Estimating and quantifying uncertainty in system parameters remains a big challenge in applied and computational mathematics. A subset of these problems includes estimating periodic parameters that have unknown dynamics. Along with their time series, the period of these parameters may also be unknown and need to be estimated. The aim of this paper is to address the periodic parameter estimation problem, with particular focus on exploring the associated uncertainty, using Monte Carlo particle methods, such as the ensemble Kalman filter. Both parameter tracking and piecewise function approximations of periodic parameters are considered, highlighting aspects of parameter uncertainty in each approach when considering factors such as the frequency of available data and the number of piecewise segments used in the approximation. Estimation of the period of the periodic parameters and related uncertainty is also analyzed in the piecewise formulation. The pros and cons of each approach are discussed relative to a numerical example estimating the external voltage parameter in the FitzHugh-Nagumo system for modeling the spiking dynamics of neurons.
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Surrogate approximation of the Grad–Shafranov free boundary problem via stochastic collocation on sparse grids
In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30.
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- Award ID(s):
- 1819115
- NSF-PAR ID:
- 10318339
- Date Published:
- Journal Name:
- Journal of computational physics
- ISSN:
- 0021-9991
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Cardiovascular simulations are increasingly used for noninvasive diagnosis of cardiovascular disease, to guide treatment decisions, and in the design of medical devices. Quantitative assessment of the variability of simulation outputs due to input uncertainty is a key step toward further integration of cardiovascular simulations in the clinical workflow. In this study, we present uncertainty quantification in computational models of the coronary circulation to investigate the effect of uncertain parameters, including coronary pressure waveform, intramyocardial pressure, morphometry exponent, and the vascular wall Young's modulus. We employ a left coronary artery model with deformable vessel walls, simulated via an Arbitrary‐Lagrangian‐Eulerian framework for fluid‐structure interaction, with a prescribed inlet pressure and open‐loop lumped parameter network outlet boundary conditions. Stochastic modeling of the uncertain inputs is determined from intra‐coronary catheterization data or gathered from the literature. Uncertainty propagation is performed using several approaches including Monte Carlo, Quasi Monte Carlo sampling, stochastic collocation, and multi‐wavelet stochastic expansion. Variabilities in the quantities of interest, including branch pressure, flow, wall shear stress, and wall deformation are assessed. We find that uncertainty in inlet pressures and intramyocardial pressures significantly affect all resulting QoIs, while uncertainty in elastic modulus only affects the mechanical response of the vascular wall. Variability in the morphometry exponent used to distribute the total downstream vascular resistance to the single outlets, has little effect on coronary hemodynamics or wall mechanics. Finally, we compare convergence behaviors of statistics of QoIs using several uncertainty propagation methods on three model benchmark problems and the left coronary simulations. From the simulation results, we conclude that the multi‐wavelet stochastic expansion shows superior accuracy and performance against Quasi Monte Carlo and stochastic collocation methods.
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1016/j.jcp.2021.110699
