Randomized Truncated SVD Levenberg-Marquardt Approach to Geothermal Natural State and History Matching

TL;DR

This paper introduces a randomized truncated SVD approach within the Levenberg-Marquardt method to enhance efficiency in geothermal reservoir model inversion, enabling faster and more scalable parameter updates.

Contribution

It develops LM variants that incorporate randomized methods for TSVD, allowing simultaneous adjoint and direct solves, improving computational efficiency over traditional Lanczos-based approaches.

Findings

Randomized methods outperform Lanczos iteration in speed.

Simultaneous solves reduce computational time.

Enhanced scalability for large reservoir models.

Abstract

The Levenberg-Marquardt (LM) method is commonly used for inverting models used to describe geothermal, groundwater, or oil and gas reservoirs. In previous studies LM parameter updates have been made tractable for highly parameterized inverse problems with large data sets by applying matrix factorization methods or iterative linear solvers to approximately solve the update equations. Some studies have shown that basing model updates on the truncated singular value decomposition (TSVD) of a dimensionless sensitivity matrix achieved using Lanczos iteration can speed up the inversion of reservoir models. Lanczos iterations only require the sensitivity matrix times a vector and its transpose times a vector, which are found efficiently using adjoint and direct simulations without the expense of forming a large sensitivity matrix. Nevertheless, Lanczos iteration has the drawback of being a serial process, requiring a separate adjoint solve and direct solve every Lanczos iteration. Randomized methods, developed for low-rank matrix approximation of large matrices, are more efficient alternatives to the standard Lanczos method. Here we develop LM variants which use randomized methods to find a TSVD of a dimensionless sensitivity matrix when updating parameters. The randomized approach offers improved efficiency by enabling simultaneous solution of all adjoint and direct problems for a parameter update.