Sparse Grid Points Research Articles

Binary Tomography Reconstruction with Limited-Data by a Convex Level-Set燤ethod

This paper proposes a new level-set-based shape recovery approach that can be applied to a wide range of binary tomography reconstructions. In this technique, we derive generic evolution equations for shape reconstruction in terms of the underlying level-set parameters. We show that using the appropriate basis function to parameterize the level-set function results in an optimization problem with a small number of parameters, which overcomes many of the problems associated with the traditional level-set approach. More concretely, in this paper, we use Gaussian functions as a basis function placed at sparse grid points to represent the parametric level-set function and provide more flexibility in the binary representation of the reconstructed image. In addition, we suggest a convex optimization method that can overcome the problem of the local minimum of the cost function by successfully recovering the coefficients of the basis function. Finally, we illustrate the performance of the proposed method using synthetic images and real X-ray CT projection data. We show that the proposed reconstruction method compares favorably to various state-of-the-art reconstruction techniques for limited-data tomography, and it is also relatively stable in the presence of modest amounts of noise. Furthermore, the shape representation using a compact Gaussian radial basis function works well.

Feasibility of DEIM for retrieving the initial field via dimensionality reduction

When parameter estimation is solved in a high-dimensional space, the dimensionality reduction strategy becomes the primary consideration for alleviating the tremendous computational cost. In the present study, the discrete empirical interpolation method (DEIM) is explored to retrieve the initial condition (IC) by combining the polynomial chaos (PC) based ensemble Kalman filter (i.e. PC-EnKF), where a non-intrusive PC expansion is considered as a surrogate model in place of the forward model in the prediction step of the ensemble Kalman filter, resulting in fewer forward model integrations but with a comparable accuracy as Monte Carlo-based approaches. The DEIM acts as a hyper-reduction tool to provide the low-dimensional input for the high-dimensional initial field, which can be reconstructed using the information on the sparse interpolation grid points that is adaptively obtained through PC-EnKF data assimilation method. Thus an innovative framework to reconstruct the IC is developed. The detailed procedure at each assimilation iteration includes: the determination of the spatial interpolation points, the estimation of the initial values on the interpolation locations using the optimal observations, and the reconstruction of IC in the full space. The current study uses the reconstruction field of initial conditions of the Navier-Stokes equations as an example to illustrate the efficacy of our method. The experimental results demonstrate the proposed algorithm achieves a satisfactory reconstruction for the initial field. The proposed method helps to extend the applicable area of DEIM in solving inverse problems.

A parallel sparse grid construction algorithm based on the shared memory architecture and its application to flash calculations

The study of sparse grids has been done in a lot of works, but none of them pay special attention on the concept of “layer”, not to mention the further research on this new concept. This work presents the layer concept at first, and then based on it a new parallel sparse grid construction algorithm is proposed, which fills the niche that few works design such kind of algorithm on shared memory architecture. With the superiorities of the layer concept, the sparse grids can be constructed in a “layer by layer” manner on a much easier structure which is the simplified tree. Different from the classical tree structures used in the former sparse grid construction algorithm, the simplified tree reduces the number of sparse grid points further and at the same time avoids many unnecessary usages of pointers. Moreover, the correctness of the algorithm is guaranteed by the two correctness criterions, and its main advantages include that it can achieve load balance among the threads easily and that it can be applied in the function domains of any kinds of configurations, which is demonstrated by theory. After that, the algorithm is tested on flash calculations which are an important application from reservoir simulations. According to the characteristics of the issue, a modified version of the algorithm is given. By comparing the tested results with those of the former algorithm, it is found that a great quantity of construction time is saved by this algorithm.

The application of high-dimensional sparse grids in flash calculations: From theory to realisation

Flash calculations are a performance bottleneck of compositional flow simulations. Some work has demonstrated the feasibility of using sparse grid techniques to remove the bottleneck, but a complete realisation of the idea is still not available. Thus, this work fills the niche. By introducing a new concept of layer to sparse grid points, the sparse grid construction can become much efficient. As a result, a much easier data structure the array can be used to store the sparse grids. Compared with the popular data structures to store the sparse grids such as the hash table and the tree, the array can minimize the space size and the traversing time, and at the same time reduce the number of points in the sparse grids by removing the architecture ancestors in the tree, which in turn makes parallelization of flash calculations come true. All of them are not only contributions to flash calculations, but also contributions to existing sparse grid techniques. Moreover, both of the sparse grid construction and interpolation algorithms can be done in parallel. Different from the former parallel algorithms in sparse grid techniques, which have troubles in decomposing the domain equally and keeping load balance among the processors, our parallel algorithm can achieve load balance easily among the threads for any sparse grid configurations. Lastly, multicomponent experiments are also carried out to demonstrate the accuracy, correctness and efficiency of the algorithms.

Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)

We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-convergence theory based on previous work by Bachmayr et al. [ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 341--363] and Chen [ESAIM Math. Model. Numer. Anal., in press, 2018, https://doi.org/10.1051/m2an/2018012] and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We specifically verify for Gauss--Hermite nodes that this assumption holds and also show algebraic convergence with respect to the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.

Nonintrusive Uncertainty Analysis of Fluid-Structure Interaction with Spatially Adaptive Sparse Grids and Polynomial Chaos Expansion

The propagation of uncertainty in physical parameters of fluid-structure interaction problems is a challenging task---both mathematically and in terms of computational workload. In this paper, we employ nonintrusive polynomial chaos expansion and model the uncertainty in five independent input parameters that characterize both fluid and structure. We propose a novel methodology to compute the expansion's coefficients using spatially adaptive sparse grids and products of one-dimensional integrals, which exploits the tensor structure of both sparse grids and probabilistic space. Furthermore, with spatial adaptivity and modified basis functions, we keep the number of sparse grid points small. We test our approach in two test cases: (i) an elastic vertical flap in a channel flow and (ii) a computationally challenging, well-established benchmark. The outputs of interest are the x-deflection and total force on the structure. In the first test case, we consider six implementations of our methodology and two established methods based on sparse grid quadrature. We evaluate the performance (in terms of number of runs of the numerical solver) and accuracy of all eight methods. The results show that one variant of our approach outperforms all the other implementations. We apply our best variant to the benchmark scenario and address the high computational demands of the resulting problem using three levels of parallelism. Importantly, our approach is not restricted to fluid-structure interaction problems. We can address a broad spectrum of computationally expensive problems, provided that sparse grid approximations can be employed.

UNCERTAINTY QUANTIFICATION FOR MAXWELL'S EQUATIONS USING STOCHASTIC COLLOCATION AND MODEL ORDER REDUCTION

Modeling and simulation are indispensable for the design process of new semiconductor structures. Difficulties arise from shrinking structure sizes, increasing working frequencies, and uncertainties of materials and geometries. Therefore, we consider the time-harmonic Maxwell's equations for the simulation of a coplanar waveguide with uncertain material parameters. To analyze the uncertainty of the system, we use stochastic collocation with Stroud and sparse grid points. The results are compared to a Monte Carlo simulation. Both methods rely on repetitive runs of a deterministic solver. To accelerate this, we compute a reduced model by means of proper orthogonal decomposition to reduce the computational cost. The Monte Carlo simulation and the stochastic collocation are both applied to the full and the reduced model. All results are compared concerning accuracy and computation time.

A Multigrid Method for Adaptive Sparse Grids

Sparse grids have become an important tool to reduce the number of degrees of freedom of discretizations of moderately high-dimensional partial differential equations; however, the reduction in degrees of freedom comes at the cost of an almost dense and unconventionally structured system of linear equations. To guarantee overall efficiency of the sparse grid approach, special linear solvers are required. We present a multigrid method that exploits the sparse grid structure to achieve an optimal runtime that scales linearly with the number of sparse grid points. Our approach is based on a novel decomposition of the right-hand sides of the coarse grid equations that leads to a reformulation in so-called auxiliary coefficients. With these auxiliary coefficients, the right-hand sides can be represented in a nodal point basis on low-dimensional full grids. Our proposed multigrid method directly operates in this auxiliary coefficient representation, circumventing most of the computationally cumbersome sparse gr...

On the Choice of Design Points for Least Square Polynomial Approximations with Application to Uncertainty Quantification

AbstractIn this work, we concern with the numerical comparison between different kinds of design points in least square (LS) approach on polynomial spaces. Such a topic is motivated by uncertainty quantification (UQ). Three kinds of design points are considered, which are the Sparse Grid (SG) points, the Monte Carlo (MC) points and the Quasi Monte Carlo (QMC) points. We focus on three aspects during the comparison: (i) the convergence properties; (ii) the stability, i.e. the properties of the resulting condition number of the design matrix; (iii) the robustness when numerical noises are present in function values. Several classical high dimensional functions together with a random ODE model are tested. It is shown numerically that (i) neither the MC sampling nor the QMC sampling introduce the low convergence rate, namely, the approach achieves high order convergence rate for all cases provided that the underlying functions admit certain regularity and enough design points are used; (ii)The use of SG points admits better convergence properties only for very low dimensional problems (say d ≤ 2); (iii)The QMC points, being deterministic, seem to be a good choice for higher dimensional problems not only for better convergence properties but also in the stability point of view.

