Susana Gómez

Multiphase equilibria calculation by direct minimization of Gibbs free energy with a global optimization method

Abstract This paper presents a new method for multiphase equilibria calculation by direct minimization of the Gibbs free energy of multicomponent systems. The methods for multiphase equilibria calculation based on the equality of chemical potentials cannot guarantee the convergence to the correct solution since the problem is non-convex (with several local minima), and they can find only one for a given initial guess. The global optimization methods currently available are generally very expensive. A global optimization method called Tunneling, able to escape from local minima and saddle points is used here, and has shown to be able to find efficiently the global solution for all the hypothetical and real problems tested. The Tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the objective function. In phase two (tunnelization), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used. The calculation of multiphase equilibria is organized in a stepwise manner, combining phase stability analysis by minimization of the tangent plane distance function with phase splitting calculations. The problems addressed here are the vapor–liquid and liquid–liquid two-phase equilibria, three-phase vapor–liquid–liquid equilibria, and three-phase vapor–liquid–solid equilibria, for a variety of representative systems. The examples show the robustness of the proposed method even in the most difficult situations. The Tunneling method is found to be more efficient than other global optimization methods. The results showed the efficiency and reliability of the novel method for solving the multiphase equilibria and the global stability problems. Although we have used here a cubic equation of state model for Gibbs free energy, any other approach can be used, as the method is model independent.

The optimal geometry of Lennard-Jones clusters: 148–309

This paper deals with the global optimization problem of determining the n-atom cluster configuration that yields the minimum Lennard-Jones potential energy. To approach this problem we propose a genetic algorithm combined with a stochastic search procedure on icosahedral lattices. Although the potentials obtained with our method for nD 148;:::;309 are in fact only upper bounds for the global minima, we believe that most of these upper bounds are tight. We provide a geometrical description of the optimal configurations found, whose structures are either icosahedral or Marks decahedral in character. Also, we were able to discover a novel morphology ‐ called FD here ‐ for Lennard-Jones atomic clusters.

Migration‐based traveltime waveform inversion of 2-D simple structures: A synthetic example

Migration-based traveltime (MBTT) formulation provides algorithms for automatically determining background velocities from full-waveform surface seismic reflection data using local optimization methods. In particular, it addresses the difficulty of the nonconvexity of the least-squares data misfit function. The method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity. We present an implementation of the MBTT approach for a 2-D finite-difference (FD) full-wave acoustic model. Numerical analysis on a 2-D synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least-squares approach, even when starting from very poor initial guesses. This enlargement of the domain of attraction for the global minima of the least-squares misfit has a price: each evaluation of the new objective function requires, besides the usual FD full-wave forward modeling, an additional full-wave prestack migration. Hence, the FD implementation of the MBTT approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size.

The triangle method for finding the corner of the L-curve

The Conjugate Gradient Method (CG) has an intrinsic regularization property when applied to systems of linear equations with ill-conditioned matrices. This regularization property is specially useful when either the right-hand side or the coefficient matrix, or both are given with errors. The regularization parameter is the iteration number, and in order to find this parameter, the L-curve is used. Here we present a novel method to find the corner of the L-curve, that determines the regularizing iteration number. Numerical results on the collection of test problems [SIAM J. Sci. Comput. 16 (1995) 506-512] are given to illustrate the potentiality of the method.

Phase stability analysis with cubic equations of state by using a global optimization method

Abstract In this paper, we propose a new method for phase stability analysis with cubic equations of state by minimization of the tangent plane distance (TPD) function. A global optimization method called tunneling, able to escape from local minima and saddle points is used here. The tunneling method has two phases. In phase one, a local bounded optimization method is used to minimize the TPD function. In phase two (tunneling), either global optimality is ascertained, or a feasible initial estimate for a new minimization is generated. For the minimization step, a limited-memory quasi-Newton method is used. The tunneling method is used for the conventional approach, and for the reduced variables approach. In the latter case, the number of independent variables does not depend on the number of components in the mixture. The solution of the minimization of TPD function should be found in a space with a significantly reduced number of dimensions. The new method is used for testing phase stability for a variety of representative systems. The examples show the robustness of the method even in the most difficult situations. The proposed method is more efficient than other global optimization methods. Furthermore, in many cases, the reduced approach is faster than the conventional approach. The results showed the efficiency and reliability of the novel method for solving the global stability problem.

Gradient-Based History-Matching with a Global Optimization Method

We investigate the application of a global optimization algorithm called the Tunneling Method to the problem of history-matching of petroleum reservoirs. Results are presented for two test cases. The first is a small synthetic case in which the global minimum is known. The second is a real field example. In both cases, a series of minima was found. The computational cost of each tunneling phase is found to be comparable with the cost of each local minimization. It is concluded that the Tunneling Method may have a practical application in history-matching as an alternative to immediate reformulation of the problem if the first minimum found does not represent an acceptable match.

A genetic algorithm for Lennard-Jones atomic clusters

In regard to the problem of determining minimum L-J configurations for clusters of n atoms, we present here a genetic algorithm able to reproduce all best-known solutions in the 13 < n < 147 size range. These include not only the classical structures adhering to the icosahedral-growth, but also seven icosahedral structures with incomplete core, six more following the Marks decahedron geometry, and the unique (n = 38) face-centered cubic configuration that has been found in this range.

Medium-Term Effects of Septal and Apical Pacing in Pacemaker-Dependent Patients: A Double-Blind Prospective Randomized Study

Pacing the right ventricle is established practice, but there remains controversy as to the optimal site to preserve hemodynamic function.

Reconstruction of capacitance tomography images of simulated two-phase flow regimes

Reconstruction of electrical capacitance tomography (ECT) images is performed using simulated gas-oil distributions. An inverse problem has to be solved to find the permittivity coefficient, using measurements of the capacitances. The least squares optimal solution is sought using a Gauss-Newton method, with a sufficient descent condition and a backtracking for the steplength. The Tikhonov regularisation method is used, to control the measurement error propagation due to the ill-posednes of the inverse problem. It is shown that the reconstruction is very sensitive to the Tikhonov regularisation parameter and the L-curve method to find its value is used. When the optimal regularisation parameter is used, convergence is attained to points where no further precision in the permittivity parameter is possible. Simulation examples using typical two-phase flow regimes are presented, and the approximated images as well as the range of values for the regularization parameter for different regimes are shown.

Archimedean polyhedron structure yields a lower energy atomic cluster

Abstract Icosahedral structures are generally accepted as the lowest energy geometries for clusters of 13 ≤ n ≤ 147 atoms of the same type. Here we propose a novel structure for the 38-atoms cluster, possessing remarkable characteristics: 1. (i) its Lennard-Jones potential energy is below the minimum reported in the literature; 2. (ii) as it can be embedded in a face-centered cubic lattice, it does not present five-fold symmetries; 3. (iii) its core is composed by six atoms at the vertices of an octahedron, decorated by 32 additional atoms to exhibit the shape of a tetrakaidecahedron , one of the beautiful Archimedean polyhedra. Our findings arose by linking the exponential tunneling method for global optimization with geometrical intuition to provide promisory starting configurations.