Polynomial Chaos Expansion Technique Research Articles

Efficient Verilog-A based time-domain variability analysis method for passive photonic integrated circuits

The performance of silicon photonic integrated circuits (PICs), especially wavelength filters, can be highly sensitive to variations in the fabrication process due to the large refractive index contrast of the silicon on insulator platform. This paper proposes an easy-to-implement and efficient time-domain variability analysis method for passive PICs. The method utilizes the polynomial chaos expansion technique to construct Verilog-A based models for estimating the statistical information of stochastic passive PICs. In comparison to existing methods, this approach is considerably easy to implement, efficient, and exhibits superior scalability, particularly as the numbers of ports and random parameters in the studied PICs increase. The technique is demonstrated via the time-domain variability analysis of a ring-resonator-based wavelength filter and a Mach-Zehnder interferometer-based demultiplexer filter.

Stochastic modeling of the elastic response of parts produced by material extrusion using ideal voids

The voids in 3D printed materials produced by the material extrusion technique are mainly located between the printed filaments. Their shape depends on material and printing parameters and can be both convex and concave. This shape influences the accuracy of the predictions of common homogenization schemes that consider spheroidal shapes. A novel approach is introduced in this study for converting real voids to ideal supercylindrical ones that approximate better the shape of voids and can be described using only one parameter. This fact leads to a fast implementation in stochastic modeling schemes where fewer samples are required. The novel approach is implemented using the polynomial chaos expansion technique to predict the elastic properties of acrylonitrile butadiene styrene. The results are compared with previously performed tensile tests and both simple and computationally intensive models to provide a holistic understanding of the advantages of the new approach.

Design computed torque control for Stewart platform with uncertainty to the rehabilitation of patients with leg disabilities

Physiotherapy is a treatment that may be required permanently by many patients. As a result, a robot that can execute physiotherapy exercises for the legs like a professional therapist with adequate performance and acceptable safety may be efficient and widely used. In this study, a robust control system for a Stewart platform with six degrees of freedom is provided. First, the Newton-Euler approach is used in conjunction with a methodology and some simplification tools to achieve explicit dynamics formulation for the Stewart platform. For the primary application of this research, which is to follow the specified trajectory of ankle rehabilitation, computed torque control law (CTCL) and polynomial chaos expansion (PCE) were used to examine and consider any uncertainty in geometric and physical parameters. In fact, this strategy integrated the uncertainties with CTCL using PCE. The suggested PCE-based CTCL eliminates the system’s nonlinearity by applying feedback linearization to evaluate generalized driving forces; hence, the nondeterministic multi-body system follows the desired direction. Uncertainties in the patient’s foot as well as the main diameter parameters of the moment of inertia of the upper platform of the Stewart robot with various uniform, beta, and normal distributions, have been analyzed. The PCE technique’s results were compared to the Monte Carlo method’s outcomes, and the strengths and weaknesses of each method were investigated. In brief, the PCE method operated far better than the Monte Carlo (MC) method in speed, accuracy, and numerical volume.

A novel sparse polynomial chaos expansion technique with high adaptiveness for surrogate modelling

In this paper, a novel adaptive modelling framework for sparse polynomial chaos expansion is proposed, which can automatically determine adequate truncation degree and training sample set simultaneously. Moreover, the curse of dimensionality issue in polynomial chaos expansion, which generally arises in dealing with high input dimension or large truncation degree, can be alleviated to a large extent. In this framework, a new basis selection strategy, which leverages basis expansion, pruning and refinement, is pursued to adaptively select the polynomial terms of proper degree during the modelling process. Besides, an outstanding sequential sampling strategy is adopted to collect samples of high quality and in relatively small quantity for training polynomial chaos expansion model, and a sparse representation method, Bayesian compressive sensing, is employed for regression calculation. To reconcile the sequential sampling and adaptive basis selection in a consistent framework, a stability evaluation process which works in parallel with the sequential sampling process is performed. The performance of the proposed adaptive modelling framework is evaluated on two benchmark functions and a physical model through comparison with an existing adaptive polynomial chaos expansion modelling technique, a sequential sampling-only approach, and several basis adaptivity strategies. Results demonstrate that the proposed method has high adaptiveness in building surrogate models for various problems. It outperforms the existing adaptive polynomial chaos expansion modelling technique in terms of modelling precision and convergence rate, and it has similar performance with the sequential sampling-only method while lessening the burden in regression calculation and enabling adaptive determination of the truncation degree. Limitations of the proposed method are also summarized after comparing it with various basis adaptivity strategies.

