Gerardo Toraldo

Daniela di Serafino, Gerardo Toraldo, Marco Viola et al.

We propose a gradient-based method for quadratic programming problems with a single linear constraint and bounds on the variables. Inspired by the GPCG algorithm for bound-constrained convex quadratic programming [J.J. Moré and G. Toraldo, SIAM J. Optim. 1, 1991], our approach alternates between two phases until convergence: an identification phase, which performs gradient projection iterations until either a candidate active set is identified or no reasonable progress is made, and an unconstrained minimization phase, which reduces the objective function in a suitable space defined by the identification phase, by applying either the conjugate gradient method or a recently proposed spectral gradient method. However, the algorithm differs from GPCG not only because it deals with a more general class of problems, but mainly for the way it stops the minimization phase. This is based on a comparison between a measure of optimality in the reduced space and a measure of bindingness of the variables that are on the bounds, defined by extending the concept of proportioning, which was proposed by some authors for box-constrained problems. If the objective function is bounded, the algorithm converges to a stationary point thanks to a suitable application of the gradient projection method in the identification phase. For strictly convex problems, the algorithm converges to the optimal solution in a finite number of steps even in case of degeneracy. Extensive numerical experiments show the effectiveness of the proposed approach.

Daniela Calvetti, Salvatore Cuomo, Monica Pragliola et al.

The Linear Multistep Method Particle Filter (LMM PF) is a method for predicting the evolution in time of a evolutionary system governed by a system of differential equations. If some of the parameters of the governing equations are unknowns, it is possible to organize the calculations so as to estimate them while following the evolution of the system in time. The underlying assumption in the approach that we present is that all unknowns are modelled as random variables, where the randomness is an indication of the uncertainty of their values rather than an intrinsic property of the quantities. Consequently, the states of the system and the parameters are described in probabilistic terms by their density, often in the form of representative samples. This approach is particularly attractive in the context of parameter estimation inverse problems, because the statistical formulation naturally provides a means of assessing the uncertainty in the solution via the spread of the distribution. The computational efficiency of the underlying sampling technique is crucial for the success of the method, because the accuracy of the solution depends on the ability to produce representative samples from the distribution of the unknown parameters. In this paper LMM PF is tested on a skeletal muscle metabolism problem, which was previously treated within the Ensemble Kalman filtering framework. Here numerical evidences are used to highlight the correlation between the main sources of errors and the influence of the linera multistep method adopted. Finally, we analyzed the effect of replacing LMM with Runge-Kutta class integration methods for supporting the PF technique.