Marielba Rojas

A New Matrix-Free Algorithm for the Large-Scale Trust-Region Subproblem

We present a new method for the large-scale trust-region subproblem. The method is matrix-free in the sense that only matrix-vector products are required. We recast the trust-region subproblem as a parameterized eigenvalue problem and compute an optimal value for the parameter. We then find the solution of the trust-region subproblem from the eigenvectors associated with two of the smallest eigenvalues of the parameterized eigenvalue problem corresponding to the optimal parameter. The new algorithm uses a different interpolating scheme than existing methods and introduces a unified iteration that naturally includes the so-called hard case. We show that the new iteration is well defined and convergent at a superlinear rate. We present computational results to illustrate convergence properties and robustness of the method.

A Trust-Region Approach to the Regularization of Large-Scale Discrete Forms of Ill-Posed Problems

We consider large-scale least squares problems where the coefficient matrix comes from the discretization of an operator in an ill-posed problem, and the right-hand side contains noise. Special techniques known as regularization methods are needed to treat these problems in order to control the effect of the noise on the solution. We pose the regularization problem as a quadratically constrained least squares problem. This formulation is equivalent to Tikhonov regularization, and we note that it is also a special case of the trust-region subproblem from optimization. We analyze the trust-region subproblem in the regularization case and we consider the nontrivial extensions of a recently developed method for general large-scale subproblems that will allow us to handle this case. The method relies on matrix-vector products only, has low and fixed storage requirements, and can handle the singularities arising in ill-posed problems. We present numerical results on test problems, on an inverse interpolation problem with field data, and on a model seismic inversion problem with field data.

An interior-point trust-region-based method for large-scale non-negative regularization

We present a new method for large-scale non-negative regularization based on a quadratically and non-negatively constrained quadratic problem. Such problems arise for example in the regularization of ill posed problems in image restoration where the matrices involved are very ill conditioned. The method is an interior-point iteration that requires the solution of a large-scale and possibly ill conditioned parametrized trust-region subproblem at each step. The method uses recently developed techniques for the large-scale trust-region subproblem. We describe the method and present preliminary numerical results on test problems and image restoration problems.

Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization

A MATLAB 6.0 implementation of the LSTRS method is presented. LSTRS was described in Rojas et al. [2000]. LSTRS is designed for large-scale quadratic problems with one norm constraint. The method is based on a reformulation of the trust-region subproblem as a parameterized eigenvalue problem, and consists of an iterative procedure that finds the optimal value for the parameter. The adjustment of the parameter requires the solution of a large-scale eigenvalue problem at each step. LSTRS relies on matrix-vector products only and has low and fixed storage requirements, features that make it suitable for large-scale computations. In the MATLAB implementation, the Hessian matrix of the quadratic objective function can be specified either explicitly, or in the form of a matrix-vector multiplication routine. Therefore, the implementation preserves the matrix-free nature of the method. A description of the LSTRS method and of the MATLAB software, version 1.2, is presented. Comparisons with other techniques and applications of the method are also included. A guide for using the software and examples are provided.

Accelerating the LSTRS Algorithm

In a process for the extraction of celluloses from lignocelluloses, the extraction is carried out by means of heating with aqueous acetic acid under pressure and the addition of formic acid, whereby there is obtained a cellulose with a very low residual lignin content, which can be bleached with ozone and peracetic acid to high grades of white, and acetic and formic acid are recovered by means of distillation, so that waste waters do not, therefore, accumulate.

Large-scale optimization techniques for nonnegative image restorations

We describe an optimization method for large-scale nonnegative regularization. The method is an interior-point iteration that requires the solution of a large-scale and possibly ill-conditioned parameterized trust-region subproblem at each step. The method relies on recently developed techniques for the large-scale trust-region subproblem. We present preliminary numerical results on image restoration problems.

A hybrid-optimization method for large-scale non-negative full regularization in image restoration

We describe a new hybrid-optimization method for solving the full-regularization problem of computing both the regularization parameter and the corresponding regularized solution in 1-norm and 2-norm Tikhonov regularizations with additional non-negativity constraints. The approach combines the simulated annealing technique for global optimization and the low-cost spectral projected gradient method for the minimization of large-scale smooth functions on convex sets. The new method is matrix-free in the sense that it relies on matrix–vector multiplications only. We describe the method and discuss some of its properties. Numerical results indicate that the new approach is a promising tool for solving large-scale image restoration problems.

Sampling techniques for monte carlo matrix multiplication with applications to image processing

Randomized algorithms for processing massive data sets have shown to be a promising alternative to deterministic techniques. Sampling strategies are an essential aspect of randomized algorithms for matrix computations. In this work, we show that strategies that are effective or even optimal in the general case, can fail when applied to ill-conditioned matrices. Our experimental study suggests that there exists a relationship between sampling performance and conditioning of the matrices involved. We present an explanation for this behavior and propose a novel, efficient, and accurate sampling strategy for randomized multiplication of affinity matrices in image segmentation.

The optimization problem in model-reduced gradient-based history matching

Abstract We present preliminary results of a performance evaluation study of several gradient-based state-of-the-art optimization methods for solving the nonlinear minimization problem arising in model-reduced gradient-based history matching. The issues discussed also apply to other areas, such as production optimization in closed-loop reservoir management.

Efficient Solution of the Optimization Problem in Model-reduced Gradient-based History Matching

Adjusting parameters in reservoir models by minimizing the discrepancy between the models predictions and actual measurements is a popular approach known as history matching. One of the most effective techniques is gradient-based history matching. For reservoir models, the number of grid blocks and therefore, the size of the problem can become very large. In recent years, model-order reduction techniques aiming to replace large, complex dynamic systems with lower-dimension models have been incorporated into history matching. In both gradient-based history matching and model-reduced gradient-based history matching, first-order optimization methods are used in order to minimize the mismatch between simulated well-production data and observed production. In this work, we investigate the performance of some optimization methods on the minimization problem in model-reduced gradient-based history matching. The methods were tested on the history matching of a small reservoir model with synthetic measurements. Our results show that fast first-order techniques such as the spectral projected gradient method can compete with the popular quasi-Newton BFGS approach.