Polynomial chaos expansion

Definition

Polynomial chaos expansion is a mathematical technique used to represent uncertain parameters in a model as a series of orthogonal polynomials. It provides a way to quantify uncertainty in a system by expressing random variables in terms of deterministic polynomials, enabling analysts to efficiently compute the effect of input uncertainties on outputs. This approach connects closely with uncertainty quantification, allowing for the analysis and propagation of uncertainties in various applications.

5 Must Know Facts For Your Next Test

1. Polynomial chaos expansion uses orthogonal polynomials like Hermite, Legendre, and Chebyshev, depending on the probability distribution of the input variables.
2. This method allows for efficient sampling techniques, reducing the computational cost when evaluating the effects of uncertainties compared to traditional Monte Carlo methods.
3. The coefficients in a polynomial chaos expansion represent the contribution of each polynomial term to the overall uncertainty in the model's output.
4. By transforming uncertain inputs into a deterministic polynomial form, polynomial chaos expansion facilitates sensitivity analysis and helps identify critical factors influencing outcomes.
5. It can be applied across various fields including engineering, finance, and environmental science, making it a versatile tool for uncertainty quantification.

Review Questions

• How does polynomial chaos expansion facilitate uncertainty quantification in complex models?
  • Polynomial chaos expansion allows uncertainty quantification by representing random inputs as deterministic polynomials. This transformation simplifies the analysis of how these uncertainties impact model outputs, providing a systematic way to assess and propagate uncertainties. By using orthogonal polynomials tailored to the probability distributions of inputs, analysts can efficiently quantify how variations influence outcomes without relying solely on extensive simulations.
• Compare polynomial chaos expansion with traditional Monte Carlo methods in terms of efficiency and application.
  • Polynomial chaos expansion is often more efficient than traditional Monte Carlo methods for uncertainty quantification because it reduces the number of simulations required to understand output variability. While Monte Carlo methods rely on random sampling to estimate effects of uncertainties, polynomial chaos can achieve similar results with far fewer samples by leveraging the deterministic nature of polynomial representation. This makes polynomial chaos particularly useful in high-dimensional problems where computational resources are limited.
• Evaluate the role of orthogonal polynomials in polynomial chaos expansion and their significance in capturing uncertainty effects.
  • Orthogonal polynomials play a crucial role in polynomial chaos expansion as they provide a structured way to represent random variables and their uncertainties. By ensuring that each polynomial term is independent, they allow for clear separation of contributions from different sources of uncertainty. The choice of orthogonal polynomials based on input distributions significantly enhances the accuracy and efficiency of the expansion, enabling precise characterization of how uncertainties propagate through models and influence outcomes.

Related terms

Uncertainty Quantification:

The process of determining how uncertain parameters affect the outcomes of a model or system.

Stochastic Processes: Mathematical objects used to describe systems or processes that evolve over time with inherent randomness.

Orthogonal Polynomials: A sequence of polynomials that are orthogonal to each other with respect to a specific inner product, often used in polynomial chaos expansions.