[Paper Review] An adaptive minimum spanning tree multi-element method for uncertainty quantification of smooth and discontinuous responses

Yous van Halder, Benjamin Sanderse+1 more|arXiv (Cornell University)|Mar 19, 2018

Probabilistic and Robust Engineering Design35 references3 citations

TL;DR

This paper proposes an adaptive minimum spanning tree multi-element (MST-ME) method for uncertainty quantification that automatically handles both smooth and discontinuous responses without prior knowledge. By combining adaptive sampling via minimum spanning trees, support vector machines for discontinuity detection, and least orthogonal interpolation on decomposed domains, the method achieves high accuracy and efficiency with minimal samples, avoiding Gibbs phenomena and enabling robust surrogate modeling for complex PDEs and engineering problems.

ABSTRACT

A novel approach for non-intrusive uncertainty propagation is proposed. Our approach overcomes the limitation of many traditional methods, such as generalised polynomial chaos methods, which may lack sufficient accuracy when the quantity of interest depends discontinuously on the input parameters. As a remedy we propose an adaptive sampling algorithm based on minimum spanning trees combined with a domain decomposition method based on support vector machines. The minimum spanning tree determines new sample locations based on both the probability density of the input parameters and the gradient in the quantity of interest. The support vector machine efficiently decomposes the random space in multiple elements, avoiding the appearance of Gibbs phenomena near discontinuities. On each element, local approximations are constructed by means of least orthogonal interpolation, in order to produce stable interpolation on the unstructured sample set. The resulting minimum spanning tree multi-element method does not require initial knowledge of the behaviour of the quantity of interest and automatically detects whether discontinuities are present. We present several numerical examples that demonstrate accuracy, efficiency and generality of the method.

Motivation & Objective

• To develop a non-intrusive uncertainty quantification method that works effectively for both smooth and discontinuous quantities of interest without requiring prior knowledge of the response behavior.
• To overcome the limitations of generalized polynomial chaos and stochastic collocation methods, particularly Gibbs phenomena near discontinuities.
• To reduce the number of expensive model evaluations by using adaptive sampling guided by probability density and gradient information.
• To enable robust surrogate modeling for complex PDEs and black-box models through automated domain decomposition and local interpolation.
• To ensure stability and accuracy in interpolation on unstructured, adaptively sampled data points.

Proposed method

• Adaptive sampling is performed using a minimum spanning tree (MST) with a custom weight function that balances probability density of input parameters and local gradient in the quantity of interest.
• Support vector machines (SVMs) are used to classify samples and detect discontinuities, enabling domain decomposition along the SVM classification boundary to isolate regions with discontinuous behavior.
• The random input space is decomposed into non-overlapping elements where each local quantity of interest is amenable to stable interpolation.
• Least orthogonal interpolation is applied within each element to construct accurate local approximations on scattered, unstructured sample sets.
• The method iteratively adds new samples based on the MST, improving resolution in high-probability and high-gradient regions until convergence is achieved.
• The entire process is fully adaptive and does not require initial assumptions about the smoothness or discontinuity of the response.

Experimental results

Research questions

• RQ1 Can an adaptive sampling strategy based on minimum spanning trees effectively reduce the number of model evaluations while maintaining accuracy in uncertainty quantification?
• RQ2 How can discontinuities in the quantity of interest be detected and localized with minimal samples using machine learning techniques?
• RQ3 Can a multi-element approach with SVM-based domain decomposition prevent Gibbs phenomena while preserving accuracy for both smooth and discontinuous responses?
• RQ4 To what extent does the combination of MST sampling, SVM classification, and least orthogonal interpolation outperform traditional stochastic collocation methods in terms of convergence and robustness?
• RQ5 Can the method automatically distinguish between smooth and discontinuous behavior without prior knowledge of the response function?

Key findings

• The MST-ME method successfully captures discontinuities in the response without introducing Gibbs phenomena, even in complex problems such as the 1D shallow water equations and 3D dam break simulations.
• For the 3D dam break problem with free surface flow, the method detected a smooth, non-linear response dependent on the height-to-length ratio, with no discontinuities observed—possibly due to the absence of gas phase modeling.
• The method achieved accurate surrogate models with only 30 samples after 6 iterations of adaptive sampling, demonstrating fast convergence and efficiency.
• The MST-ME method automatically detects whether the response is smooth or discontinuous, eliminating the need for prior knowledge about the response behavior.
• Least orthogonal interpolation provided stable local approximations on unstructured sample sets, and the method remained robust even when discontinuities were present.
• The method is well-suited for parametric solutions of PDEs and can be used as a surrogate for Monte Carlo-based uncertainty quantification or inverse problems.