Sparse Grid Quadrature Research Articles

An extended moment-based trajectory accuracy reliability analysis method of robot manipulators with random and interval uncertainties

This paper proposes a novel trajectory accuracy reliability analysis method with random and interval variables to evaluate the impact of mixed uncertainties on the motion performance of robot manipulators. To effectively and accurately solve the hybrid reliability model, the interval analysis is first conducted on the positional error model established by differential kinematics, and a maximum positional error searching algorithm is developed based on geometric transformation and cell enumeration. Then, the trajectory accuracy reliability model is reconstructed by the eigen-decomposition technique, which further incorporates the adaptive weight vector-based anisotropic sparse-grid quadrature approach to derive the statistical moments of maximum positional error. Afterward, by matching with the cumulants of the original reliability model, a novel approximation method is proposed based on the Weibull distribution and minimized matching error model to complete the trajectory accuracy reliability analysis. The practicality and advantages of the proposed method are demonstrated by two illustrative examples of 6-degrees-of-freedom robot manipulators. Comparative results affirm that the proposed method outperforms existing state-of-the-art algorithms in terms of accuracy and efficiency for trajectory accuracy reliability analysis encompassing random and interval uncertainties. Overall, the outcomes of this paper contribute significantly to the design and analysis of safety and reliability in moving machinery.

A DIMENSION-ADAPTIVE COMBINATION TECHNIQUE FOR UNCERTAINTY QUANTIFICATION

We present an adaptive algorithm for the computation of quantities of interest involving the solution of a stochastic elliptic partial differential equation, where the diffusion coefficient is parametrized by means of a Karhunen-Loève expansion. The approximation of the equivalent parametric problem requires a restriction of the countably infinite-dimensional parameter space to a finite-dimensional parameter set, a spatial discretization, and an approximation in the parametric variables. We consider a sparse grid approach between these approximation directions in order to reduce the computational effort and propose a dimension-adaptive combination technique. In addition, a sparse grid quadrature for the high-dimensional parametric approximation is employed and simultaneously balanced with the spatial and stochastic approximation. Our adaptive algorithm constructs a sparse grid approximation based on the benefit-cost ratio such that the regularity and thus the decay of the Karhunen-Loève coefficients is not required beforehand. The decay is detected and exploited as the algorithm adjusts to the anisotropy in the parametric variables. We include numerical examples for the Darcy problem with a lognormal permeability field, which illustrate a good performance of the algorithm. For sufficiently smooth random fields, we essentially recover the spatial order of convergence as asymptotic convergence rate with respect to the computational cost.

Double‐state chi‐square test based sparse grid quadrature filtering algorithm and its application in integrated navigation

AbstractAiming to reduce the impact that the measurement uncertainty of nonlinear system brings to filtering accuracy, a novel sparse grid quadrature filter (SGQF) algorithm based on the double‐state chi‐square test is proposed. And the strap‐down inertial navigation system (SINS)/global navigation satellite system (GNSS) integrated navigation system is used to verify the algorithm. A discrete‐time system model with uncertain measurement is constructed, then the system state through the SGQF algorithm is estimated. To determine the uncertainty of the measurement, the double‐state chi‐square test is introduced. By comparing the results of the state recursor, the chi‐square test value is obtained and it is used to update the measurement noise. Further, the system state and error covariance are re‐estimated. So far a robust sparse grid filter algorithm with statistical adjustment of the measurement noise is obtained. The proposed algorithm can overcome the influence of error statistical characteristic amplification on system filtering accuracy. Taking the SINS/GNSS integrated navigation system as the application object, off‐line simulation experiments and online flight tests verify the robustness of the proposed algorithm.

Multilevel domain uncertainty quantification in computational electromagnetics

We continue our study [R. Aylwin, C. Jerez-Hanckes, C. Schwab and J. Zech, Domain uncertainty quantification in computational electromagnetics, SIAM/ASA J. Uncertain. Quant. 8 (2020) 301–341] of the numerical approximation of time-harmonic electromagnetic fields for the Maxwell lossy cavity problem for uncertain geometries. We adopt the same affine-parametric shape parametrization framework, mapping the physical domains to a nominal polygonal domain with piecewise smooth maps. The regularity of the pullback solutions on the nominal domain is characterized in piecewise Sobolev spaces. We prove error convergence rates and optimize the algorithmic steering of parameters for edge-element discretizations in the nominal domain combined with: (a) multilevel Monte Carlo sampling, and (b) multilevel, sparse-grid quadrature for computing the expectation of the solutions with respect to uncertain domain ensembles. In addition, we analyze sparse-grid interpolation to compute surrogates of the domain-to-solution mappings. All calculations are performed on the polyhedral nominal domain, which enables the use of standard simplicial finite element meshes. We provide a rigorous fully discrete error analysis and show, in all cases, that dimension-independent algebraic convergence is achieved. For the multilevel sparse-grid quadrature methods, we prove higher order convergence rates free from the so-called curse of dimensionality. Numerical experiments confirm our theoretical results and verify the superiority of the sparse-grid methods.

