Generalized Polynomial Chaos-Based Extended Kalman Filter: Improvement and Expansion

2013 ◽  
Author(s):  
Jeremy Kolansky ◽  
Corina Sandu
Keyword(s):  
Kalman Filter ◽  
State Space ◽  
Extended Kalman Filter ◽  
Polynomial Chaos ◽  
Computationally Efficient ◽  
Generalized Polynomial Chaos ◽  
Parameter Estimations ◽  
Generalized Polynomial ◽  
State Estimations ◽  
Polynomial Chaos Expansions

The generalized polynomial chaos (gPC) method for propagating uncertain parameters through dynamical systems (previously developed at Virginia Tech) has been shown to be very computationally efficient. This method seems also to be ideal for real-time parameter estimation when merged with the Extended Kalman Filter (EKF). The resulting technique is shown in the present paper for systems in state-space representations, and then expanded to systems in regressions formulations. Due to the way the filter interacts with the polynomial chaos expansions, the covariance matrix is forced to zero in finite time. This problem shows itself as an inability to perform state estimations and causes the parameters to converge to incorrect values for state space systems. In order to address this issue, improvements to the method are implemented and the updated method is applied to both state space and regression systems. The resultant technique shows high accuracy of both state and parameter estimations.

Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ

2018 ◽  
Vol 17 (01) ◽  
pp. 19-55 ◽  
Author(s):  
Christoph Schwab ◽  
Jakob Zech
Keyword(s):  
Polynomial Chaos ◽  
Expressive Power ◽  
Response Surfaces ◽  
Holomorphic Maps ◽  
Generalized Polynomial Chaos ◽  
Generalized Polynomial ◽  
Number Formula ◽  
Expansion Coefficients ◽  
Polynomial Chaos Expansions ◽  
Parametric Maps

We estimate the expressive power of certain deep neural networks (DNNs for short) on a class of countably-parametric, holomorphic maps [Formula: see text] on the parameter domain [Formula: see text]. Dimension-independent rates of best [Formula: see text]-term truncations of generalized polynomial chaos (gpc for short) approximations depend only on the summability exponent of the sequence of their gpc expansion coefficients. So-called [Formula: see text]-holomorphic maps [Formula: see text], with [Formula: see text] for some [Formula: see text], are known to allow gpc expansions with coefficient sequences in [Formula: see text]. Such maps arise for example as response surfaces of parametric PDEs, with applications in PDE uncertainty quantification (UQ) for many mathematical models in engineering and the sciences. Up to logarithmic terms, we establish the dimension independent approximation rate [Formula: see text] for these functions in terms of the total number [Formula: see text] of units and weights in the DNN. It follows that certain DNN architectures can overcome the curse of dimensionality when expressing possibly countably-parametric, real-valued maps with a certain degree of sparsity in the sequences of their gpc expansion coefficients. We also obtain rates of expressive power of DNNs for countably-parametric maps [Formula: see text], where [Formula: see text] is the Hilbert space [Formula: see text].

Enhanced Polynomial Chaos-Based Extended Kalman Filter Technique for Parameter Estimation

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Jeremy Kolansky ◽  
Corina Sandu
Keyword(s):  
Parameter Estimation ◽  
Kalman Filter ◽  
Extended Kalman Filter ◽  
Parameter Uncertainty ◽  
Polynomial Chaos ◽  
Recursive Least Squares ◽  
Mathematical Technique ◽  
Generalized Polynomial ◽  
State Tracking ◽  
Parameter Distributions

The generalized polynomial chaos (gPC) mathematical technique, when integrated with the extended Kalman filter (EKF) method, provides a parameter estimation and state tracking method. The truncation of the series expansions degrades the link between parameter convergence and parameter uncertainty which the filter uses to perform the estimations. An empirically derived correction for this problem is implemented, which maintains the original parameter distributions. A comparison is performed to illustrate the improvements of the proposed approach. The method is demonstrated for parameter estimation on a regression system, where it is compared to the recursive least squares (RLS) method.

A Statistical Imputation Method for Handling Missing Values in Generalized Polynomial Chaos Expansions

2019 ◽  
Author(s):  
Mark Andrews ◽  
Gavin Jones ◽  
Brian Leyde ◽  
Lie Xiong ◽  
Max Xu ◽  
...  
Keyword(s):  
Missing Data ◽  
Missing Values ◽  
Polynomial Chaos ◽  
Imputation Method ◽  
Sparse Grid ◽  
Sampling Point ◽  
Generalized Polynomial Chaos ◽  
Sensitivity Indices ◽  
Generalized Polynomial ◽  
Polynomial Chaos Expansions

Abstract Generalized Polynomial Chaos Expansion (gPCE) is widely used in uncertainty quantification and sensitivity analysis for applications in the aerospace industry. gPCE uses the spectrum projection to fit a polynomial model, the gPCE model, to a sparse grid Design of Experiments (DOEs). The gPCE model can be used to make predictions, analytically determine uncertainties, and calculate sensitivity indices. However, the model’s accuracy is very dependent on having complete DOEs. When a sampling point is missing from the sparse grid DOE, this severely impacts the accuracy of the gPCE analysis and often necessitates running a new DOE. Missing data points are a common occurrence in engineering testing and simulation. This problem complicates the use of the gPCE analysis. In this paper, we present a statistical imputation method for addressing this missing data problem. This methodology allows gPCE modeling to handle missing values in the sparse grid DOE. Using a series of numerical results, the study demonstrates the convergence characteristics of the methodology with respect to reaching steady state values for the missing points. The article concludes with a discussion of the convergence rate, advantages, and feasibility of using the proposed methodology.

scholarly journals On the convergence of generalized polynomial chaos expansions

2011 ◽  
Vol 46 (2) ◽  
pp. 317-339 ◽  
Author(s):  
Oliver G. Ernst ◽  
Antje Mugler ◽  
Hans-Jörg Starkloff ◽  
Elisabeth Ullmann
Keyword(s):  
Polynomial Chaos ◽  
Generalized Polynomial Chaos ◽  
Generalized Polynomial ◽  
Polynomial Chaos Expansions
Sign in / Sign up

Export Citation Format

Share Document