Stochastic Collocation Method Research Articles

Stochastic-deterministic boundary element modelling of transcranial electric stimulation using a three layer head model

This paper deals with the boundary element (BE) approach to modelling of transcranial electric stimulation as an alternative to the widely used finite element method (FEM). The advantages of the BE approach are listed in the paper and demonstrated on a computational example. The formulation is based on the quasi-static approximation of currents and voltages induced in living tissues while the head is represented by a three layered model consisting of skin, skull and brain tissues. Another contribution is the fact that the uncertainty present in the tissue conductivity values is taken into account by modelling them as uniformly distributed random variables. The stochastic collocation method (SCM) is applied for propagation of the uncertainty to the output electric scalar potential. Accordingly, stochastic moments are computed and sensitivity analysis is carried out using the ANalysis Of VAriance approach (ANOVA). The results given in the paper show the efficiency of the BE-SCM combination. Inspecting the results obtained from the proposed BE-SCM approach it is clear that the confidence intervals are appreciably larger in the interior tissues. The impact of the skull's conductivity is shown to be negligible for most of the observation points while the skin and brain conductivities have a significant impact on the output value.

On the performance evaluation of stochastic finite elements in linear and nonlinear problems

Due to the inherent uncertainties in various systems, deterministic approaches may not be able to satisfactorily characterize their response. In such cases, stochastic approaches that can systematically consider uncertainties have to be employed. In the past, many stochastic finite element analysis (SFEA) methods have been developed for uncertainty quantification (UQ), among which the perturbation methods, intrusive and non-intrusive polynomial chaos expansion (IPCE/NIPCE) methods and stochastic collocation (SC) methods have received considerable attention. However, in mechanics, most of the applications of these methods are confined to relatively simple problems, and the applicability and performance of these methods to complex nonlinear mechanics problems are not clear. To this end, this study carried out an investigation on the performance of different SFEA methods in linear and nonlinear problems. Numerical studies show that the NIPCE and SC methods are superior in terms of accuracy among other methods in the linear elastic case. The NIPCE method is also used for UQ in stochastic models with plasticity and nonlocal elastoplastic damage. The results demonstrate that the stochastic averages can be significantly different from the deterministic results, which indicates the necessity of considering UQ for improving response predictions.

Sensitivity analysis of the direct time domain analytical solution for transient impedance of the horizontal grounding electrode using ANOVA approach

The paper deals with the sensitivity analysis of the transient impedance of the horizontal grounding electrode. Simple and efficient direct time domain analytical solution for the calculation of transient impedance derived from the Generalized Telegrapher's Equation for spatial distribution of the scattered voltage in the frequency domain is used as a deterministic model. Stochastic collocation method is subsequently applied by taking into account extensive list of various random parameters (electrode length and depth, soil resistivity and permittivity, lightning pulse parameters) having effect on the efficiency of the grounding electrode. Sensitivity analysis is performed to determine the influence of each variable as well as the combination of variables on the transient impedance.

Mechanical‐probabilistic formulation of the soil–structure interaction, accounting for the average shear wave velocity

SummaryThis work proposes a mechanical‐probabilistic formulation of the soil–structure interaction, taking into account the variability of the mean velocity of shear wave seismic load. This work aims at studying the sensitivity of the seismic response of a reinforced concrete frame structure by accounting for different seismic scenarios. For this purpose, the simplified model of a two‐story structure system has been developed. This model is based both on a macro‐element approach and on a stochastic collocation method. The sensitivity of the maximum displacement of the structure under different average shear wave velocities is studied. This aim in view, Gaussian or lognormal random variables are arbitrarily chosen to model this velocity. Analysis of the seismic vulnerability of the soil–structure system leads to cumulative distribution functions, which sometimes show significant differences for the same soil class in the sense of the European standard for the design of structures for earthquake resistance.

Efficient Quantification of Model Uncertainties When De-boarding a Train

It is difficult to provide live simulation systems for decision support. Time is limited and uncertainty quantification requires many simulation runs. We combine a surrogate model with the stochastic collocation method to overcome time and storage restrictions and show a proof of concept for a de-boarding scenario of a train.

Uncertainty Analysis for Return Trajectory of Vertical Takeoff and Vertical Landing Reusable Launch Vehicle

The uncertainties during the return trajectory of vertical takeoff and vertical landing reusable launch vehicle weaken the ability of precision landing and make the return process more challenging. This paper is devoted to quantifying the probability uncertainty of return trajectory with uncertain parameters. The uncertainty model of return multi-flight-phase under the uncertainties of initial flight path angle, axial aerodynamic coefficient, and atmospheric density is established using the generalized polynomial chaos expansion method. By parameterizing random uncertainties and introducing random parameters into the uncertainty model, the uncertainty analysis problem of return trajectory is transformed into stochastic trajectory approximation problem. The coefficients of the polynomial basis function are solved by the stochastic collocation method. Then state solutions, statistical properties, and global sensitivity with Sobol index are established based on coefficients. The simulation results show the efficiency and accuracy of this method compared with the Monte Carlo method, the evolution process of main output parameters under random parameters, and relative importance for random parameters. Through the uncertainty analysis of the return trajectory, the robustness of return trajectory can be quantified, which is contributed to improving the safety, reliability, and robustness of recovery and landing mission.