UNCERTAINTY QUANTIFICATION IN DYNAMIC SIMULATIONS OF LARGE-SCALE POWER SYSTEM MODELS USING THE HIGH-ORDER PROBABILISTIC COLLOCATION METHOD ON SPARSE GRIDS

This paper proposes a probabilistic collocation method (PCM) to quantify the uncertainties with dynamic simulations in power systems. The appraoch was tested on a single-machine-infinite-bus system and the over 15,000 -bus Western Electricity Coordinating Council (WECC) system. Comparing to classic Monte-Carlo (MC) method, the proposed PCM applies the Smolyak algorithm to reduce the number of simulations that have to be performed. Therefore, the computational cost can be greatly reduced using PCM. The algorithm and procedures are described in the paper. Comparison was made with MC method on the single machine as well as the WECC system. The simulation results shows that using PCM only a small number of sparse grid points need to be sampled even when dealing with systems with a relatively large number of uncertain parameters. PCM is, therefore, computationally more efficient than MC method.

Comparative study of broadband damage localization methods applied to test data

This application-oriented paper presents comparison of various broadband frequency based structural damage localization indices using experimental data from a full-scale structure known as the I-40 Bridge. First, three ‘damage-sensitive’ response parameters determined in the time and/or spectral domain, modal domain and wavelet domain are presented for damage localization in the context of a ‘non-model-based’ damage identification approach. Secondly, experimental modal data (namely natural frequencies, mode shapes and modal damping) obtained from this full-scale bridge subjected to various damage conditions is employed for assessment of the robustness of these methods on ‘real-world’ applications. Consequently, the results obtained are compared with those obtained from existing resonance frequency based damage identification methods. The results obtained demonstrate the improved capability of the broadband methods to localize damage in a full-scale structure despite sparse modal information and limited measurement grid points.

Uncertainty quantification via random domain decomposition and probabilistic collocation on sparse grids

Quantitative predictions of the behavior of many deterministic systems are uncertain due to ubiquitous heterogeneity and insufficient characterization by data. We present a computational approach to quantify predictive uncertainty in complex phenomena, which is modeled by (partial) differential equations with uncertain parameters exhibiting multi-scale variability. The approach is motivated by flow in random composites whose internal architecture (spatial arrangement of constitutive materials) and spatial variability of properties of each material are both uncertain. The proposed two-scale framework combines a random domain decomposition (RDD) and a probabilistic collocation method (PCM) on sparse grids to quantify these two sources of uncertainty, respectively. The use of sparse grid points significantly reduces the overall computational cost, especially for random processes with small correlation lengths. A series of one-, two-, and three-dimensional computational examples demonstrate that the combined RDD–PCM approach yields efficient, robust and non-intrusive approximations for the statistics of diffusion in random composites.

An efficient, high-order probabilistic collocation method on sparse grids for three-dimensional flow and solute transport in randomly heterogeneous porous media

In this study, a probabilistic collocation method (PCM) on sparse grids is used to solve stochastic equations describing flow and transport in three-dimensional, saturated, randomly heterogeneous porous media. The Karhunen–Loève decomposition is used to represent log hydraulic conductivity Y = ln K s . The hydraulic head h and average pore-velocity v are obtained by solving the continuity equation coupled with Darcy’s law with random hydraulic conductivity field. The concentration is computed by solving a stochastic advection–dispersion equation with stochastic average pore-velocity v computed from Darcy’s law. The PCM approach is an extension of the generalized polynomial chaos (gPC) that couples gPC with probabilistic collocation. By using sparse grid points in sample space rather than standard grids based on full tensor products, the PCM approach becomes much more efficient when applied to random processes with a large number of random dimensions. Monte Carlo (MC) simulations have also been conducted to verify accuracy of the PCM approach and to demonstrate that the PCM approach is computationally more efficient than MC simulations. The numerical examples demonstrate that the PCM approach on sparse grids can efficiently simulate solute transport in randomly heterogeneous porous media with large variances.