Principal Component Geostatistical Approach for large-dimensional inverse problems

Area of Science:

• Geostatistics
• Inverse Problems
• Computational Science

Background:

• Quasi-linear geostatistical methods are used for weakly nonlinear inverse problems.
• Current implementations require Jacobian matrix computation, which is computationally expensive for large datasets.
• Existing methods have a high computational cost, often scaling with m^2*n.

Purpose of the Study:

• To present a matrix-free Gauss-Newton method for geostatistical inverse problems.
• To improve the scalability and reduce the computational cost of these methods.
• To provide an efficient alternative for large-scale inverse problems.

Main Methods:

• Utilized a matrix-free Gauss-Newton method.
• Reduced the need for explicit Jacobian matrix computation.
• Implemented an iterative approach requiring K forward problem runs per iteration, where K << m and K < n.

Main Results:

• Achieved a computational and storage cost scaling roughly linearly with m, instead of m^2.
• Demonstrated a dramatic reduction in computational cost for problems with very large m.
• Validated the approach's effectiveness and provided insights into optimal performance conditions.

Conclusions:

• The matrix-free Gauss-Newton method significantly enhances the scalability of geostatistical inverse problems.
• This approach offers substantial computational savings compared to traditional methods.
• The method is particularly beneficial for large-scale applications in fields like Hydraulic and Electrical Resistivity Tomography.