handle arbitrarily large disturbance but has large complexity in time/space. The differences in robustness and complexity of the l_infty and LQ controllers motivate the design of a hybrid controller that interpolates between the two: The l_infty controller is applied when the disturbance is not too large (normal mode) and the LQ controller is resorted to otherwise (acute mode). We characterize the switching behavior between the normal and acute modes. Using theoretical bounds and supplementary numerical experiments, we show that the hybrid controller can achieve a sweet spot in robustness-complexity tradeoff, ie, reject occasional large disturbance while operating with low complexity most of the time. 
    more » « less