Uncertainty propagation of frequency response of viscoelastic damping structures using a modified high-dimensional adaptive sparse grid collocation method

A comparative assessment of efficient uncertainty analysis techniques for environmental fate and transport models: application to the FACT model

We present a sensitivity analysis of the optimization of the probe placement in radiofrequency (RF) ablation which takes the uncertainty associated with biophysical tissue properties (electrical and thermal conductivity) into account. Our forward simulation of RF ablation is based upon a system of partial differential equations (PDEs) that describe the electric potential of the probe and the steady state of the induced heat. The probe placement is optimized by minimizing a temperature-based objective function such that the volume of destroyed tumor tissue is maximized. The resulting optimality system is solved with a multilevel gradient descent approach. By evaluating the corresponding optimality system for certain realizations of tissue parameters (i.e., at certain, well-chosen points in the stochastic space) the sensitivity of the system can be analyzed with respect to variations in the tissue parameters. For the interpolation in the stochastic space we use an adaptive sparse grid collocation (ASGC) approach presented by Ma and Zabaras. We underscore the significance of the approach by applying the optimization to CT data obtained from a real RF ablation case.

Summary A sparse grid approach to orbit uncertainty propagation is presented. Efficient and accurate uncertainty propagation methods for nonlinear dynamic systems have been of enormous interest to space object tracking. Recent methods include those based on the time evolution of the probability density function, the statistical moments, the random samples, or a sum of Gaussian components. The idea of the sparse grid method for orbit uncertainty propagation is to represent the initial uncertainty by a sparse grid, propagate the sparse grid points individually through the nonlinear orbit dynamics, and compute the statistical moments from the propagated sparse grid points. The Smolyak rule is used to generate the sparse grid through linear combinations of low-level tensor products; the number of the resultant sparse grid points is polynomial in the dimension of the state. The sparse grid based uncertainty propagation may be considered a generalization of the Unscented Transformation (UT) of the well-known Unscented Kalman Filter but can achieve much higher accuracy levels than UT. Two examples of orbit propagation in different dynamic environments are used to show the efficacy of the sparse grid approach.

Spatial variability of Darcy velocity is presented due to the heterogeneity of aquifer parameters. The uncertainty qualification of velocity suffers great challenge in the complex porous media. This work focuses on the use of sparse grid collocation method in velocity simulation. Since the sparse grid collocation method provides a non-intrusive way to incorporate any existing deterministic solver, the mixed finite element method is combined as the deterministic solver to retain the local continuity of Darcy velocity. We decompose the error of the velocity into three components, and illustrate that the Karhunen-Loeve truncation brings more error into velocity approximation than into head. The convergence properties of velocity moments restrict the application of sparse grid collocation method in the problems with small correlation lengths. This work provides insights towards the application of sparse grid collocation method to velocity modeling. It is demonstrated that for which problems the sparse grid collocation method is expected to be competitive with the Monte Carlo simulation. Further work about the anisotropic sparse grid collocation method should be extended to circumvent the obstacle of dimensionality.

This paper presents an uncertainty propagation analysis of viscoelastic sandwich laminated structures using a stochastic collocation method based on Smolyak’s sparse grids approach. Shear deformation in the viscoelastic layers is accurately captured by means of layer-wise plate finite elements. Modal properties are determined by solving nonlinear eigenvalue problems due to the frequency-dependent nature of the considered viscoelastic materials. Variations in material properties and layer thicknesses are both treated in a probabilistic way. The variabilities of natural frequencies and modal loss factors of two different viscoelastic laminated plate structures (plane and cylindrical) are studied. Results show that the present method converges quickly with small sample sizes when compared to the classical Monte-Carlo method. The influence of the variability of several parameters on viscoelastic structure modal properties is commented.