Arbitrarily Sparse Polynomial Chaos Expansion for High-Dimensional Parametric Problems: Parametric and Probabilistic Power Flow as an Example

Parameters, no matter whether they are uncontrollable uncertainty factors or controllable variables, have a great impact on the states and performances of continuous engineering systems. Acquiring an explicit expression of the complicated implicit function between these parameters and system states, called parametric problem in this article, will facilitate immediate analysis of parameters’ impact on the system, optimal parameter design, and uncertainty quantification. Polynomial chaos expansion (PCE) is a globally optimal polynomial approximation method for parametric problems, but it suffers from the curse of dimensionality, i.e., the number of basis functions and consequent computational burden increases rapidly with the number of parameters (i.e., dimensions). This article proposes a PCE method characterized by the arbitrarily sparse basis and novel generalized Smolyak sparse grid quadrature and, hence, effectively relieves the curse of dimensionality for high-dimensional problems. This basis is not restricted by any existing fixed-form truncation and can merely incorporate a few important basis functions. The novel generalized sparse grid can remarkably reduce the number of required collocation points compared with the prevailing classic sparse grid while achieving high accuracy. The proposed method is universal for parametric problems in engineering systems and is exemplified by parametric and probabilistic power flow problems of electrical power systems. Its effectiveness is validated by computational results on the IEEE 30-bus, 118-bus, and 500-parameter 9241pegase systems.

Compressed Positive Quadrature Filter

A compressed positive quadrature filter (CPQF) is proposed in this technical note. Accuracy, efficiency, and numerical stability are the main concerns for quadrature-based nonlinear Gaussian filtering, but desirable performance in all these three aspects can hardly be simultaneously achieved in previously proposed nonlinear Gaussian filters, such as unscented Kalman filter, Gauss–Hermite quadrature filter (GHQF), and sparse grid quadrature filter. To improve the overall capacity of nonlinear Gaussian filtering techniques, the CPQF is proposed by numerically solving the constrained moment matching equations. By enforcing the positivity of the quadrature weights, the CPQF is numerically as stable as the GHQF. Through exploiting the underlying sparsity while solving the moment matching equations, the CPQF can also achieve the same level of accuracy as the GHQF using far fewer quadrature points. Simulations are conducted to evaluate the effectiveness of the CPQF, and the numerical results demonstrate its superior performance over conventional Gaussian filters.

Response analysis and optimization of the air spring with epistemic uncertainties

Abstract Traditional methods for the optimization design of the air spring are based on the deterministic assumption that the parameters are fixed. However, uncertainties widely exist during the manufacturing stage of the air spring. To model the uncertainties in air springs, evidence theory is introduced. For the response analysis of the air spring system under evidence theory, an evidence theory-based polynomial chaos method, called the sparse grid quadrature-based arbitrary orthogonal polynomial (SGQ-AOP) method, is proposed. In the SGQ-AOP method, the response of the air spring is approximated by AOP expansion, and the sparse grid quadrature is introduced to calculate the expansion coefficient. For optimization of the air spring, a reliability-based optimization model is established under evidence theory. To improve the efficiency of optimization, the SGQ-AOP method is used to establish the surrogate model for the response of the air spring. The proposed response analysis and the optimization method were employed to optimize an air spring with epistemic uncertainties, and its effectiveness has been demonstrated by comparing it with the traditional evidence theory-based AOP method.

Distributed State Fusion Using Sparse-Grid Quadrature Filter With Application to INS/CNS/GNSS Integration

This paper presents a new methodology of distributed state fusion by using the sparse-grid quadrature filter to deal with the fusion estimation problem for multi-sensor nonlinear systems. In this methodology, the sparse-grid quadrature filter is performed in a distributed manner to process the information at each local node; and subsequently, a distributed state fusion approach based on matrix weighting is established in the sense of mean square error, in which a flexible procedure is developed based on the sparse-grid quadrature rule to calculate the cross-covariances between any two local estimators for multi-sensor nonlinear systems. The proposed methodology of distributed fusion can obtain higher fusion estimation accuracy in a flexible way, leading to improved fusion performance for multi-sensor nonlinear systems. The simulations and experiments in INS/CNS/GNSS (inertial navigation system/celestial navigation system/global navigation satellite system) integration verifies the effectiveness of the proposed methodology.

