gPC Approach Research Articles

Entropy-Conservative Discontinuous Galerkin Methods for the Shallow Water Equations with Uncertainty

In this paper, we develop an entropy-conservative discontinuous Galerkin (DG) method for the shallow water (SW) equation with random inputs. One of the most popular methods for uncertainty quantification is the generalized Polynomial Chaos (gPC) approach which we consider in the following manuscript. We apply the stochastic Galerkin (SG) method to the stochastic SW equations. Using the SG approach in the stochastic hyperbolic SW system yields a purely deterministic system that is not necessarily hyperbolic anymore. The lack of the hyperbolicity leads to ill-posedness and stability issues in numerical simulations. By transforming the system using Roe variables, the hyperbolicity can be ensured and an entropy-entropy flux pair is known from a recent investigation by Gerster and Herty (Commun. Comput. Phys. 27(3): 639–671, 2020). We use this pair and determine a corresponding entropy flux potential. Then, we construct entropy conservative numerical two-point fluxes for this augmented system. By applying these new numerical fluxes in a nodal DG spectral element method (DGSEM) with flux differencing ansatz, we obtain a provable entropy conservative (dissipative) scheme. In numerical experiments, we validate our theoretical findings.

Hemodynamics and stresses in numerical simulations of the thoracic aorta: Stochastic sensitivity analysis to inlet flow-rate waveform

Numerical simulations of the blood flow inside a patient-specific thoracic aorta in presence of an aneurysm are considered. We focus on the impact on the numerical predictions of the inlet flow-rate waveform. First, the results obtained by using an idealized and a MRI-measured flow-rate waveform are compared. The measured boundary condition produces significantly higher wall shear stresses than those obtained in the idealized case. Discrepancies are reduced but they are still present even if the idealized inlet waveform is rescaled in order to match the stroke volume.This motivates a systematic sensitivity analysis of numerical predictions to the shape of the inlet flow-rate waveform that is carried out in the second part of the paper. Two parameters are selected to describe the inlet waveform: the stroke volume and the period of the cardiac cycle. A stochastic approach based on the generalized Polynomial Chaos (gPC) approach, in which continuous response surfaces of the quantities of interest in the parameter space can be obtained from a limited number of simulations, is used. For both selected uncertain parameters, we use beta PDFs reproducing clinical data. The two selected input parameters appear to have a significant influence on wall shear stresses as well as on the velocity distribution in vessel regions characterized by large curvature. This confirms the need of using patient-specific inlet conditions to obtain reliable hemodynamic predictions.

A modified perturbation method for mathematical models with randomness: An analysis through the steady‐state solution to Burgers' partial differential equation

The variability of the data and the incomplete knowledge of the true physics require the incorporation of randomness into the formulation of mathematical models. In this setting, the deterministic numerical methods cannot capture the propagation of the uncertainty from the inputs to the model output. For some problems, such as the Burgers' equation (simplification to understand properties of the Navier–Stokes equations), a small variation in the parameters causes nonnegligible changes in the output. Thus, suitable techniques for uncertainty quantification must be used. The generalized polynomial chaos (gPC) method has been successfully applied to compute the location of the transition layer of the steady‐state solution, when a small uncertainty is incorporated into the boundary. On the contrary, the classical perturbation method does not give reliable results, due to the uncertainty magnitude of the output. We propose a modification of the perturbation method that converges and is comparable with the gPC approach in terms of efficiency and rate of convergence. The method is even applicable when the input random parameters are dependent random variables.

Influence of Head Tissue Conductivity Uncertainties on EEG Dipole Reconstruction.

Reliable EEG source analysis depends on sufficiently detailed and accurate head models. In this study, we investigate how uncertainties inherent to the experimentally determined conductivity values of the different conductive compartments influence the results of EEG source analysis. In a single source scenario, the superficial and focal somatosensory P20/N20 component, we analyze the influence of varying conductivities on dipole reconstructions using a generalized polynomial chaos (gPC) approach. We find that in particular the conductivity uncertainties for skin and skull have a significant influence on the EEG inverse solution, leading to variations in source localization by several centimeters. The conductivity uncertainties for gray and white matter were found to have little influence on the source localization, but a strong influence on the strength and orientation of the reconstructed source, respectively. As the CSF conductivity is most accurately determined of all conductivities in a realistic head model, CSF conductivity uncertainties had a negligible influence on the source reconstruction. This small uncertainty is a further benefit of distinguishing the CSF in realistic volume conductor models.

