Details
| Original language | English |
|---|---|
| Article number | 117887 |
| Number of pages | 25 |
| Journal | Computer Methods in Applied Mechanics and Engineering |
| Volume | 439 |
| Early online date | 13 Mar 2025 |
| Publication status | Published - 1 May 2025 |
Abstract
Effectively estimating output probability distributions in stochastic static and dynamic systems with a limited number of simulations is a significant challenge, especially for complex distributions with multi-modality and heavy tails. To address this challenge, this work explores the potential of the Laplace Transform (LT) and its inversion. First, the statistical information embedded in the derivatives of the LT is analysed, establishing the theoretical foundation for recovering output probability distributions. Subsequently, a novel analytical expression for the response probability density function (PDF) is derived by decomposing its inverse LT (ILT) using Euler's formula. Building on the numerically estimated LT, a non-parametric numerical solution, termed the Numerical Decomposed ILT (NDILT) algorithm, is developed to flexibly estimate the main body of complex PDFs with limited samples. Second, the Taylor expansion of the real component of LT (RCLT) reveals its rich statistical content. Exploiting this property, another parametric method, the LT-based Maximum Entropy Method (LT-MEM), is proposed, incorporating estimated RCLT as constraints of the maximum entropy principle. By solving an optimization problem, LT-MEM can effectively reconstruct complex PDFs across their entire distribution domain using a small sample size. The proposed methods rediscover and harness the power of the LT and ILT to reconstruct complex-shaped probability distributions, offering a valuable alternative. Parameter selection strategies for NDILT and LT-MEM are provided, and their robust accuracy is validated through analytical and numerical examples across various challenging distributions.
Keywords
- Complex distribution, Inverse Laplace transform, Laplace transform, Maximum entropy, Probability distribution estimation, Stochastic system
ASJC Scopus subject areas
- Engineering(all)
- Computational Mechanics
- Engineering(all)
- Mechanics of Materials
- Engineering(all)
- Mechanical Engineering
- Physics and Astronomy(all)
- General Physics and Astronomy
- Computer Science(all)
- Computer Science Applications
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In: Computer Methods in Applied Mechanics and Engineering, Vol. 439, 117887, 01.05.2025.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Output probability distribution estimation of stochastic static and dynamic systems using Laplace transform and maximum entropy
AU - Zhang, Yang
AU - Dang, Chao
AU - Xu, Jun
AU - Beer, Michael
N1 - Publisher Copyright: © 2025 The Authors
PY - 2025/5/1
Y1 - 2025/5/1
N2 - Effectively estimating output probability distributions in stochastic static and dynamic systems with a limited number of simulations is a significant challenge, especially for complex distributions with multi-modality and heavy tails. To address this challenge, this work explores the potential of the Laplace Transform (LT) and its inversion. First, the statistical information embedded in the derivatives of the LT is analysed, establishing the theoretical foundation for recovering output probability distributions. Subsequently, a novel analytical expression for the response probability density function (PDF) is derived by decomposing its inverse LT (ILT) using Euler's formula. Building on the numerically estimated LT, a non-parametric numerical solution, termed the Numerical Decomposed ILT (NDILT) algorithm, is developed to flexibly estimate the main body of complex PDFs with limited samples. Second, the Taylor expansion of the real component of LT (RCLT) reveals its rich statistical content. Exploiting this property, another parametric method, the LT-based Maximum Entropy Method (LT-MEM), is proposed, incorporating estimated RCLT as constraints of the maximum entropy principle. By solving an optimization problem, LT-MEM can effectively reconstruct complex PDFs across their entire distribution domain using a small sample size. The proposed methods rediscover and harness the power of the LT and ILT to reconstruct complex-shaped probability distributions, offering a valuable alternative. Parameter selection strategies for NDILT and LT-MEM are provided, and their robust accuracy is validated through analytical and numerical examples across various challenging distributions.
AB - Effectively estimating output probability distributions in stochastic static and dynamic systems with a limited number of simulations is a significant challenge, especially for complex distributions with multi-modality and heavy tails. To address this challenge, this work explores the potential of the Laplace Transform (LT) and its inversion. First, the statistical information embedded in the derivatives of the LT is analysed, establishing the theoretical foundation for recovering output probability distributions. Subsequently, a novel analytical expression for the response probability density function (PDF) is derived by decomposing its inverse LT (ILT) using Euler's formula. Building on the numerically estimated LT, a non-parametric numerical solution, termed the Numerical Decomposed ILT (NDILT) algorithm, is developed to flexibly estimate the main body of complex PDFs with limited samples. Second, the Taylor expansion of the real component of LT (RCLT) reveals its rich statistical content. Exploiting this property, another parametric method, the LT-based Maximum Entropy Method (LT-MEM), is proposed, incorporating estimated RCLT as constraints of the maximum entropy principle. By solving an optimization problem, LT-MEM can effectively reconstruct complex PDFs across their entire distribution domain using a small sample size. The proposed methods rediscover and harness the power of the LT and ILT to reconstruct complex-shaped probability distributions, offering a valuable alternative. Parameter selection strategies for NDILT and LT-MEM are provided, and their robust accuracy is validated through analytical and numerical examples across various challenging distributions.
KW - Complex distribution
KW - Inverse Laplace transform
KW - Laplace transform
KW - Maximum entropy
KW - Probability distribution estimation
KW - Stochastic system
UR - http://www.scopus.com/inward/record.url?scp=86000619508&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2025.117887
DO - 10.1016/j.cma.2025.117887
M3 - Article
AN - SCOPUS:86000619508
VL - 439
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
SN - 0045-7825
M1 - 117887
ER -