Intrusive acceleration strategies for uncertainty quantification for hyperbolic systems of conservation laws

Methods for quantifying the effects of uncertainties in hyperbolic problems can be divided into intrusive and non-intrusive techniques. Non-intrusive methods allow the usage of a given deterministic solver in a black-box manner, while being embarrassingly parallel. On the other hand, intrusive modifications allow for certain acceleration techniques. Moreover, intrusive methods are expected to reach a given accuracy with a smaller number of unknowns compared to non-intrusive techniques. This effect is amplified in settings with high dimensional uncertainty. A downside of intrusive methods is the need to guarantee hyperbolicity of the resulting moment system. In contrast to stochastic-Galerkin (SG), the Intrusive Polynomial Moment (IPM) method is able to maintain hyperbolicity at the cost of solving an optimization problem in every spatial cell and every time step.In this work, we propose several acceleration techniques for intrusive methods and study their advantages and shortcomings compared to the non-intrusive Stochastic Collocation method. When solving steady problems with IPM, the numerical costs arising from repeatedly solving the IPM optimization problem can be reduced by using concepts from PDE-constrained optimization. Integrating the iteration from the numerical treatment of the optimization problem into the moment update reduces numerical costs, while preserving local convergence. Additionally, we propose an adaptive implementation and efficient parallelization strategy of the IPM method. The effectiveness of the proposed adaptations is demonstrated for multi-dimensional uncertainties in fluid dynamics applications, resulting in the observation of requiring a smaller number of unknowns to achieve a given accuracy when using intrusive methods. Furthermore, using the proposed acceleration techniques, our implementation reaches a given accuracy faster than Stochastic Collocation.

Propagation of uncertainty in a rotating pipe mechanism to generate an impinging swirling jet flow for heat transfer from a flat plate

In Computational Fluid Dynamics (CFD) studies composed of the coupling of different simulations, the uncertainty in one stage may be propagated to the following stage and affect the accuracy of the prediction. In this paper, a framework for uncertainty quantification is applied to the two-step simulation of the mechanical design of a swirling jet flow generated by a rotating pipe (\textit{Simulation 1}) impinging on a flat plate to provide convective heat transfer (\textit{Simulation 2}). The first approach is the Stochastic Collocation Method (SCM) with Clenshaw-Curtis sparse grids. The conclusion drawn from the analysis is that the simulated system does not exhibit a significant sensitivity to stochastic variations of model input parameters, over the tested uncertainty ranges. Additionally, a set of non-linear regression models for the stochastic velocity and turbulent profiles for the pipe nozzle are created and tested, since impinging jets at Reynolds number of $Re=23000$ are very frequent in the literature, but stochastic inlet conditions have never been provided. Numerical results demonstrate a negligible difference in the predicted convective heat transfer with respect to the use of the profiles simulated via CFD. These suggested surrogate models can be directly embedded onto other CFD applications (e.g arrays of jets or jet flows impinging on plates with different shapes) in which a realistic swirling flow under uncertainty can be of interest.

The effects of clinically-derived parametric data uncertainty in patient-specific coronary simulations with deformable walls.

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.

Stochastic dosimetry of a three compartment head model

The deterministic three-compartment head model based on the hybrid finite element/boundary element method (FEM/BEM) formalism is coupled with the nonintrusive stochastic collocation method (SCM) in this paper. The SCM based on sparse grid (SG) interpolation is used to assess the stochastic moments of the electric field induced in the head exposed to high frequency (HF) plane wave. The conductivity and relative permittivity of the scalp, skull and brain tissue, respectively, are modelled as random variables with uniform distribution. The stochastic mean and variance of the internal field in three tissues are computed and the sensitivity analysis of the tissue parameters is carried out, as well. The convergence of the SCM method is shown to be satisfactory. Presented approach provides an insight into the behaviour of the model output with respect to input parameters variation. The analysis showed that the highest impact pertains to scalp permittivity, while skull conductivity impact can be considered rather negligible. The results obtained using the three-compartment head model confirm that both brain permittivity and conductivity are the parameters most significantly influencing the variance of the induced field inside the brain.

