A weighted -minimization approach for sparse polynomial chaos expansions

Abstract

This work proposes a method for sparse polynomial chaos (PC) approximation of high-dimensional stochastic functions based on non-adapted random sampling. We modify the standard ℓ 1 -minimization algorithm, originally proposed in the context of compressive sampling, using a priori information about the decay of the PC coefficients and refer to the resulting algorithm as weighted ℓ 1 -minimization. We provide conditions under which we may guarantee recovery using this weighted scheme. Numerical tests are used to compare the weighted and non-weighted methods for the recovery of solutions to two differential equations with high-dimensional random inputs: a boundary value problem with a random elliptic operator and a 2-D thermally driven cavity flow with random boundary condition.

Review of Polynomial Chaos-Based Methods for Uncertainty Quantification in Modern Integrated Circuits

“…Hence, a popular strategy is to choose a priori the order of the expansion P for the problem at hand. Alternatively, an adaptive strategy can be adopted based on cross-validation in the estimation of stochastic moments, such as [39], or methods computing sparse PC models, where only the significant terms out of all the M + 1 elements in (3) are used, see, for example, [40,41].…”

“…The least angle regression (LAR) algorithm proposed in [40] iteratively determines the significant basis sets, improving the overall accuracy of the PC model. A weighted 1 -minimization approach was proposed in [41] to obtain sparse PC expansions suitable to solve differential equations with high-dimensional random inputs.…”

A Kriging and Stochastic Collocation ensemble for uncertainty quantification in engineering applications

“…Recently, the PC expansion has also been combined with Model Order Reduction techniques [24,25], in order to study system described by a large set of equations. In [26], a sparse PC expansion was introduced to efficiently detect significant coefficients of PC expansion based on the least angle regression algorithm, while a weighted 1 -minimization approach was proposed in [27] to obtain sparse PC expansions suitable to solve differential equations with high-dimensional random inputs.…”

Research on Robot Motion Control based on Artificial Potential Method

“…And inside of each basis motion control module includes joint controller, joint driver and several sensors. The structure has fully taken into account the situation that joints in close position and similar functions of humanoid robot may be tightly coupled and has carried out appropriate concentration of functions, which not only retains the intelligent and flexible advantage of distributed structure but also reduces the burden of data communication so as to adapt to the control and sensing requirements of humanoid robot in a better manner [1][2][3][4][5].…”