Option pricing with a direct adaptive sparse grid approach

Sparse grids provide a very efficient method for the multilinear approximation of functions, especially in higher-dimensional spaces. In the d-dimensional space, the nodal multilinear basis on a grid with mesh size $h = 2^{ - n} $ consists of $O(2^{nd} )$ basis functions and leads to an $L\_2$-error of order $O(4^{ - n} )$ and an $H\_1$-error of order $O(2^{ - n} )$. -With sparse grids we get an $L\_2$-error of order $O(4^{ - n} n^{d - 1} )$ and an $H\_1$-error of order $O(2^{ - n} )$ with only $O(2^n n^{d - 1} )$ basis functions, if the function u fulfills the condition $\frac{{\partial ^{2d} }} {{\partial x_1^2 \partial x_2^2 \cdots \partial x_d^2 }}u < \infty $. Therefore, we can achieve much more accurate approximations with the same amount of storage. A data structure for the sparse grid representation of functions defined on cubes of arbitrary dimension and a finite element approach for the Helmholtz equation with sparse grid functions are introduced. Special emphasis is taken in the development of an efficient algorithm for the multiplication with the stiffness matrix. With an appropriate preconditioned conjugate gradient method (cg-method), the linear systems can be solved efficiently. Numerical experiments are presented for Helmholtz equations and eigenvalue problems for the Laplacian in two and three dimensions, and for a six-dimensional Poisson problem. The results support the assertion that the $L\_2$-error bounds for the sparse-grid approximation are also valid for sparse grid finite element solutions of elliptic differential equations. Problems with nonsmooth solutions are treated with adaptive sparse grids.

Initial orbit determination is highly enabling for operational awareness and safety in space. Reliable orbit determination in non-Keplerian systems, such as multi-body orbital mechanics, is challenging due to chaotic and unpredictable motion of these systems over long time scales. Typical approaches to non-Keplerian orbit determination use explicit integration with numerical shooting methods to propagate candidate trajectories. These methods often break down for measurement gaps on the order of days or even hours. This work presents a novel algorithm for non-Keplerian orbit determination using sparse grid collocation methods. The approach provides many benefits, including a large region of convergence without a close initial guess, as well as increased stability and performance over long observing time scales. A further advancement is made for estimating periodic orbits, in which a repeating orbit and its natural period are determined simultaneously from angles-only measurements. Orbit determination using collocation may significantly reduce sensor tasking requirements and enhance capabilities in complex orbital environments such as cislunar space.

This paper presents non-intrusive, efficient stochastic approaches for predicting uncertainties associated with petroleum reservoir simulations. The Monte Carlo simulation method, which is the most common and straightforward approach for uncertainty quantification in the industry, requires to perform a large number of reservoir simulations and is thus computationally expensive especially for large-scale problems. We propose an efficient and accurate alternative through the collocation-based stochastic approaches. The reservoirs are considered to exhibit randomly heterogeneous flow properties. The underlying random permeability field can be represented by the Karhunen-Loeve expansion (or principal component analysis), which reduces the dimensionality of random space. Two different collocation-based methods are introduced to propagate uncertainty of the reservoir response. The first one is the probabilistic collocation method that deals with the random reservoir responses by employing the orthogonal polynomial functions as the bases of the random space and utilizing the collocation technique in the random space. The second one is the sparse grid collocation method that is based on the multi-dimensional interpolation and high-dimensional quadrature techniques. They are non-intrusive in that the resulting equations have the exactly the same form as the original equations and can thus be solved with existing reservoir simulators. These methods are efficient since only a small number of simulations are required and the statistical moments and probability density functions of the quantities of interest in the oil reservoirs can be accurately estimated. The proposed approaches are demonstrated with a 3D reservoir model originating from the 9th SPE comparative project. The accuracy, efficiency, and compatibility are compared against Monte Carlo simulations. This study reveals that, compared to traditional Monte Carlo simulations, the collocation-based stochastic approaches can accurately quantify uncertainty in petroleum reservoirs and greatly reduce the computational cost.

In this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

In this paper we use a splitting technique to develop new multiscale basis functions for the multiscale finite element method (MsFEM). The multiscale basis functions are iteratively generated using a Green's kernel. The Green's kernel is based on the first differential operator of the splitting. The proposed MsFEM is applied to deterministic elliptic equations and stochastic elliptic equations, and we show that the proposed MsFEM can considerably reduce the dimension of the random parameter space for stochastic problems. By combining the method with sparse grid collocation methods, the need for a prohibitive number of deterministic solves is alleviated. We rigorously analyze the convergence of the proposed method for both the deterministic and stochastic elliptic equations. Computational complexity discussions are also offered to supplement the convergence analysis. A number of numerical results are presented to confirm the theoretical findings.