Uncertainty Quantification in Small-Timescale Model-Based Fatigue Crack Growth Analysis Using a Stochastic Collocation Method

Due to the uncertainties originating from the underlying physical model, material properties and the measurement data in fatigue crack growth (FCG) processing, the prediction of fatigue crack growth lifetime is still challenging. The objective of this paper was to investigate a methodology for uncertainty quantification in FCG analysis and probabilistic remaining useful life prediction. A small-timescale growth model for the fracture mechanics-based analysis and predicting crack-growth lifetime is studied. A stochastic collocation method is used to alleviate the computational difficulties in the uncertainty quantification in the small-timescale model-based FCG analysis, which is derived from tensor products based on the solution of deterministic FCG problems on sparse grids of collocation point sets in random space. The proposed method is applied to the prediction of fatigue crack growth lifetime of Al7075-T6 alloy plates and verified by fatigue crack-growth experiments. The results show that the proposed method has the advantage of computational efficiency in uncertainty quantification of remaining life prediction of FCG.

Multilevel double loop Monte Carlo and stochastic collocation methods with importance sampling for Bayesian optimal experimental design

SummaryAn optimal experimental set‐up maximizes the value of data for statistical inferences. The efficiency of strategies for finding optimal experimental set‐ups is particularly important for experiments that are time‐consuming or expensive to perform. In the situation when the experiments are modeled by partial differential equations (PDEs), multilevel methods have been proven to reduce the computational complexity of their single‐level counterparts when estimating expected values. For a setting where PDEs can model experiments, we propose two multilevel methods for estimating a popular criterion known as the expected information gain (EIG) in Bayesian optimal experimental design. We propose a multilevel double loop Monte Carlo, which is a multilevel strategy with double loop Monte Carlo, and a multilevel double loop stochastic collocation, which performs a high‐dimensional integration on sparse grids. For both methods, the Laplace approximation is used for importance sampling that significantly reduces the computational work of estimating inner expectations. The values of the method parameters are determined by minimizing the computational work, subject to satisfying the desired error tolerance. The efficiencies of the methods are demonstrated by estimating EIG for inference of the fiber orientation in composite laminate materials from an electrical impedance tomography experiment.

Nested sparse-grid Stochastic Collocation Method for uncertainty quantification of blade stagger angle

In the present paper, the Nested Sparse-grid Stochastic Collocation Method (NSSCM) is utilized to investigate the uncertain effects of stochastic blade stagger angles on the aerodynamic performance of the turbine blade. Two typical test functions are used to validate the calculation efficiency and precision of the NSSCM in Uncertainty Quantification (UQ). Based on the validation, non-deterministic CFD (Computational Fluid Dynamics) simulations are applied to estimate the impact on the aerodynamic performance of 2D (two-dimensional) turbine blades cases, and the blade stagger angles are considered as the stochastic variables in a Gaussian distribution. Four schemes of a single blade, two adjacent blades, two separated blades, and five blades with stochastic blade stagger angle errors are studied. The calculation results show that the aerodynamic performance was significantly influenced, and the stochastic fluctuations of total pressure loss caused by uncertain stagger angles can be up to 40%. The uncertain errors of the blade stagger angle have a significant effect on the suction side, and the wake region, and the 30%–50% axial region on the blade suction side is the sensitive region. The effects on the flow field propagate from the pressure side to the suction side.

A posteriori error analysis and adaptive non-intrusive numerical schemes for systems of random conservation laws

This article considers one-dimensional random systems of hyperbolic conservation laws. Existence and uniqueness of random entropy admissible solutions for initial value problems of conservation laws, which involve random initial data and random flux functions, are established. Based on these results an a posteriori error analysis for a numerical approximation of the random entropy solution is presented. For the stochastic discretization, a non-intrusive approach, namely the Stochastic Collocation method is used. The spatio-temporal discretization relies on the Runge–Kutta Discontinuous Galerkin method. The a posteriori estimator is derived using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework this yields computable error bounds for the entire space-stochastic discretization error. The estimator admits a splitting into a stochastic and a deterministic (space-time) part, allowing for a novel residual-based space-stochastic adaptive mesh refinement algorithm. The scaling properties of the residuals are investigated and the efficiency of the proposed adaptive algorithms is illustrated in various numerical examples.

Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs

Convergence Analysis on Stochastic Collocation Methods for the Linear Schrödinger Equation with Random Inputs

SIMPLEX STOCHASTIC COLLOCATION FOR PIECEWISE SMOOTH FUNCTIONS WITH KINKS

Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty quantification of quantities of interest for gas networks. This is due to the regulation of the gas flow, pressure, or temperature. But, one can exploit that for each sample in the parameter space it is known if a regulator was active or not, which can be obtained from the result of the corresponding numerical solution. This information can be exploited in a stochastic collocation method. We approximate the function separately on each smooth region by polynomial interpolation and obtain an approximation to the kink. Note that we do not need information about the exact location of kinks, but only an indicator assigning each sample point to its smooth region. We obtain a global order of convergence of $(p+1)/d$, where $p$ is the degree of the employed polynomials and $d$ the dimension of the parameter space.

