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Preconditioned Subspace Descent Method for Nonlinear Systems of Equations

Abstract Nonlinear least squares iterative solver is considered for real-valued sufficiently smooth functions. The algorithm is based on successive solution of orthogonal projections of the linearized equation on a sequence of appropriately chosen low-dimensional subspaces. The bases of the latter are constructed using only the first-order derivatives of the function. The technique based on the concept of the limiting stepsize along normalized direction (developed earlier by the author) is used to guarantee the monotone decrease of the nonlinear residual norm. Under rather mild conditions, the convergence to zero is proved for the gradient and residual norms. The results of numerical testing are presented, including not only small-sized standard test problems, but also larger and harder examples, such as algebraic problems associated with canonical decomposition of dense and sparse 3D tensors as well as finite-difference discretizations of 2D nonlinear boundary problems for 2nd order partial differential equations.