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We present a derivative-free algorithm for solving bound constrained systems of nonlinear monotone equations. The algorithm generates feasible iterates using in a systematic way the residual as search direction and a suitable step-length closely related to the Barzilai-Borwein choice. A convergence analysis is described. We also present one application in solving problems related with the study of reaction-diffusion processes that can be described by nonlinear partial differential equations of elliptic type. Numerical experiences are included to highlight the efficacy of proposed algorithm.

In this paper the use of recent residual algorithms for the simulation of the oxygen distribution in a tumor tissue in 2-D is proposed. The oxygen distribution in a tumor is considered a reaction-diffusion problem in steady state, whose mathematical model is a nonlinear partial differential equation, which is numerically solved using the traditional methods for systems of nonlinear equations (Newton's method, Broyden method, inexact Newton methods, etc.). Unlike of these traditional methods that require the use of derivatives and a large memory storage capacity, the proposed residual algorithms are derivative-free methods with low memory storage. The preliminary numerical results indicate that the proposed methods allows efficiently determine the distribution of oxygen in a tumor tissue to synthetic problems.

This paper presents a residual approach of the square root balanced truncation algorithm for model order reduction of continuous, linear and time-invariante compartmental systems. Specifically, the new approach uses a residual method to approximate the controllability and observability gramians, whose resolution is an essential step of the square root balanced truncation algorithm, that requires a great computational cost. Numerical experiences are included to highlight the efficacy of the proposed approach.