Sparse Grid Approach to Orbit Uncertainty Propagation

This work proposes and analyzes an anisotropic sparse grid stochastic collocation
\nmethod for solving partial differential equations with random coefficients and forcing terms (input
\ndata of the model). The method consists of a Galerkin approximation in the space variables and a
\ncollocation, in probability space, on sparse tensor product grids utilizing either Clenshaw–Curtis or
\nGaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution
\nof uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori
\nand a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem.
\nThese procedures seem to be very effective for the problems under study. The proposed method
\ncombines the advantages of isotropic sparse collocation with those of anisotropic full tensor product
\ncollocation: the first approach is effective for problems depending on random variables which weigh
\napproximately equally in the solution, while the benefits of the latter approach become apparent when
\nsolving highly anisotropic problems depending on a relatively small number of random variables, as
\nin the case where input random variables are Karhunen–Lo`eve truncations of “smooth” random
\nfields. This work also provides a rigorous convergence analysis of the fully discrete problem and
\ndemonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in
\nthe preasymptotic regime, with respect to the total number of collocation points. It also shows
\nthat the anisotropic approximation breaks the curse of dimensionality for a wide set of problems.
\nNumerical examples illustrate the theoretical results and are used to compare this approach with
\nseveral others, including the standard Monte Carlo. In particular, for moderately large-dimensional
\nproblems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and
\nsuperior to all examined methods.

These days sparse grids are of increasing interest in numerical simulations. Based upon hierarchical tensor product bases, the sparse grid approach is a very efficient one improving the ratio of invested storage and computing time to the achieved accuracy for many problems in the area of numerical solution of differential equations, for instance in numerical fluid mechanics. The particle tracing algorithms that are available so far cannot cope with sparse grids. Now we present an approach that directly works on sparse grids. As a second aspect in this paper, we suggest to use sparse girds as a data compression method in order to visualize huge data sets even on small workstations. Because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workstations can continue to handle these data sets.

The sparse grid approach uses basis functions which are tensor products of univariate basis functions. This gives a connection between sparse grids and the tensor space setting. On the other hand, in the tensor calculus one is interested in suitable representations of tensors (cf. Hackbusch, Tensor spaces and numerical tensor calculus, Monograph in preparation). Then the storage size is in particular described by certain rank parameters. Limited rank requires an approximation of the tensors (functions). In particular, function approximations like polynomial interpolation may lead to such representations. Here, we exploit the sparse grid approximation. We show that the involved rank is not the sparse grid complexity \(O(N{\log }^{d-1}N),\) but \(O({N{}^{(d-1)/d}\log }^{d-3}N).\) KeywordsSparse GridTensor SpaceNumerical Tensor CalculusUnivariate Basis FunctionsTucker FormatThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Option pricing with a direct adaptive sparse grid approach

Comparative study of broadband damage localization methods applied to test data

This paper proposes a novel integer-mixed probabilistic optimal power flow (IM-POPF) strategy for frequency regulation in islanded microgrids under uncertain operating conditions. Existing load-shedding approaches face critical limitations: continuous frameworks fail to reflect the discrete nature of actual load disconnections, while deterministic models inadequately capture the stochastic behavior of renewable generation and load variations. The proposed approach formulates load shedding as an integer optimization problem where variables are categorized as integer (load disconnection decisions at specific nodes) and continuous (voltages, power generation, and steady-state frequency), better reflecting practical power system operations. The key innovation combines integer load-shedding optimization with efficient uncertainty propagation through Unscented Transformation, eliminating the computational burden of Monte Carlo simulations while maintaining accuracy. Load and renewable uncertainties are modeled as normally distributed variables, and probabilistic constraints ensure operational limits compliance with predefined confidence levels. The methodology integrates Differential Evolution metaheuristics with Unscented Transformation for uncertainty propagation, requiring only 137 deterministic evaluations compared to 5000 for Monte Carlo methods. Validation on an IEEE 33-bus radial distribution system configured as an islanded microgrid demonstrates significant advantages over conventional approaches. Results show 36.5-fold computational efficiency improvement while achieving 95.28% confidence level compliance for frequency limits, compared to only 50% for deterministic methods. The integer formulation requires minimal additional load shedding (21.265%) compared to continuous approaches (20.682%), while better aligning with the discrete nature of real-world operational decisions. The proposed IM-POPF framework successfully minimizes total load shedding while maintaining frequency stability under uncertain conditions, providing a computationally efficient solution for real-time microgrid operation.

