7 research outputs found
Simulation of non-stationary and non-Gaussian random processes by 3rd-order Spectral Representation Method: Theory and POD implementation
This paper introduces the -order Spectral Representation Method for
simulation of non-stationary and non-Gaussian stochastic processes. The
proposed method extends the classical -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 -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 . 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
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