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79 records · Page 5

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DOE OSTI2012

New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection-reaction equation. By using a Fourier-Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2020

Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

We use a conditional Karhunen-Lo` eve (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, Y (x). We assume that a small number of Y (x) measurements are available and model Y (x) with a KL expansion. We achieve reduction in uncertainty by conditioning the KL expansion coefficients on measurements. We consider two approaches for conditioning the KL expansion: In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condition it. We employ the conditional KL expansion together with Monte Carlo and sparse grid collocation methods to compute the moments of the solution of the SPDE problem. Uncertainty of the problem is further reduced by adaptively selecting additional observation locations using two active learning methods. Method 1 minimizes the variance of the PDE coefficient, while Method 2 minimizes the variance of the solution of the PDE. We demonstrate that conditioning leads to dimension reduction of the KL representation of Y (x). For a linear diffusion SPDE with uncertain log-normal coefficient, we show that Approach 1 provides a more accurate approximation of the conditional log-normal coefficient and solution of the SPDE than Approach 2 for the same number of random dimensions in a conditional KL expansion. Furthermore, Approach 2 provides a good estimate for the number of terms of the truncated KL expansion of the conditional field of Approach 1. Finally, we demonstrate that active learning based on Method 2 is more efficient for uncertainty reduction in the SPDE’s states (i.e., it leads to a larger reduction of the variance) than active learning using Method 2.

Conditioned Karhunen-Lo` eve expanion, machine lea↗

DOE OSTI2016

SAMBA: Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos

A new arbitrary Polynomial Chaos (aPC) method is presented for moderately high-dimensional problems characterised by limited input data availability. The proposed methodology improves the algorithm of aPC and extends the method, that was previously only introduced as tensor product expansion, to moderately high-dimensional stochastic problems. The fundamental idea of aPC is to use the statistical moments of the input random variables to develop the polynomial chaos expansion. This approach provides the possibility to propagate continuous or discrete probability density functions and also histograms (data sets) as long as their moments exist, are finite and the determinant of the moment matrix is strictly positive. For cases with limited data availability, this approach avoids bias and fitting errors caused by wrong assumptions. In this work, an alternative way to calculate the aPC is suggested, which provides the optimal polynomials, Gaussian quadrature collocation points and weights from the moments using only a handful of matrix operations on the Hankel matrix of moments. It can therefore be implemented without requiring prior knowledge about statistical data analysis or a detailed understanding of the mathematics of polynomial chaos expansions. The extension to more input variables suggested in this work, is an anisotropic and adaptive version of Smolyak's algorithm that is solely based on the moments of the input probability distributions. It is referred to as SAMBA (PC), which is short for Sparse Approximation of Moment-Based Arbitrary Polynomial Chaos. It is illustrated that for moderately high-dimensional problems (up to 20 different input variables or histograms) SAMBA can significantly simplify the calculation of sparse Gaussian quadrature rules. SAMBA's efficiency for multivariate functions with regard to data availability is further demonstrated by analysing higher order convergence and accuracy for a set of nonlinear test functions with 2, 5 and 10 different input distributions or histograms.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2015

Constructing Surrogate Models of Complex Systems with Enhanced Sparsity: Quantifying the influence of conformational uncertainty in biomolecular solvation

Biomolecules exhibit conformational fluctuations near equilibrium states, inducing uncertainty in various biological properties in a dynamic way. We have developed a general method to quantify the uncertainty of target properties induced by conformational fluctuations. Using a generalized polynomial chaos (gPC) expansion, we construct a surrogate model of the target property with respect to varying conformational states. We also propose a method to increase the sparsity of the gPC expansion by defining a set of conformational “active space” random variables. With the increased sparsity, we employ the compressive sensing method to accurately construct the surrogate model. We demonstrate the performance of the surrogate model by evaluating fluctuation-induced uncertainty in solvent-accessible surface area for the bovine trypsin inhibitor protein system and show that the new approach offers more accurate statistical information than standard Monte Carlo approaches. Further more, the constructed surrogate model also enables us to directly evaluate the target property under various conformational states, yielding a more accurate response surface than standard sparse grid collocation methods. In particular, the new method provides higher accuracy in high-dimensional systems, such as biomolecules, where sparse grid performance is limited by the accuracy of the computed quantity of interest. Our new framework is generalizable and can be used to investigate the uncertainty of a wide variety of target properties in biomolecular systems.