Application and Comparison of Uncertainty Quantification Methods for Railway Vehicle Dynamics with Random Mechanical Parameters

This paper aims to investigate uncertainties in railway vehicle suspension components and the implement of uncertainty quantification methods in railway vehicle dynamics. The sampling-based method represented by Latin Hypercube Sampling (LHS) and generalized polynomial chaos approaches including the stochastic Galerkin and Collocation methods (SGM and SCM) are employed to analyze the propagation of uncertainties from the parameters input in a vehicle-track mathematical model to the results of running dynamics. In order to illustrate the performance qualities of SGM, SCM and LHS, a stochastic wheel model with uncertainties of the stiffness and damping is firstly formulated to study the vertical displacement of wheel. Numerical results show that SCM, which can be easily implemented by means of the existing deterministic model, has explicit advantages over SGM and LHS in terms of the efficiency and accuracy. Furthermore, a simplified stochastic bogie model with three random suspension parameters is also established by means of SCM and LHS to analyze the critical speed, which is affected obviously by the parametric uncertainties. Finally, a stochastic vertical vehicle-track coupled model with parametric uncertainties is built comprehensively on the basis of SCM, by which the impact behavior of wheel-rail interaction under a rail defect is investigated and the dynamic response of vehicles under the track irregularity is explored in terms of the Sperling index. It concludes that the uncertainties of parameters have a significant influence on P2 force and Sperling index from the view of the running quality.

A hyperbolicity-preserving discontinuous stochastic Galerkin scheme for uncertain hyperbolic systems of equations

Intrusive Uncertainty Quantification methods such as stochastic Galerkin are gaining popularity, whereas the classical stochastic Galerkin approach is not ensured to preserve hyperbolicity of the underlying hyperbolic system. We apply a modification of this method that uses a slope limiter to retain admissible solutions of the system, while providing high-order approximations in the physical and stochastic space. This is done using a spatial discontinuous Galerkin scheme and a Multi-Element stochastic Galerkin ansatz in the random space. We analyze the convergence of the resulting scheme and apply it to the compressible Euler equations with various uncertain initial states in one and two spatial domains with up to three uncertainties. The performance in multiple stochastic dimensions is compared to the non-intrusive Stochastic Collocation method. The numerical results underline the strength of our method, especially if discontinuities are present in the uncertainty of the solution.

A bi-fidelity method for the multiscale Boltzmann equation with random parameters

In this paper, we study the multiscale Boltzmann equation with multi-dimensional random parameters by a bi-fidelity stochastic collocation (SC) method developed in [52,70,71]. By choosing the compressible Euler system as the low-fidelity model, we adapt the bi-fidelity SC method to combine computational efficiency of the low-fidelity model with high accuracy of the high-fidelity (Boltzmann) model. With only a small number of high-fidelity asymptotic-preserving solver runs for the Boltzmann equation, the bi-fidelity approximation can capture well the macroscopic quantities of the solution to the Boltzmann equation in the random space. A priori estimate on the accuracy between the high- and bi-fidelity solutions together with a convergence analysis is established. Finally, we present extensive numerical experiments to verify the efficiency and accuracy of our proposed method.

Stochastic Delay Characterization for Multicoupled RLC Interconnects Under Process Variations

The characterization of interconnect delay metrics in terms of process variations is an important but complicated task for statistical timing analysis in today’s integrated circuits (ICs). This paper presents a stochastic delay characterization framework for multicoupled interconnects in the presence of process variation. The proposed method starts with deriving the stochastic nodal equations for the RLC network that models multicoupled interconnects. By employing polynomial chaos (PC) expansion, a nonsampling-based stochastic prediction method, we further represent the voltage responses of network nodes as a series of orthogonal polynomials of random variables. During the expansion procedure, we use an adaptive approximation algorithm to reduce the number of required sampling points. We then use a stochastic collocation method to estimate the coefficients in the PC expansion model. With the voltage response determined as an expression of a multi-dimensional polynomial of random variables, the stochastic properties of the delay of multicoupled interconnects can be predicted. The proposed method not only takes into account the strong correlations among process variations, but also extracts an explicit delay representation for multicoupled interconnects in terms of process variations. Experimental results demonstrate that the delay characteristics predicted by the proposed method match well with the results by the brute-force Monte-Carlo method. Moreover, a significant speedup over the Monte-Carlo method has been achieved by the proposed delay characterization framework.