Uncertainty quantification for nonlinear difference equations with dependent random inputs via a stochastic Galerkin projection technique

Discrete stochastic systems model discrete response data on some phenomenon with inherent uncertainty. The main goal of uncertainty quantification is to derive the probabilistic features of the stochastic system. This paper deals with theoretical and computational aspects of uncertainty quantification for nonlinear difference equations with dependent random inputs. When the random inputs are independent random variables, a generalized Polynomial Chaos (gPC) approach has been usually used to computationally quantify the uncertainty of stochastic systems. In the gPC technique, the stochastic Galerkin projections are done onto linear spans of orthogonal polynomials from the Askey–Wiener scheme or from Gram–Schmidt orthonormalization procedures. In this regard, recent results have established the algebraic or exponential convergence of these Galerkin projections to the solution process. In this paper, as the random inputs of the difference equation may be dependent, we perform Galerkin projections directly onto linear spans of canonical polynomials. The main contribution of this paper is to study the spectral convergence of these Galerkin projections for the solution process of general random difference equations. Spectral convergence is important to derive the main statistics of the response process at a cheap computational expense. In this regard, the numerical experiments bring to light the theoretical discussion of the paper.

Application of multiplicative dimensional reduction method for uncertainty quantification and sensitivity analysis of MEMS electrostatic actuators

The effect of electrostatic actuation allows MEMS devices to have specific physical movements. They have several advantages including low power, low cost, and small size and are used widely in variable capacitors, micro-accelerometers, etc. In this study, we consider a MEMS actuator consisting of a moveable plate and a fixed plate in the presence of an applied electric field. The gap between the two plates can normally be changed by voltage control. It is known that as the gap reduces to two thirds of the original gap, the so-called “pull-in effect” tends to occur, causing the plates to collide (resulting in dielectric breakdown and actuator failure). It is therefore important to predict the onset of the “pull-in effect”. As it is practically impossible to obtain the model parameters precisely, this prediction should account for the presence of uncertainties. Sampling methods such as Monte Carlo and Quasi Monte Carlo are easy to use with the caveat of low accuracy and high computational cost. The other popular method is polynomial chaos. It has high accuracy and low computational cost under smoothness assumption for problems with small number of uncertain parameters. In this study, we consider a two-stage approach to quantify the parametric uncertainty of MEMS electrostatic actuators with a moderate number of causal stochastic factors. In the first stage, a multiplicative dimensional reduction method is used to approximate the variance-based global sensitivity measures in order to simplify the model for the uncertainty quantification stage. The second stage involves the use of the generalized polynomial chaos (gPC) approach to quantify uncertainty of the simplified model from the first stage.

Multidimensional quantification of uncertainty and application to a turbulent mixing model

SummaryThis paper concerns the implementation of the generalized polynomial chaos (gPC) approach for parametric studies, including the quantification of uncertainty (UQ), of turbulence modelling. The method is applied to Richtmyer‐Meshkov turbulent mixing. The K−L turbulence model has been chosen as a prototypical example, and parametric studies have been performed to examine the effects of closure coefficients and initial conditions on the flow results. It is shown that the proposed method can be used to obtain a relation between the uncertain inputs and the monitored flow quantities, thus efficiently performing parametric studies. It allows the simultaneous calibration and quantification of uncertainty in an efficient numerical framework.

Uncertainty quantification and global sensitivity analysis of complex chemical process using a generalized polynomial chaos approach

Uncertainties are ubiquitous and unavoidable in process design and modeling. Because they can significantly affect the safety, reliability and economic decisions, it is important to quantify these uncertainties and reflect their propagation effect to process design. This paper proposes the application of generalized polynomial chaos (gPC)-based approach for uncertainty quantification and sensitivity analysis of complex chemical processes. The gPC approach approximates the dependence of a process state or output on the process inputs and parameters through expansion on an orthogonal polynomial basis. All statistical information of the interested quantity (output) can be obtained from the surrogate gPC model. The proposed methodology was compared with the traditional Monte-Carlo and Quasi Monte-Carlo sampling-based approaches to illustrate its advantages in terms of the computational efficiency. The result showed that the gPC method reduces computational effort for uncertainty quantification of complex chemical processes with an acceptable accuracy. Furthermore, Sobol’s sensitivity indices to identify influential random inputs can be obtained directly from the surrogated gPC model, which in turn further reduces the required simulations remarkably. The framework developed in this study can be usefully applied to the robust design of complex processes under uncertainties.