Multiobjective design optimization of a quadrupole resonator under uncertainties

For a precise determination of the rf properties of superconducting materials, a calorimetric measurement is carried out with the aid of a so-called quadrupole resonator (QPR). This procedure is affected by certain systematic measurement errors with various sources of uncertainties. In this paper, to reduce the impact of geometrical uncertainties on the measurement bias, the modified steepest descent method is used for the multiobjective shape optimization of a QPR in terms of an expectation measure. Thereby, variations of geometrical parameters are modeled by the polynomial chaos expansion technique. Then, the resulting Maxwell's eigenvalue problem with random input data is solved using the polynomial chaos-based stochastic collocation method. Furthermore, to assess the contribution of the particular geometrical parameters, the variance-based sensitivity analysis is proposed. This allows for modifying the steepest descent algorithm, which results in reducing the computational load needed to find optimal solutions. Finally, optimization results in the form of an efficient approximation of the Pareto front for a three-dimensional model of the QPR are shown.

Optimized sparse polynomial chaos expansion with entropy regularization

Sparse Polynomial Chaos Expansion (PCE) is widely used in various engineering fields to quantitatively analyse the influence of uncertainty, while alleviating the problem of dimensionality curse. However, current sparse PCE techniques focus on choosing features with the largest coefficients, which may ignore uncertainties propagated with high order features. Hence, this paper proposes the idea of selecting polynomial chaos basis based on information entropy, which aims to retain the advantages of existing sparse techniques while considering entropy change as output uncertainty. A novel entropy-based optimization method is proposed to update the state-of-the-art sparse PCE models. This work further develops an entropy-based synthetic sparse model, which has higher computational efficiency. Two benchmark functions and a computational fluid dynamics (CFD) experiment are used to compare the accuracy and efficiency between the proposed method and classical methods. The results show that entropy-based methods can better capture the features of uncertainty propagation, improving accuracy and reducing sparsity while avoiding over-fitting problems.

Uncertainty Analysis for Hydrological Models With Interdependent Parameters: An Improved Polynomial Chaos Expansion Approach

AbstractThe use of polynomial chaos expansion (PCE) has gained a lot of attention due to its ability to efficiently estimate the effects of parameter uncertainty on model outputs. The traditional PCE technique requires the studied parameters to be independent. In hydrological modeling, although model parameters are often assumed to be independent for simplicity of computation, such an assumption is not always valid. Neglecting parameter correlations could significantly affect the analysis of uncertainty, leading to distorted modeling results. In this study, an improved PCE approach is proposed to address this issue and support the uncertainty analysis for hydrological models with correlated parameters. The proposed approach is based on the integration of principle component analysis (PCA) and PCE, where PCA is used to transform correlated parameters into orthogonal independent components. To demonstrate the applicability of this approach, the Soil & Water Assessment Tool (SWAT) model is applied to the Guadalupe River Watershed in Texas, US, and the integrated PCA‐PCE framework is used to assess the propagation of uncertainty of SWAT's interdependent parameters. A traditional Monte‐Carlo (MC) simulation is also used to address the uncertainty in the developed SWAT model. The results show that PCA‐PCE could generate similar probabilistic flow results compared to MC while maintaining a very high computational efficiency. The coefficients of determination (R2) for the mean and variance are 0.998 and 0.973, respectively, and the computational requirement is reduced by 99% using the developed PCA‐PCE approach. It is shown that the PCA‐PCE approach is reliable and efficient in assessing uncertainties in hydrological models with interdependent parameters.

Characterizing the Influence of Multiple Uncertainties on Predictions of Contaminant Discharge in Groundwater Within a Lagrangian Stochastic Formulation

AbstractAssessing the risks associated with transport of contaminants in hydrogeological systems requires the characterization of multiple sources of uncertainty. This paper examines the impact of the uncertainty in the source zone mass release rate, aquifer recharge, and the spatial structure of the hydraulic conductivity on transport predictions. Through the use of the Lagrangian framework, we develop semianalytical solutions for the first two moments of the total solute discharge through a control plane while accounting for source zone release conditions and recharge. We employ global sensitivity analysis (GSA) to investigate how the predictive uncertainty of the mass discharge is affected by uncertainty in source zone mass release rate, recharge, and the variance of the log‐conductivity field. The semianalytical solutions are employed with the polynomial chaos expansion technique to perform a GSA. Our results reveal the relative influence of each source of uncertainty on the robustness of model predictions, which is critical for site managers to allocate resources and design mitigation strategies.