With the recent growth in volume and complexity of available data has come a renewed interest in the problem of estimating multivariate probability density functions. However, traditional methods encounter the curse of dimensionality (complexity grows exponentially with dimension). Here we provide an outline of a convergence analysis of a sparse grid based probability density estimation, which supports the use of the method for moderately complex (up to 15 dimensions) data sets, as has already been demonstrated for sparse grid quadrature and interpolation. References Hans-Joachim Bungartz and Michael Griebel. A note on the complexity of solving {P}oisson's equation for spaces of bounded mixed derivatives. J. Complexity , 15(2):167--199, 1999. http://dx.doi.org/10.1006/jcom.1999.0499 Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica , 13:147--269, 2004. http://dx.doi.org/10.1017/S0962492904000182 J. Garcke. Sparse grid tutorial. http://www.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf [29 January, 2009], 2008. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing , 67:225--253, 2001. http://dx.doi.org/10.1007/s006070170007 D. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization . Wiley, New York, 1992. C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations: Proceedings of the Sixth GAMM-Seminar, Kiel, 1990 , volume 31 of Notes on Num. Fluid Mech. , pages 241--251. Vieweg, 1991.

With the recent growth in volume and complexity of available data has come a renewed interest in the problem of estimating multivariate probability density functions. However, traditional methods encounter the curse of dimensionality (complexity grows exponentially with dimension). Here we provide an outline of a convergence analysis of a sparse grid based probability density estimation, which supports the use of the method for moderately complex (up to 15 dimensions) data sets, as has already been demonstrated for sparse grid quadrature and interpolation. References Hans-Joachim Bungartz and Michael Griebel. A note on the complexity of solving {P}oisson's equation for spaces of bounded mixed derivatives. J. Complexity, 15(2):167--199, 1999. http://dx.doi.org/10.1006/jcom.1999.0499 Hans-Joachim Bungartz and Michael Griebel. Sparse grids. Acta Numerica, 13:147--269, 2004. http://dx.doi.org/10.1017/S0962492904000182 J. Garcke. Sparse grid tutorial. http://www.math.tu-berlin.de/ garcke/paper/sparseGridTutorial.pdf [29 January, 2009], 2008. J. Garcke, M. Griebel, and M. Thess. Data mining with sparse grids. Computing, 67:225--253, 2001. http://dx.doi.org/10.1007/s006070170007 D. Scott. Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, New York, 1992. C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations: Proceedings of the Sixth GAMM-Seminar, Kiel, 1990, volume 31 of Notes on Num. Fluid Mech., pages 241--251. Vieweg, 1991.

Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures. One challenge in solving this problem is enormous computation cost from the UP analysis. This paper aims to develop a novel computational effective method based on the combination of the adaptive sparse grid collocation (ASGC) method and the high dimensional model representation (HDMR) technique to evaluate the variability of the frequency response of viscoelastic damping structures. First, a well-validated layer-wise finite element method is employed to model the viscoelastic damping structures. The direct frequency response (DFR) method is utilized to calculate the response. Then, a modified adaptive strategy using expectation increments as the sampling indicator is proposed for cost-effective sparse grid construction. Lastly, high dimensional model representation (HDMR) technique is introduced to address the difficulty in moderate and high random dimensional situations. Two numerical examples are provided to assess the performances of the proposed method. Variations in constitutive parameters of viscoelastic material and thicknesses of the viscoelastic layer are considered. Numerical results show that, compared to the original high-dimensional ASGC method and Monte Carlo Simulation (MCS), the proposed method accurately predicts the variability of frequency responses and significantly improves the computational efficiency.

Simulation evaluation of emerging estimation techniques for multinomial probit models

Space activities are continuously at risk due to the ever-increasing amount of uncontrolled space objects, called space debris, which can collide with operational spacecraft. Therefore, space agencies have started space situational awareness (SSA) programs, which aim to provide timely and accurate data regarding the space environment and hazards to infrastructure in orbit and on the ground. An essential need for SSA is the orbit determination of as many space objects as possible, in order to predict and prevent collisions. In this work, techniques for orbit determination of objects observed on short arcs are developed. This scenario frequently occurs with space debris, which are characterised by long observational gaps due to observability constraints. The classical least squares method, used for orbit determination, is revisited by studying the effect of nonlinearities in the mapping between observations and state. Differential algebra techniques are exploited, which enable the computation of high-order Taylor expansions of the residuals with respect to the state. Approximations of differential correction methods can then be avoided and the confidence region of the solution accurately characterized with a nonlinear approach. This uncertainty region can be reduced when more observations of the same objects are acquired. Thus, a method that both recognizes associated observations and sequentially reduces the solution uncertainty when two or more sets of observations are associated is proposed. The six dimensional ($6D$) association problem is addressed as a series of $2D$ and $4D$ problems to alleviate the computational cost. In case of successful association, part of the initial uncertainty is pruned away. The least squares solution and its uncertainty region can finally be used to initialize a sequential filter. A particle filter is implemented using differential algebra, so that system and measurement equations are substituted by polynomial evaluations and the resulting computational cost of the algorithm is alleviated. A population-based stochastic optimizer, the multiobjective particle swarm optimizer, is exploited to generate particles that best represent the probability distribution over solution orbits.