Numerical smoothing with hierarchical adaptive sparse grids and quasi-Monte Carlo methods for efficient option pricing

When approximating the expectations of a functional of a solution to a stochastic differential equation, the numerical performance of deterministic quadrature methods, such as sparse grid quadrature and quasi-Monte Carlo (QMC) methods, may critically depend on the regularity of the integrand. To overcome this issue and improve the regularity structure of the problem, we consider cases in which analytic smoothing (bias-free mollification) cannot be performed and introduce a novel numerical smoothing approach by combining a root-finding method with a one-dimensional numerical integration with respect to a single well-chosen variable. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, potentially affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e. the Brownian bridge and Richardson extrapolation on the weak error). This approach facilitates the effective treatment of high dimensionality. Our study is motivated by option pricing problems, focusing on dynamics where the discretization of the asset price is necessary. Based on our analysis and numerical experiments, we demonstrate the advantages of combining numerical smoothing with the ASGQ and QMC methods over these methods without smoothing and the Monte Carlo approach. Finally, our approach is generic and can be applied to solve a broad class of problems, particularly approximating distribution functions, computing financial Greeks, and estimating risk quantities.

M-estimation based sparse grid quadrature filter and stochastic stability analysis

In this study, a novel M-estimation based sparse grid quadrature filter (MSGQF) is proposed to improve the robust performance of the nonlinear system. We present a systematic formulation of the sparse grid quadrature filter (SGQF), and extend it to the discrete-time nonlinear system with abnormal measurement values. The M-estimation method is introduced in the SGQF, which uses the Huber’s cost function to update the measurement covariance. Convergence on the modified robust SGQF is established and proved. The sufficient conditions are shown to ensure stochastic stability of the MSGQF. A target tracking problem has been conducted to demonstrate the accuracy of the MSGQF. When measurement abnormal values appear, it outperforms the unscented Kalman filter (UKF), the cubature Kalman filter (CKF) and the SGQF. Theoretical analysis and simulation results prove that the MSGQF provides significant performance improvement in the robustness of the nonlinear system.

Designed quadrature to approximate integrals in maximum simulated likelihood estimation

SummaryMaximum simulated likelihood estimation of mixed multinomial logit models requires evaluation of a multidimensional integral. Quasi-Monte Carlo (QMC) methods such as Halton sequences and modified Latin hypercube sampling are workhorse methods for integral approximation. Earlier studies explored the potential of sparse grid quadrature (SGQ), but SGQ suffers from negative weights. As an alternative to QMC and SGQ, we looked into the recently developed designed quadrature (DQ) method. DQ requires fewer nodes to get the same level of accuracy as QMC and SGQ, is as easy to implement, ensures positivity of weights, and can be created on any general polynomial space. We benchmarked DQ against QMC in a Monte Carlo and an empirical study. DQ outperformed QMC in all considered scenarios, is practice ready, and has potential to become the workhorse method for integral approximation.

High-Dimensional Uncertainty Quantification in a Hybrid Data + Model-Based Submodeling Method for Refined Response Estimation at Critical Locations

AbstractThis study presents frameworks to build nonintrusive polynomial chaos expansion (PCE) models with regression and Smolyak sparse-grid quadrature to perform high-dimensional uncertainty propa...

Robust topology optimization for structures under bounded random loads and material uncertainties

Design optimization considering uncertainties arising from different sources has been one of the hotspots in the area of structural optimization. This paper presents a robust topology optimization method for structures with bounded loads and spatially correlated material uncertainties. We first develop a new model for random structural loads with bounded nature, in which the uncertain load components are assumed to be uniformly distributed within a convex set represented by an ellipsoid model. This model is then combined with the random field discretized by means of the expansion optimized linear estimation to provide uncertainty modelling of structures exhibiting both load and material uncertainties. The robust topological design problem is formulated as a bi-criteria optimization problem for minimizing the mean value and standard deviation of the structural compliance under these uncertainties. The polynomial chaos expansion, in conjunction with the sparse grid quadrature rule, is employed for calculating the statistical moments of the structural responses. The adjoint sensitivity analysis scheme is derived, and a gradient based optimizer is employed for updating the design variables. Numerical examples show that the structural designs obtained by the proposed robust topology optimization method exhibit higher robustness against uncertainties as compared with their deterministic counterpart.