Uncertainty Quantification in Energy Management Procedures

Complex energy systems are made up of a number of components interacting together via different energy vectors. The assessment of their performance under dynamic working conditions, where user demand and energy prices vary over time, requires a simulation tool. Regardless of the accuracy of this procedure, the uncertainty in data, obtained both by measurements or by forecasting, is usually non-negligible and requires the study of the sensitivity of results versus input data. In this work, polynomial chaos expansion technique is used to evaluate the variation of cogeneration plant performance with respect to the uncertainty of energy prices and user requests. The procedure allows to obtain this information with a much lower computational cost than that of usual Monte-Carlo approaches. Furthermore, all the tools used in this paper, which were developed in Python, are published as free and open source software.

A surrogate-based optimization method for the issuance of passenger evacuation orders under ship fires

In order to reduce assembly time and improve the computational efficiency under ship fires, a surrogate-based optimization technique is proposed to estimate the optimal time of the issuance of evacuation orders, which determines the passenger response times. The Legendre polynomial chaos expansion technique is employed to construct the surrogate model of passenger assembly time with response time parameters, and validated with the original computer evacuation model. The genetic algorithm (GA) is subsequently applied to find the optimal response time combination. A 3-storey passenger ship is presented and four different fire scenarios that involve typical fire locations are considered to address the validity of this proposed method. The results suggest that this proposed method can be used to determine the optimal and the worst response time parameters with a significantly reduced number of evaluation samples. The relative error of assembly time from the surrogate model follows the normal distribution. For evacuation zone with the longest average travel distance, passengers should be informed of the immediate evacuation. For evacuation zones with short average travel distances, the optimal response time may be larger than the worst solution. This work provides a new method to make a reasonable evacuation planning in ship fires.

Efficient Variability Analysis of Photonic Circuits by Stochastic Parametric Building Blocks

This paper presents a method to build stochastic parametric building blocks to be used in photonic process design kits. The building blocks are based on parametric macro-models computed by means of the generalized Polynomial Chaos Expansion technique. These macro-models can be built upfront and stored in process design kits. Being parametric, they do not have to be recalculated if the value of their design or statistical parameters change. It is shown that a single deterministic simulation performed with a classical circuit simulator is sufficient to perform the statistical analysis of any arbitrary photonic circuit realized combining these building blocks with different parameters, without the need of time-consuming Monte Carlo approach. Relevant numerical examples are used to demonstrate that the proposed macro-models are truly parametric, inherently stochastic and have greater simulation efficiency compared to Monte Carlo.

Dimension-adaptive algorithm-based PCE for models with many model parameters

Purpose To present the models with many model parameters by polynomial chaos expansion (PCE), and improve the accuracy, this paper aims to present dimension-adaptive algorithm-based PCE technique and verify the feasibility of the proposed method through taking solid rocket motor ignition under low temperature as an example. Design/methodology/approach The main approaches of this work are as follows: presenting a two-step dimension-adaptive algorithm; through computing the PCE coefficients using dimension-adaptive algorithm, improving the accuracy of PCE surrogate model obtained; and applying the proposed method to uncertainty quantification (UQ) of solid rocket motor ignition under low temperature to verify the feasibility of the proposed method. Findings The result indicates that by means of comparing with some conventional non-invasive method, the proposed method is able to raise the computational accuracy significantly on condition of meeting the efficiency requirement. Originality/value This paper proposes an approach in which the optimal non-uniform grid that can avoid the issue of overfitting or underfitting is obtained.

Stochastic PEEC Method Based on Polynomial Chaos Expansion

The new stochastic partial element equivalent circuit (PEEC) method is proposed for uncertainty quantification in electromagnetic problems when material parameters are considered as random variables. The proposed formulation is derived using polynomial chaos expansion (PCE) and Galerkin projection. For the first time, the well-known advantages of PEEC are combined with those of PCE techniques, which can tackle also large variations in the random parameters. Volume conductive media are first considered in the formulation, which is then extended to dielectric, magnetic, and surface conductive media.

