  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.