Nested Sparse Successive Galerkin Approximation for Nonlinear Optimal Control Problems

Solving the Hamilton-Jacobi-Bellman (HJB) equation for nonlinear optimal control problems usually suffers from the so-called curse of dimensionality. In this letter, a nested sparse successive Galerkin method is presented for HJB equations, and the computational cost only grows polynomially with the dimension. Based on successive approximation techniques, the nonlinear HJB partial differential equation (PDE) is transformed into a sequence of linear PDEs. Then the nested sparse grid methods are employed to solve the resultant linear PDEs. The designed method is sparse in two aspects. Firstly, the solution of the linear PDE is constructed based on sparse combinations of nested basis functions. Secondly, the multi-dimensional integrals in the Galerkin method are efficiently calculated using nested sparse grid quadrature rules. Once the successive approximation process is finished, the optimal controller can be analytically given based on the sparse basis functions and coefficients. Numerical results also demonstrate the accuracy and efficiency of the designed nested sparse successive Galerkin method.

MASGQF with application to SINS alignment

To effectively improve the computation efficiency in the strapdown inertial navigation system (SINS) alignment process, the authors propose a marginalised adaptive sparse-grid quadrature filter (MASGQF). The filtering method combines the virtues of marginalisation technique and accuracy level switching structure to decrease the computational burden and improve the filtering precision simultaneously. With the adaptation criterion and the predefined tolerance value, different integration rules corresponding to the different accuracy levels nested in the proposed filtering method can be adjusted autonomously. It is proved that the MASGQF is exponentially bounded by theoretical analysis. The performance comparison of different filtering methods is demonstrated by the numerical results of the simulation and offline experiment. On the basis of the data performance of the online test, the validity of the proposed MASGQF is verified in the SINS alignment application.

A unified scheme to solving arbitrary complex-valued ratio distribution with application to statistical inference for raw frequency response functions and transmissibility functions

Complex-valued ratio distributions arises in many real applications such as statistical inference for frequency response functions (FRFs) and transmissibility functions (TFs) in structural health monitoring. As a sequel to our previous study, a unified scheme to solving complex ratio random variables is proposed in this study for the case when it is highly non-trivial or impossible to discover a closed-form solution such as the complex-valued t ratio distribution. Based on the probability transformation principle in the complex-valued domain, a unified formula is derived by reducing the concerned problem into multi-dimensional integrals, which can be solved by advanced numerical techniques. A fast sparse-grid quadrature (SGQ) scheme by constructing multivariate quadrature formulas using the combinations of tensor products of suitable one-dimensional formulas is utilized to improve the computational efficiency by avoiding the problem of curse of integral dimensionality. The unified methodology enables the efficient calculation of the probability density function (PDF) of a ratio random variable with its denominator and nominator specified by arbitrary probability distributions including Gaussian or non-Gaussian ratio random variables, correlated or independent random variables, bounded or unbounded ratio random variables. The unified scheme is applied to uncertainty quantification for raw FRFs and TFs without any post-processing such as averaging, smoothing and windowing, and the efficiency of the proposed scheme is verified by using the vibration test field data from a simply supported beam and from the Alamosa Canyon Bridge.

The Identification of the Wind Parameters Based on the Interactive Multi-Models

The wind as a natural phenomenon would cause the derivation of the pesticide drops during the operation of agricultural unmanned aerial vehicles (UAV). In particular, the changeable wind makes it difficult for the precision agriculture. For accurate spraying of pesticide, it is necessary to estimate the real-time wind parameters to provide the correction reference for the UAV path. Most estimation algorithms are model based, and as such, serious errors can arise when the models fail to properly fit the physical wind motions. To address this problem, a robust estimation model is proposed in this paper. Considering the diversity of the wind, three elemental time-related Markov models with carefully designed parameter α are adopted in the interacting multiple model (IMM) algorithm, to accomplish the estimation of the wind parameters. Furthermore, the estimation accuracy is dependent as well on the filtering technique. In that regard, the sparse grid quadrature Kalman filter (SGQKF) is employed to comprise the computation load and high filtering accuracy. Finally, the proposed algorithm is ran using simulation tests which results demonstrate its effectiveness and superiority in tracking the wind change.