Orbit uncertainty propagation plays an important role in the analysis of a space mission. The accuracy and computation expense are two critical essences of uncertainty propagation. Repeated evaluations of the objective model are required to improve the preciseness of prediction, especially for long-term propagation. To balance the computational complexity and accuracy, an adaptive Gaussian mixture model using virtual sample generation (AGMM-VSG) is proposed. First, an unscented transformation and Cubature rule (UT-CR) based splitting method is employed to adaptive update the weights of Gaussian components for nonlinear dynamics. The Gaussian mixture model (GMM) approximation is applied to better approximate the original probability density function. Second, instead of the pure expensive evaluations by conventional GMM methods, virtual samples are generated using a new active-sampling-based Kriging (AS-KRG) method to improve the propagation efficiency. Three cases of uncertain orbital dynamical systems are used to verify the accuracy and efficiency of the proposed manuscript. The likelihood agreement measure (LAM) criterion and the number of expense evaluations prove the performance.

Uncertainty propagation (UP) is crucial for assessing the impact of input uncertainty on structural responses, holding significant importance in engineering applications. However, achieving accurate and efficient UP remains challenging, especially for highly nonlinear structures. Bayesian neural networks (BNN) have gained attention for addressing UP issues, yet current BNN models only utilize input samples and corresponding structural responses for training. However, incorporating gradients of structural responses with respect to input samples provides valuable information. This study proposes a novel approach called gradient-enhanced Bayesian neural networks (GEBNN) to tackle UP tasks. In the GEBNN, a modified evidence lower bound (MELBO) loss is developed to consider both structural responses and gradient information simultaneously. This includes disparities between actual and predicted responses, as well as disparities between actual and predicted derivatives. Additionally, a gradient screening strategy based on the marginal probability density functions (PDFs) of input samples is established to identify significant derivative data for GEBNN training. Once the GEBNN is configured, it is utilized to replace the computationally intensive finite element model to efficiently provide UP results. Various applications, including nonlinear numerical examples, and mechanical, civil, and aeronautical structures, are presented to demonstrate the effectiveness of the GEBNN. The results show that the GEBNN significantly enhances the computational accuracy of the BNN in solving UP tasks.

Modeling of Microblogging Social Networks: Dynamical System vs. Random Dynamical System

Uncertainty, Sensitivity and Geostatistical Studies of Flow and Contaminant Transport in Heterogeneous Unsaturated Zone by Feng Pan Dr. Zhongbo Yu, Examination Committee Chair Associate Professor of Hydrology University of Nevada Las Vegas Dr. Jianting Zhu, Examination Committee Co-Chair Associate Professor of Hydrologic Science Desert Research Institute The objectives of this study are: (1) to develop a methodology of estimating probability density functions (PDFs) of unsaturated hydraulic parameters when field samples are sparse, (2) to evaluate the predictive uncertainties in flow and contaminant transport due to parameter uncertainties in the layerand local-scale heterogeneities of hydraulic parameters in unsaturated zone (UZ), (3) to investigate the contributions of the parameter uncertainties to the flow and transport uncertainties, and (4) to estimate the spatial correlation structures of hydraulic parameters by incorporating prior information and site measurements. At layer scale, the uncertainty assessment of flow and contaminant transport in UZ entails PDFs of the hydraulic parameters. A non-conventional maximum likelihood (ML) approach is used in this study to estimate the PDFs of water retention parameters (e.g., van Genuchten a and ri) for situations common in field scale applications where core samples are sparse and prior PDFs of the parameters are unknown. This study also investigates the effects of the uncertainties in the water retention parameters on the predictive uncertainties in flow and transport in UZ. By comparing the predictive uncertainties with and without incorporating the random water retention parameters, it is found that the random water retention parameters have limited effects on the mean predictions of the state variables including percolation flux, normalized cumulative mass arrival, and contaminant travel time. However, incorporating the uncertainties in the water retention parameters significantly increases the magnitude and spatial extent of predictive uncertainties of the state variables. The layer-scale uncertainty is specific to hydrogeologic layers, while the localscale heterogeneity refers to the spatial variation of hydraulic properties within a layer. The local-scale heterogeneity is important in predicting flow path, velocity, and travel time of contaminants, but it is often neglected in modeling practices. This study incorporates the local-scale heterogeneity and examines its relative effects to the layerscale uncertainty on flow and transport uncertainties in UZ. Results illustrate that localscale heterogeneity significantly increases predictive uncertainties in the percolation fluxes and contaminant plumes, whereas the mean predictions are only slightly affected by the local-scale heterogeneity. Layer-scale uncertainty is more important than localscale heterogeneity for simulating overall contaminant travel time, suggesting that it would be more cost-effective to reduce the layer-scale parameter uncertainty in order to reduce predictive uncertainty in contaminant transport.