Nonlinear magnetoquasistatic interface problem in a permanent-magnet machine with stochastic partial differential equation constraints

ABSTRACTThis study discusses an application of the stochastic collocation method for the solution of a nonlinear magnetoquasistatic interface problem that is constrained by a partial differential equation with random input data. Special attention is given to finding the robust design of a rotor and a stator of an electrical machine for the reduction of electromagnetic losses and a cogging torque under material uncertainties. For this reason, variations of material properties are modelled using the polynomial chaos expansion technique. To develop the domain derivative of a robust cost functional, the velocity method is applied with an adjoint technique. The utility and efficiency of the proposed method has been successfully demonstrated by applying it to the design of a permanent-magnet synchronous machine used in electric vehicles.

Uncertainty analysis for seawater intrusion in fractured coastal aquifers: Effects of fracture location, aperture, density and hydrodynamic parameters

In this study we use polynomial chaos expansion (PCE) to perform uncertainty analysis for seawater intrusion (SWI) in fractured coastal aquifers (FCAs) which is simulated using the coupled discrete fracture network (DFN) and variable-density flow (VDF) models. The DFN-VDF model requires detailed discontinuous analysis of the fractures. In real field applications, these characteristics are usually uncertain which may have a major effect on the predictive capability of the model. Thus, we perform global sensitivity analysis (GSA) to provide a preliminary assessment on how these uncertainties can affect the model outputs. As our conceptual model, we consider fractured configurations of the Henry Problem which is widely used to understand SWI processes. A finite element DFN-VDF model is developed in the framework of COMSOL Multiphysics®. We examine the uncertainty of several SWI metrics and salinity distribution due to the incomplete knowledge of fracture characteristics. PCE is used as a surrogate model to reduce the computational burden. A new sparse PCE technique is used to allow for high polynomial orders at low computational cost. The Sobol’ indices (SIs) are used as sensitivity measures to identify the key variables driving the model outputs uncertainties. The proposed GSA methodology based on PCE and SIs is useful for identifying the source of uncertainties on the model outputs with an affordable computational cost and an acceptable accuracy. It shows that fracture hydraulic conductivity is the first source of uncertainty on the salinity distribution. The imperfect knowledge of fracture location and density affects mainly the toe position and the total flux of saltwater entering the aquifer. Marginal effects based on the PCE are used to understand the effects of fracture characteristics on SWI. The findings provide a technical support for monitoring, controlling and preventing SWI in FCAs.

Variance-Based Robust Optimization of a Permanent Magnet Synchronous Machine

This paper focuses on the application of the variance-based global sensitivity analysis for a topology derivative method in order to solve a stochastic nonlinear time-dependent magnetoquasistatic interface problem. To illustrate the approach a permanent magnet synchronous machine has been considered. Our key objective is to provide a robust design of the rotor poles and of the tooth base in a stator for the reduction of the torque ripple and electromagnetic losses, while taking material uncertainties into account. Input variations of material parameters are modeled using the polynomial chaos expansion technique, which is incorporated into the stochastic collocation method in order to provide a response surface model. Additionally, we can benefit from the variance-based sensitivity analysis. This allows us to reduce the dimensionality of the stochastic optimization problems, described by the random-dependent cost functional. Finally, to validate our approach, we provide the two-dimensional simulations and analysis, which confirm the usefulness of the proposed method and yield a novel topology of a permanent magnet synchronous machine.

Quantifying Roughness Coefficient Uncertainty in Urban Flooding Simulations through a Simplified Methodology

A methodology is presented which can be used in the evaluation of parametric uncertainty in urban flooding simulation. Due to the fact that such simulations are time consuming, the following methodology is proposed: (a) simplification of the description of the physical process; (b) derivation of a training data set; (c) development of a data-driven surrogate model; (d) use of a forward uncertainty propagation scheme. The simplification comprises the following steps: (a) unit hydrograph derivation using a 2D hydrodynamic model; (b) calculation of the losses in order to determine the effective rainfall depth; (c) flood event simulation using the principle of the proportionality and superposition. The above methodology was implemented in an urban catchment located in the city of Athens, Greece. The model used for the first step of the simplification was FLOW-R2D, whereas the well-known SWMM software (US Environmental Protection Agency, Washington, DC, USA) was used for the second step of the simplification. For the training data set derivation, an ensemble of 100 Unit Hydrographs was derived with the FLOW-R2D model. The parameters which were modified in order to produce this ensemble were the Manning coefficients in the two friction zones (residential and urban open space areas). The surrogate model used to replicate the unit hydrograph derivation, using the Manning coefficients as an input, was based on the Polynomial Chaos Expansion technique. It was found that, although the uncertainties in the derived results have to be taken into account, the proposed methodology can be a fast and efficient way to cope with dynamic flood simulation in an urban catchment.