coarse graining↗

DOE OSTI2018

A state estimation approach based on stochastic expansions

This paper presents a new approach for state estimation problems. It is based on the representation of random variables using stochastic functions. Its main idea is to expand the state variables in terms of the noise variables of the system, and then estimate the unnoisy value of the state variables by taking the mean value of the stochastic expansion. Moreover, it was shown that in some situations, the proposed approach may be adapted to the determination of the probability distribution of the state noise. For the determination of the coefficients of the expansions, we present three approaches: moment matching (MM), collocation (COL) and variational (VAR). In the numerical analysis section, three examples are analyzed including a discrete linear system, the Influenza in a boarding school and the state estimation problem in the Hodgkin–Huxley’s model. In all these examples, the proposed approach was able to estimate the values of the state variables with precision, i.e., with very low RMS values.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2012

Image Mapping and Visual Attention on the Sensory Ego-Sphere

The Sensory Ego-Sphere (SES) is a short-term memory for a robot in the form of an egocentric, tessellated, spherical, sensory-motor map of the robot s locale. Visual attention enables fast alignment of overlapping images without warping or position optimization, since an attentional point (AP) on the composite typically corresponds to one on each of the collocated regions in the images. Such alignment speeds analysis of the multiple images of the area. Compositing and attention were performed two ways and compared: (1) APs were computed directly on the composite and not on the full-resolution images until the time of retrieval; and (2) the attentional operator was applied to all incoming imagery. It was found that although the second method was slower, it produced consistent and, thereby, more useful APs. The SES is an integral part of a control system that will enable a robot to learn new behaviors based on its previous experiences, and that will enable it to recombine its known behaviors in such a way as to solve related, but novel, task problems with apparent creativity. The approach is to combine sensory-motor data association and dimensionality reduction to learn navigation and manipulation tasks as sequences of basic behaviors that can be implemented with a small set of closed-loop controllers. Over time, the aggregate of behaviors and their transition probabilities form a stochastic network. Then given a task, the robot finds a path in the network that leads from its current state to the goal. The SES provides a short-term memory for the cognitive functions of the robot, association of sensory and motor data via spatio-temporal coincidence, direction of the attention of the robot, navigation through spatial localization with respect to known or discovered landmarks, and structured data sharing between the robot and human team members, the individuals in multi-robot teams, or with a C3 center.

Fleming, Katherine Achim↗

DOE OSTI2010

Uncertainty Quantification via Random Domain Decomposition and Probabilistic Collocation on Sparse Grids

Due to lack of knowledge or insufficient data, many physical systems are subject to uncertainty. Such uncertainty occurs on a multiplicity of scales. In this study, we conduct the uncertainty analysis of diffusion in random composites with two dominant scales of uncertainty: Large-scale uncertainty in the spatial arrangement of materials and small-scale uncertainty in the parameters within each material. A general two-scale framework that combines random domain decomposition (RDD) and probabilistic collocation method (PCM) on sparse grids to quantify the large and small scales of uncertainty, respectively. Using sparse grid points instead of standard grids based on full tensor products for both the large and small scales of uncertainty can greatly reduce the overall computational cost, especially for random process with small correlation length (large number of random dimensions). For one-dimensional random contact point problem and random inclusion problem, analytical solution and Monte Carlo simulations have been conducted respectively to verify the accuracy of the combined RDD-PCM approach. Additionally, we employed our combined RDD-PCM approach to two- and three-dimensional examples to demonstrate that our combined RDD-PCM approach provides efficient, robust and nonintrusive approximations for the statistics of diffusion in random composites.

97 MATHEMATICS AND COMPUTING↗