Efficient hyperparameter estimation in Bayesian inverse problems using sample average approximation.

Area of Science:

• Applied mathematics
• Computational geophysics
• Bayesian inference

Background:

• Bayesian inverse problems often require estimating hyperparameters for priors and noise models from data.
• Linear inverse problems with Gaussian noise and Matérn covariance priors necessitate computationally intensive maximum a posteriori (MAP) estimation involving log determinants.

Purpose of the Study:

• To develop computationally efficient methods for hyperparameter estimation in Bayesian inverse problems.
• To address the computational burden of log determinant calculations in MAP estimation.

Main Methods:

• Utilizing a stochastic average approximation (SAA) of the objective function.
• Employing the preconditioned Lanczos method for efficient function and gradient approximations.
• Proposing a novel, cheaply updatable preconditioner for hyperparameters.
• Developing an approach to approximate gradient evaluations by reusing information from function evaluations.

Main Results:

• The proposed methods significantly reduce the computational cost associated with hyperparameter estimation.
• The new preconditioner allows for efficient updates as hyperparameter values change.
• Information reuse in gradient approximation further enhances computational efficiency.

Conclusions:

• The developed techniques provide a computationally feasible approach for hyperparameter estimation in complex Bayesian inverse problems.
• The methods are successfully demonstrated on static and dynamic seismic tomography problems, showcasing their practical applicability.