Uncertainty Quantification in Control Problems for Flocking Models

The optimal control of flocking models with random inputs is investigated from a numerical point of view. The effect of uncertainty in the interaction parameters is studied for a Cucker-Smale type model using a generalized polynomial chaos (gPC) approach. Numerical evidence of threshold effects in the alignment dynamic due to the random parameters is given. The use of a selective model predictive control permits steering of the system towards the desired state even in unstable regimes.

Fuzzy Predictive Control for a Tubular Heat Exchanger System

The paper deals with the problem of controlling the outlet temperature of a tubular heat exchanger system by means of flow pressure. The usual industrial system behavior must precisely be modeled and appropriate control action needs to be obtained based on novel techniques. A new multiple models control strategy using the well-known linear generalized predictive control (LGPC) scheme has been proposed in this paper. Then the best model of the system is accurately identified by an intelligent decision mechanism (IDM) which is organized based on both new recursive weight generator and fuzzy adaptive Kalman filter approaches. Simulations are all done and the results are also compared with those obtained using a nonlinear GPC (NLGPC) approach that is realized based on the Wiener model of the system. The results can verify the validity of the proposed control scheme.

Multi-element stochastic spectral projection for high quantile estimation

We investigate quantile estimation by multi-element generalized Polynomial Chaos (gPC) metamodel where the exact numerical model is approximated by complementary metamodels in overlapping domains that mimic the model’s exact response. The gPC metamodel is constructed by the non-intrusive stochastic spectral projection approach and function evaluation on the gPC metamodel can be considered as essentially free. Thus, large number of Monte Carlo samples from the metamodel can be used to estimate α-quantile, for moderate values of α. As the gPC metamodel is an expansion about the means of the inputs, its accuracy may worsen away from these mean values where the extreme events may occur. By increasing the approximation accuracy of the metamodel, we may eventually improve accuracy of quantile estimation but it is very expensive. A multi-element approach is therefore proposed by combining a global metamodel in the standard normal space with supplementary local metamodels constructed in bounded domains about the design points corresponding to the extreme events. To improve the accuracy and to minimize the sampling cost, sparse-tensor and anisotropic-tensor quadratures are tested in addition to the full-tensor Gauss quadrature in the construction of local metamodels; different bounds of the gPC expansion are also examined. The global and local metamodels are combined in the multi-element gPC (MEgPC) approach and it is shown that MEgPC can be more accurate than Monte Carlo or importance sampling methods for high quantile estimations for input dimensions roughly below N=8, a limit that is very much case- and α-dependent.

SU-F-BRCD-08: Uncertainty Quantification by Generalized Polynomial Chaos for MR-Guided Laser Induced Thermal Therapy.

Magnetic resonance-guided laser induced thermal therapy (MRgLITT) is a minimally invasive thermal treatment for metastatic brain lesions, offering an alternative to conventional surgery. The purpose of this investigation is to incorporate uncertainty quantification (UQ) into the biothermal parameters used in the Pennes bioheat transfer equation (BHT), in order to account for imprecise values available in the literature. The BHT is a partial differential equation commonly used in thermal therapy models. MRgLITT was performed on an in vivo canine brain in a previous investigation. The canine MRgLITT was modeled using the BHT. The BHT has four parameters'" microperfusion, conductivity, optical absorption, and optical scattering'"which lack precise measurements in living brain and tumor. The uncertainties in the parameters were expressed as probability distribution functions derived from literature values. A univariate generalized polynomial chaos (gPC) expansion was applied to the stochastic BHT. The gPC approach to UQ provides a novel methodology to calculate spatio-temporal voxel-wise means and variances of the predicted temperature distributions. The performance of the gPC predictions were evaluated retrospectively by comparison with MR thermal imaging (MRTI) acquired during the MRgLITT procedure in the canine model. The comparison was evaluated with root mean square difference (RMSD), isotherm contours, spatial profiles, and z-tests. The peak RMSD was ∼1.5 standard deviations for microperfusion, conductivity, and optical absorption, while optical scattering was ∼2.2 standard deviations. Isotherm contours and spatial profiles of the simulation's predicted mean plus or minus two standard deviations demonstrate the MRTI temperature was enclosed by the model's isotherm confidence interval predictions. An a = 0.01 z-test demonstrates agreement. The application of gPC for UQ is a potentially powerful means for providing predictive simulations despite poorly known input parameters. gPC provides an output that represents the probable distribution of outcomes for MRgLITT.