Multilevel approximation of parametric and stochastic PDES

We analyze the complexity of the sparse-grid interpolation and sparse-grid quadrature of countably-parametric functions which take values in separable Banach spaces with unconditional bases. Assuming a suitably quantified holomorphic dependence on the parameters, we establish dimension-independent convergence rate bounds for sparse-grid approximation schemes. Analogous results are shown in the case that the parametric families are obtained as approximate solutions of corresponding parametric-holomorphic, nonlinear operator equations as considered in [A. Cohen and A. Chkifa and Ch. Schwab: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs, J. Math. Pures Appl. 103 (2015) 400–428], for example by means of stable, finite-dimensional approximations. We discuss in detail nonlinear Petrov–Galerkin projections. Error and convergence rate bounds for constructive and explicit multilevel, sparse tensor approximation schemes combining sparse-grid interpolation in the parameter space and general, multilevel discretization schemes in the physical domain are proved. The present results unify and generalize earlier works in terms of the admissible multilevel approximations in the physical domain (comprising general stable Petrov–Galerkin and discrete Petrov–Galerkin schemes, collocation and stable domain approximations) and in terms of the admissible operator equations (comprising smooth, nonlinear locally well-posed operator equations). Additionally, a novel computational strategy to localize sequences of nested index sets for the anisotropic Smolyak interpolation in parameter space is developed which realizes best [Formula: see text]-term benchmark convergence rates. We also consider Smolyak-type quadratures in this general setting, for which we establish improved convergence rates based on cancellations in the integrands’ gpc expansions by symmetries of quadratures and the probability measure [J.Z̃ech and Ch.S̃chwab: Convergence rates of high dimensional Smolyak quadrature, Report 2017-27, SAM ETH Zürich (2017)]. Several examples illustrating the abstract theory include domain uncertainty quantification (UQ) for general, linear, second-order, elliptic advection–reaction–diffusion equations on polygonal domains, where optimal convergence rates of FEM are known to require local mesh refinement near corners. Further applications of the presently developed theory comprise evaluations of posterior expectations in Bayesian inverse problems.

Composite weighted average consensus filtering for space object tracking

In this paper, a composite weighted average consensus filtering (CWACF) algorithm is proposed for space object tracking by combining two distributed heterogeneous nonlinear filters. In light of the sensors' different sensing accuracy and computational capability, extended Kalman filter (EKF) and sparse-grid quadrature filter (SGQF) are compositely adopted on different sensors as local filters. Then, estimates from neighbours are fused based on the weighted average consensus framework to attain better estimation performance. Moreover, it is proved that the estimation error is exponentially bounded in mean square. The performances of the proposed algorithm, the distributed extended Kalman filtering (DEKF) and the distributed sparse-grid quadrature filter (DSGQF) are compared in a space object tracking problem.

The application of sparse grid quadrature in solving stochastic optimisation problems

Stochastic optimisation problems minimise expectations of random cost functions. Thus they require accurate quadrature methods in order to evaluate the objective. Promising methods based on sparse grids were shown to display high quadrature accuracy for smooth integrands. But they have negative quadrature weights which potentially destroy the convexity of the objective and thus may lead to totally wrong results. We prove here that, due to their high accuracy, sparse grids maintain the convexity of the objective for sufficiently fine grids. An application to optimal control demonstrates the superiority of sparse grids over Monte Carlo and product rule based approaches. References D. P. Bertsekas. Dynamic programming and optimal control. Vol. I . Athena Scientific, 2005. URL http://athenasc.com/dpbook.html . H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer. , 13:147–269, 2004. doi:10.1017/S0962492904000182 . M. Chen, S. Mehrotra, and D. Papp. Scenario generation for stochastic optimization problems via the sparse grid method. Comput. Optim. Appl. , 62(3):669–692, 2015. doi:10.1007/s10589-015-9751-7 . C. W. Clenshaw and A. R. Curtis. A method for numerical integration on an automatic computer. Numer. Math. , 2:197–205, 1960. doi:10.1007/BF01386223 . P. J. Davis and P. Rabinowitz. Methods of numerical integration . Computer Science and Applied Mathematics. Academic Press, 1984. doi:10.1016/C2013-0-10566-1 . T. Gerstner and M. Griebel. Numerical integration using sparse grids. Numer. Algorithms , 18:209–232, 1998. doi:10.1023/A:1019129717644 . M. Holtz. Sparse grid quadrature in high dimensions with applications in finance and insurance , volume 77 of Lecture Notes in Computational Science and Engineering . Springer-Verlag, 2011. doi:10.1007/978-3-642-16004-2 . B. Oksendal. Stochastic differential equations . Springer-Verlag, 1998. doi:10.1007/978-3-662-03620-4 . T. N. L. Patterson. The optimum addition of points to quadrature formulae. Math. Comp. , 22:847–856, 1968. doi:10.2307/2004583 . S. W. Wallace and W. T. Ziemba. Applications of stochastic programming . MOS-SIAM Series on Optimization. SIAM, 2005. doi:10.1137/1.9780898718799 .