Grain size distribution uncertainty quantification in volcanic ash dispersal and deposition from weak plumes

AbstractWe present the results of uncertainty quantification and sensitivity analysis applied to volcanic ash dispersal from weak plumes with focus on the uncertainties associated to the original grain size distribution of the mixture. The Lagrangian particle model Lagrangian Particles Advection Code is used to simulate the transport of inertial particles under the action of realistic atmospheric conditions. The particle motion equations are derived by expressing the particle acceleration as the sum of forces acting along its trajectory, with the drag force calculated as a function of particle diameter, density, shape, and Reynolds number. Simulations are representative of a weak plume event of Mount Etna (Italy) and aimed at quantifying the effect on the dispersal process of the uncertainty in the mean and standard deviation of a lognormal function describing the initial grain size distribution and in particle sphericity. In order to analyze the sensitivity of particle dispersal to these uncertain variables with a reasonable number of simulations, response surfaces in the parameter space are built by using the generalized polynomial chaos expansion technique. The mean diameter and standard deviation of particle size distribution, and their probability density functions, at various distances from the source, both airborne and on ground, are quantified. Results highlight that uncertainty ranges in these quantities are drastically reduced with distance from source, making them largely dependent just on the location. Moreover, at a given distance from source, the distribution is mostly controlled by particle sphericity, particularly on the ground, whereas in air also mean diameter and sorting play a main role.

A Computational Spectral Approach to Interest Rate Models

The Polynomial Chaos Expansion (PCE) technique recovers a finite second order random variable exploiting suitable linear combinations of orthogonal polynomials which are functions of a given stochastic quantity $\xi$, hence acting as a kind of random basis. The PCE methodology has been developed as a mathematically rigorous Uncertainty Quantification (UQ) method which aims at providing reliable numerical estimates for some uncertain physical quantities defining the dynamic of certain engineering models and their related simulations. In the present paper we exploit the PCE approach to analyze some equity and interest rate models considering, without loss of generality, the one dimensional case. In particular we will take into account those models which are based on the Geometric Brownian Motion (gBm), e.g. the Vasicek model, the CIR model, etc. We also provide several numerical applications and results which are discussed for a set of volatility values. The latter allows us to test the PCE technique on a quite large set of different scenarios, hence providing a rather complete and detailed investigation on PCE-approximation's features and properties, such as the convergence of statistics, distribution and quantiles. Moreover we give results concerning both an efficiency and an accuracy study of our approach by comparing our outputs with the ones obtained adopting the Monte Carlo approach in its standard form as well as in its enhanced version.

Evaluated Kinetics of the Reactions of H and CH3 with n-Alkanes: Experiments with n-Butane and a Combustion Model Reaction Network Analysis.

Presented is a combined experimental and modeling study of the kinetics of the reactions of H and CH3 with n-butane, a representative aliphatic fuel. Abstraction of H from n-alkane fuels creates alkyl radicals that rapidly decompose at high temperatures to alkenes and daughter radicals. In combustion and pyrolysis, the branching ratio for attack on primary and secondary hydrogens is a key determinant of the initial olefin and radical pool, and results propagate through the chemistry of ignition, combustion, and byproduct formation. Experiments to determine relative and absolute rate constants for attack of H and CH3 have been carried out in a shock tube between 859 and 1136 K for methyl radicals and 890 to 1146 K for H atoms. Pressures ranged from 140 to 410 kPa. Appropriate precursors are used to thermally generate H and CH3 in separate experiments under dilute and well-defined conditions. A mathematical design algorithm has been applied to select the optimum experimental conditions. In conjunction with postshock product analyses, a network analysis based on the detailed chemical kinetic combustion model JetSurf 2 has been applied. Polynomial chaos expansion techniques and Monte Carlo methods are used to analyze the data and assess uncertainties. The present results provide the first experimental measurements of the branching ratios for attack of H and CH3 on primary and secondary hydrogens at temperatures near 1000 K. Results from the literature are reviewed and combined with the present data to generate evaluated rate expressions for attack on n-butane covering 300 to 2000 K for H atoms and 400 to 2000 K for methyl radicals. Values for generic n-alkanes and related hydrocarbons are also recommended. The present experiments and network analysis further demonstrate that the C-H bond scission channels in butyl radicals are an order of magnitude less important than currently indicated by JetSurf 2. Updated rate expressions for butyl radical fragmentation reactions are provided.