State Uncertainty Propagation in the Presence of Parametric Uncertainty and Additive White Noise

Abstract The focus of this work is on the development of a framework permitting the unification of generalized polynomial chaos (gPC) with the linear moment propagation equations, to accurately characterize the state distribution for linear systems subject to initial condition uncertainty, Gaussian white noise excitation and parametric uncertainty which is not required to be Gaussian. For a fixed value of parameters, an ensemble of moment propagation equations characterize the distribution of the state vector resulting from Gaussian initial conditions and stochastic forcing, which is modeled as Gaussian white noise. These moment equations exploit the gPC approach to describe the propagation of a combination of uncertainties in model parameters, initial conditions and forcing terms. Sampling the uncertain parameters according to the gPC approach, and integrating via quadrature, the distribution for the state vector can be obtained. Similarly, for a fixed realization of the stochastic forcing process, the gPC approach provides an output distribution resulting from parametric uncertainty. This approach can be further combined with moment propagation equations to describe the propagation of the state distribution, which encapsulates uncertainties in model parameters, initial conditions and forcing terms. The proposed techniques are illustrated on two benchmark problems to demonstrate the techniques’ potential in characterizing the non-Gaussian distribution of the state vector.

Fuzzy predictive control based multiple models strategy for a tubular heat exchanger system

This work deals with the problem of controlling the outlet temperature of a tubular heat exchanger system by means of flow pressure. The usual industrial case is to try to control the outlet temperature by either the temperature or the flow of the fluid, which flows through the shell tube. But, in some situations, this is not possible, due to the fact that the whole of system coefficients variation cannot quite be covered by control action. In this case, the system behavior must precisely be modeled and appropriate control action needs to be obtained based on novel techniques. A new multiple models control strategy using the well-known linear generalized predictive control (LGPC) scheme has been proposed, in this paper. The main idea of the proposed control strategy is to represent the operating environments of the system, which have a wide range of variation with respect to time by multiple explicit linear models. In this strategy, the best model of the system is accurately identified, at each instant of time, by an intelligent decision mechanism (IDM), which is organized based on both new recursive weight generator and fuzzy adaptive Kalman filter approaches. After that, the adaptive algorithm is implemented on the chosen model. Finally, for having a good tracking performance, the generalized predictive control is instantly updated and its control action is also applied to the system. For demonstrating the effectiveness of the proposed approach, simulations are all done and the results are also compared with those obtained using a nonlinear GPC (NLGPC) approach that is realized based on the Wiener model of the system. The results can verify the validity of the proposed control scheme.

AN ADAPTIVE GPC APPROACH TO LOW-FLOW ANAESTHESIA

An adaptive method is developed to control the end-tidal isoflurane concentration using a low-flow method. The isoflurane concentration is calculated based on measurements of the end-tidal isoflurane concentration and mean arterial pressure (MAP) using a model-based multivariable GPC algorithm. The control design allows explicit handling of input and output constrains, accommodates model uncertainty and takes into account possible MAP disturbances caused by surgical stimulation and blood loss. The proposed approach was tested on a complex nonlinear compartmental model of the anaesthesia system. The simulations demonstrate that the proposed control approach provides fast response, accurate set-point tracking and has good disturbance rejection behaviour. The advantages and drawbacks of low-flow over high flow anaesthesia techniques were evaluated using control-related performance criteria.

A numerically robust state-space approach to stable-predictive control strategies

Recent work with predictive control strategies has shown that the stable GPC (SGPC) approach has significant computational and numerical advantages. However, SGPC is cast in the transfer function framework which limits its application. Here we develop a means of extending the results to state-space models and show that improved numerical conditioning can be obtained for many stable-predictive control strategies.

GPC Control of a Pulp Dryer

In this paper we present the different stages is the implementation of an adaptive predictive controller in a temperature control loop in a dryer of a beet sugar factory. Firstly, the process is described and a computer aided design, based on an identified model of the plant, of a PID is made. Because of the nature of the process this controller seems to be inadecuate, so that an explicit adaptive scheme based on the GPC approach is introduced. Results of its use in the plant and implementation details are given.

Generalized Predictive Control—Part II Extensions and interpretations

The original GMV self-tuner was later extended to provide a general framework which included feedforward compensation and user-chosen polynomials with detuned model-reference, optimal Smith predictor and load-disturbance tailoring objectives. This paper adds similar refinements to the GPC algorithm which are illustrated by a set of simulations. The relationship between GPC and LQ designs is investigated to show the computational advantage of the new approach. The roles of the output and control horizons are explored for processes with nonminimum-phase, unstable and variable dead-time models. The robustness of the GPC approach to model over- and under-parameterization and to fast sampling rates is demonstrated by further simulations. An appendix derives stability results showing that certain choices of control and output horizons in GPC lead to cheap LQ, “mean-level”, state-dead-beat and pole-placement controllers.