Simulation of non-stationary and non-Gaussian random processes by 3rd-order Spectral Representation Method: Theory and POD implementation

This paper introduces the $3^{rd}$-order Spectral Representation Method for simulation of non-stationary and non-Gaussian stochastic processes. The proposed method extends the classical $2^{nd}$-order Spectral Representation Method to expand the stochastic process from an evolutionary bispectrum and an evolutionary power spectrum, thus matching the process completely up to third-order. A Proper Orthogonal Decomposition (POD) approach is further proposed to enable an efficient FFT-based implementation that reduces computational cost significantly. Two examples are presented, including the simulation of a fully non-stationary seismic ground motion process, highlighting the accuracy and efficacy of the proposed method.Comment: 38 pages, 9 figures, 2 table

Simulation of non-Gaussian wind field as a 3rd3^{rd}-order stochastic wave

This paper presents a methodology for the simulation of non-Gaussian wind field as a stochastic wave using the 3rd-order Spectral Representation Method. Traditionally, the wind field is modeled as a stochastic vector process at discrete locations in space. But the simulation of vector process is well-known to be computationally challenging and numerically unstable when modeling wind at a large number of discrete points in space. Recently, stochastic waves have been used to model the field as a continuous process indexed both in time and space. We extend the classical Spectral Representation Method for simulation of Gaussian stochastic waves to a third-order representation modeling asymmetrically skewed non-Gaussian stochastic waves from a prescribed power spectrum and bispectrum. We present an efficient implementation using the fast Fourier transform, which reduces the computational time dramatically. We then apply the method for simulation of a non-Gaussian wind velocity field along a long-span bridge.Comment: 22 pages, 4 figure, 1 tabl

Efficient simulation of higher-order non-Gaussian/non-stationary stochastic processes: An FFT approach

The theory of stochastic processes and their generations are indispensable to characterize wind fluctuations, ocean waves, and earthquake excitations among other quantities in engineering. To computationally analyze and simulate these stochastic systems, practical realization of samples of stochastic processes is essential. The objective of this thesis is to generalize the third-order Spectral Representation Method for the simulating non-Gaussian stochastic fields, vector processes, non-stationary processes and waves. Along with developing the simulation formulae for these cases, computationally efficient implementations using fast Fourier transform are also presented. This improves the simulation time for all the cases and makes simulation computationally tractable. In Chapter 2, a framework for the simulation of multi-dimensional non-Gaussian random fields is derived. The increments satisfying higher-order orthogonality conditions are explicitly defined. The formula for general 2, 3, and d-dimensional random field using the increments are first presented and a simpler case for quadrant random fields is also presented. An FFT based implementation is presented which reduces the simulation time exponentially. Examples of 2-dimensional and 3-dimensional random fields highlighting the salient features of the proposed methodology are presented. In Chapter 3, simulation of non-Gaussian vector processes by the 3rd-Order Spectral Representation Method is presented. A frequency indexing trick to make such vector processes ergodic is also introduced. FFT based implementation is introduced to improve the simulation time. Simulation of wind velocities along a vertical profile is performed using the proposed simulation formula. In Chapter 4, a stochastic waves approach for the 3rd-order Spectral Representation Method is presented. While vector processes can be used to simulate time histories along multiple points of a long-span bridge or a tall building. Simulation of vector processes involves the Cholesky or Singular Value Decomposition of the cross power spectral density, which can pose challenges for large simulation. It also involves tensorial product of decomposed components of the cross power spectral density with the cross bispectral density. Stochastic waves are a computationally viable alternative to simulate time histories along a large number of points. A simulation formula along with a FFT based implementation is presented. In Chapter 5, the 3rd-Order Spectral Representation Method is extended for the simulation of non-stationary and non-Gaussian processes. Firstly, increments satisfying the evolutionary power spectrum and the evolutionary bispectrum simultaneously are derived. A formula for the simulation of non-stationary non-Gaussian processes is proposed using the increments. Since non-stationary processes have coupled time and frequency terms, fast Fourier Transform cannot be applied directly. To overcome this challenge, Proper Orthogonal Decomposition is used to appropriately decompose the evolutionary spectra. Fast fourier transform can then be used to improve the computational time for the generation of non-stationary processes using the 3rd-order Spectral Representation Method. Lastly, future directions and potential applications for this research are presented. One potential application is the investigation of importance of non-Gaussian stochastic processes in the context of non-linear structural systems. The simulation framework can also be extended to simulate processes of order $> 3$. This would involve deriving increments to incorporate the information contained in higher order spectral Trispectrum and beyond. Finally, another future direction for this research is the extending the proposed framework for simulating higher-order processes on a generalized manifold

A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems

Constructing surrogate models for uncertainty quantification (UQ) on complex partial differential equations (PDEs) having inherently high-dimensional $\mathcal{O}(10^{\ge 2})$ stochastic inputs (e.g., forcing terms, boundary conditions, initial conditions) poses tremendous challenges. The curse of dimensionality can be addressed with suitable unsupervised learning techniques used as a pre-processing tool to encode inputs onto lower-dimensional subspaces while retaining its structural information and meaningful properties. In this work, we review and investigate thirteen dimension reduction methods including linear and nonlinear, spectral, blind source separation, convex and non-convex methods and utilize the resulting embeddings to construct a mapping to quantities of interest via polynomial chaos expansions (PCE). We refer to the general proposed approach as manifold PCE (m-PCE), where manifold corresponds to the latent space resulting from any of the studied dimension reduction methods. To investigate the capabilities and limitations of these methods we conduct numerical tests for three physics-based systems (treated as black-boxes) having high-dimensional stochastic inputs of varying complexity modeled as both Gaussian and non-Gaussian random fields to investigate the effect of the intrinsic dimensionality of input data. We demonstrate both the advantages and limitations of the unsupervised learning methods and we conclude that a suitable m-PCE model provides a cost-effective approach compared to alternative algorithms proposed in the literature, including recently proposed expensive deep neural network-based surrogates and can be readily applied for high-dimensional UQ in stochastic PDEs.Comment: 45 pages, 14 figure

UQpy v4.1: Uncertainty Quantification with Python

This paper presents the latest improvements introduced in Version 4 of the UQpy, Uncertainty Quantification with Python, library. In the latest version, the code was restructured to conform with the latest Python coding conventions, refactored to simplify previous tightly coupled features, and improve its extensibility and modularity. To improve the robustness of UQpy, software engineering best practices were adopted. A new software development workflow significantly improved collaboration between team members, and continous integration and automated testing ensured the robustness and reliability of software performance. Continuous deployment of UQpy allowed its automated packaging and distribution in system agnostic format via multiple channels, while a Docker image enables the use of the toolbox regardless of operating system limitations

UQpy v4.1: Uncertainty quantification with Python

This paper presents the latest improvements introduced in Version 4 of the UQpy, Uncertainty Quantification with Python, library. In the latest version, the code was restructured to conform with the latest Python coding conventions, refactored to simplify previous tightly coupled features, and improve its extensibility and modularity. To improve the robustness of UQpy, software engineering best practices were adopted. A new software development workflow significantly improved collaboration between team members, and continuous integration and automated testing ensured the robustness and reliability of software performance. Continuous deployment of UQpy allowed its automated packaging and distribution in system agnostic format via multiple channels, while a Docker image enables the use of the toolbox regardless of operating system limitations