In this work, we investigate a novel two-level discretization method for the elliptic equations with random input data. Motivated by the two-grid method for deterministic nonlinear partial differential equations introduced by Xu [36], our two-level discretization method uses a two-grid finite element method in the physical space and a two-scale stochastic collocation method with sparse grid in the random domain. Specifically, we solve a semilinear equations on a coarse mesh 𝒯 H ⁢ ( D ) {\mathcal{T}_{H}(D)} with small scale of sparse collocation points η ⁢ ( L , N ) {\eta(L,N)} and solve a linearized equations on a fine mesh 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} using large scale of sparse collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} (where η ⁢ ( L , N ) , η ⁢ ( ℓ , N ) {\eta(L,N),\eta(\ell,N)} are the numbers of sparse grid with respect to different levels L , ℓ {L,\ell} in N dimensions). Moreover, an error correction on the coarse mesh with large scale of collocation points is used in the method. Theoretical results show that when h ≈ H 3 , η ⁢ ( ℓ , N ) ≈ ( η ⁢ ( L , N ) ) 3 {{h\approx H^{3},\eta(\ell,N)\approx(\eta(L,N))^{3}}} , the novel two-level discretization method achieves the same convergence accuracy in norm ∥ ⋅ ∥ ℒ ρ 2 ⁢ ( Γ ) ⊗ ℒ 2 ⁢ ( D ) {\|\cdot\|_{\mathcal{L}_{\rho}^{2}(\Gamma)\otimes\mathcal{L}^{2}(D)}} ( ℒ ρ 2 ⁢ ( Γ ) {\mathcal{L}_{\rho}^{2}(\Gamma)} is the weighted ℒ 2 {\mathcal{L}^{2}} space with ρ a probability density function) as that for the original semilinear problem directly by sparse grid stochastic collocation method with 𝒯 h ⁢ ( D ) {\mathcal{T}_{h}(D)} and large scale collocation points η ⁢ ( ℓ , N ) {\eta(\ell,N)} in random spaces.

Efficient robust topology optimization of piezoelectric energy harvesters with smooth boundaries and reduced computational cost

Nonlinear filtering algorithm is the key technology for dealing with complex systems in sensor data processing. To improve the filtering accuracy of the nonlinear filtering algorithm in the non‐Gaussian case, an improved version of the Gaussian sum algorithm, the Gaussian sum adaptive sparse grid quadrature filter (GSASQF), is proposed. The proposed algorithm overcomes the challenges by introducing the Gaussian sum principle, which converts the non‐Gaussian state and noise in the system into the form of weighted sum of Gaussian components. Based on the Bayesian filtering framework, a three‐level sparse grid sampling rule is introduced, with the sparse grid orthogonal filtering algorithm serving as the sub‐filter. By determining the sampling point parameters, the filtering process for each combination of Gaussian components is implemented, thereby ensuring the filtering accuracy of each group. In addition, in combination with the ideal of data‐driven, the weight of each Gaussian component combination is adaptively updated inversely by the values of the sensor measurement, which improves the global filtering accuracy of nonlinear system under non‐Gaussian noise. The combination of these three improvements enables high‐precision filtering of non‐Gaussian non‐linear systems. Theoretical analysis and simulation confirm that the proposed GSASQF algorithm provides advantages in filtering accuracy for nonlinear non‐Gaussian filtering problems.

Investigating the interplay between muscular activity and motion is the basis to improve our understanding of healthy or diseased musculoskeletal systems. To be able to analyze the musculoskeletal systems, computational models are used. Albeit some severe modeling assumptions, almost all existing musculoskeletal system simulations appeal to multibody simulation frameworks. Although continuum-mechanical musculoskeletal system models can compensate for some of these limitations, they are essentially not considered because of their computational complexity and cost. The proposed framework is the first activation-driven musculoskeletal system model, in which the exerted skeletal muscle forces are computed using 3-dimensional, continuum-mechanical skeletal muscle models and in which muscle activations are determined based on a constraint optimization problem. Numerical feasibility is achieved by computing sparse grid surrogates with hierarchical B-splines, and adaptive sparse grid refinement further reduces the computational effort. The choice of B-splines allows the use of all existing gradient-based optimization techniques without further numerical approximation. This paper demonstrates that the resulting surrogates have low relative errors (less than 0.76%) and can be used within forward simulations that are subject to constraint optimization. To demonstrate this, we set up several different test scenarios in which an upper limb model consisting of the elbow joint, the biceps and triceps brachii, and an external load is subjected to different optimization criteria. Even though this novel method has only been demonstrated for a 2-muscle system, it can easily be extended to musculoskeletal systems with 3 or more muscles.