Bayesian adaptation of chaos representations using variational inference and sampling on geodesics.

Area of Science:

• Computational mathematics
• Applied probability
• Numerical analysis

Background:

• Polynomial chaos expansions are crucial for uncertainty quantification.
• Existing adaptation methods in Homogeneous Chaos spaces have limitations.
• Bayesian inference offers a robust framework for parameter estimation.

Purpose of the Study:

• To develop a novel Bayesian approach for constructing polynomial chaos representations.
• To incorporate an adaptation rotation matrix as a new parameter for Bayesian inference.
• To enhance the accuracy of scalar quantities of interest (QoI) predictions.

Main Methods:

• Bayesian formulation to characterize posterior distributions of series coefficients and adaptation matrix.
• Variational inference for approximating posterior distributions of coefficients.
• Geodesic Monte Carlo sampling (Hamiltonian Monte Carlo on Stiefel manifold) for the rotation matrix posterior.
• Application to multiphase flow in heterogeneous porous media.

Main Results:

• The proposed method successfully estimates the adaptation rotation matrix and series coefficients.
• Demonstrated performance through numerical examples, including complex fluid dynamics problems.
• Improved accuracy in polynomial chaos representations through adaptive Bayesian inference.

Conclusions:

• The developed Bayesian framework provides an effective method for adaptive polynomial chaos expansions.
• The integration of an adaptation matrix improves the modeling of input-output relationships.
• This approach offers significant potential for uncertainty quantification in complex scientific and engineering problems.