Uncertainty Quantification
In uncertainty quantification (UQ), the impact of randomness on the behavior of the responses is assessed. In reality, various sources of uncertainties affect the response such as inherent randomness, modeling error, and lack of knowledge. The sources of uncertainties and how they influence the response should be studied appropriately to have a proper confidence in the prediction of stochastic responses. However, performing UQ is challenging and computationally exhaustive even with state-of-the-art computational power. Hence, efficient algorithms should be utilized while performing UQ of expensive functions. Furthermore, since there are many software available for deterministic simulations, the priority is to use the available software for finite element analysis, computational fluid dynamics, with less number of calls.
My main specialization in a spectral approach called Polynomial Chaos Expansion (PCE) for UQ, which is known for its mean-square convergence. However, like other computational approaches for UQ, it is also affected by the "Curse of dimensionality". To tackle this problem, some of my recently proposed works include adaptive PCE and adaptive sampling using regression based PCE. More recently, L1-minimization approaches for sparse PCE has been implemented, and Karhunen-Loeve Expansion (KLE) is used to randomness provided as random field. The PCE approximation thus built can be used for estimation of the statistics of the stochastic responses or used for reliability estimation.
Furthermore, I am interested in data-driven approaches for UQ, and in this regard I have implemented arbitrary PCE to perform UQ when the data regarding the random inputs is available. For example, the construction of arbitrary orthogonal polynomials based on the wind data is provided below.
Furthermore, I am interested in data-driven approaches for UQ, and in this regard I have implemented arbitrary PCE to perform UQ when the data regarding the random inputs is available. For example, the construction of arbitrary orthogonal polynomials based on the wind data is provided below.
Global Sensitivity Analysis
In uncertainty analysis, it is paramount to understand the influence or contribution of random parameters on the stochastic response. To this end, global sensitivity analysis (GSA) can be utilized. Two of the main approaches are: variance-based (Sobol Indices), and moment-independent (Borgonov Indices).
The variance-based GSA can be obtained as a post-processing once the PCE of the stochastic responses. So, it has been widely popular in finding the significant random parameters that will guide the dimension reduction and help in calibration process.
The variance-based GSA can be obtained as a post-processing once the PCE of the stochastic responses. So, it has been widely popular in finding the significant random parameters that will guide the dimension reduction and help in calibration process.
Anisotropic Sparse Polynomial Chaos based on Global Sensitivity Analysis
In this study, I proposed an anisotropic approach for polynomial basis expansion. To mitigate the factorial growth of PCE terms for construction of PCE with a large number of random inputs, a global sensitivity analysis based anisotropic basis expansion approach was proposed. Using this approach, the random input dimensions with high values of Sobol Indices from GSA are considered, and a dimension adaptive PCE is implemented. Doing so helps to mitigate the factorial growth of basis terms and make the problem tractable. It also has benefits in terms of computational cost regarding the number of function evaluations.
doi: doi.org/10.1016/j.ress.2022.108354
doi: doi.org/10.1016/j.ress.2022.108354
Example: The demonstration of anisotropic basis expansion is demonstrated in the first three figures whereas the non-adaptive total order expansion is demonstrated in the fourth figure below. In the first three figures, the basis are included only for the influential dimensions which mitigates the number of terms to be supplied for l1-minimization algorithms.
Another example of basis expansion based on the GSA is demonstrated here for a high dimensional problem with 36 random inputs. As shown in the figure, the basis expansion is performed only along the few influential dimensions (eight random inputs) thereby requiring only 120 additional basis terms while increasing the PCE order from p=2 to p=3; whereas it would have required a total of 8436 basis terms if using the usual isotropic approach.
Classifier based Basis Selection for High-Dimensional Polynomial Chaos Expansion
In this study, I proposed a classifier based approach for basis selection. In order to mitigate the factorial growth of basis terms and to keep the training cost minimal for l1-minimization based sparse PCE, a classifier is considered based on the multi-indices for filtering and selecting the new basis candidates. This helps in maintaining the mutual coherence of the information matrix during l1-minimization and avoid the degradation of accuracy of l1-minimization based higher order PCE.
Example: As shown below, the multi-indices are first labelled as influential and non-influential, which are then used to build a classifier- support vector machine (SVM). The SVM classifier can then be used to filter the new basis candidates and make a decision on whether to include the new higher-order basis terms for l1-minimization or not.
Several algorithms and programs for UQ using PCE and GSA have been developed at the lab and can be provided upon a suitable request for collaboration.