Chao-Yang Gau

Reliable nonlinear parameter estimation in VLE modeling

Abstract The reliable solution of nonlinear parameter estimation problems is an important computational problem in the modeling of vapor–liquid equilibrium (VLE). Conventional solution methods may not be reliable since they do not guarantee convergence to the global optimum sought in the parameter estimation problem. We demonstrate here a technique that is based on interval analysis, which can solve the nonlinear parameter estimation problem with complete reliability, and provides a mathematical and computational guarantee that the global optimum is found. As an example, we consider the estimation of parameters in the Wilson equation, using VLE data sets from a variety of binary systems. Results indicate that several sets of parameter values published in the DECHEMA VLE Data Collection correspond to local optima only, with new globally optimal parameter values found by using the interval approach. When applied to VLE modeling, the globally optimal parameters can provide significant improvements in predictive capability. For example, in one case, when the previously published locally optimal parameters are used, the Wilson equation does not predict experimentally observed homogeneous azeotropes, but, when the globally optimal parameters are used, the azeotropes are predicted.

Reliable nonlinear parameter estimation using interval analysis: error-in-variable approach

Abstract Parameter estimation is a key problem in the development of process models, both steady- and unsteady-state, and thus is an important issue in both process design and control. The error-in-variable (EIV) approach differs distinctly from the standard approach in that measurement errors in both dependent and independent system variables are taken into account when formulating the objective function in the parameter estimation problem. It is not uncommon for the objective function in nonlinear parameter estimation problems to have multiple local optima. However, the usual methods used to solve these problems are local methods that offer no guarantee that the global optimum, and thus the best set of model parameters, has been found. We demonstrate here a technique, based on interval analysis, that can solve the EIV parameter estimation problem with complete reliability, providing a mathematical and computational guarantee that the global optimum is found. As examples, we consider the estimation of parameters in both steady and unsteady-state models, including a vapor—liquid equilibrium (VLE) model, a CSTR model, and a reaction kinetics model.

New interval methodologies for reliable chemical process modeling

The use of interval methods, in particular interval-Newton/generalized-bisection (IN/GB) techniques, provides an approach that is mathematically and computationally guaranteed to reliably solve difficult nonlinear equation solving and global optimization problems, such as those that arise in chemical process modeling. The most significant drawback of the currently used interval methods is the potentially high computational cost that must be paid to obtain the mathematical and computational guarantees of certainty. New methodologies are described here for improving the efficiency of the interval approach. In particular, a new hybrid preconditioning strategy, in which a simple pivoting preconditioner is used in combination with the standard inverse-midpoint method, is presented, as is a new scheme for selection of the real point used in formulating the interval-Newton equation. These techniques can be implemented with relatively little computational overhead, and lead to a large reduction in the number of subintervals that must be tested during the interval-Newton procedure. Tests on a variety of problems arising in chemical process modeling have shown that the new methodologies lead to substantial reductions in computation time requirements, in many cases by multiple orders of magnitude.

Dynamic load balancing for parallel interval-Newton using message passing

Branch-and-prune and branch-and-bound techniques are commonly used for intelligent search in finding all solutions, or the optimal solution, within a space of interest. The corresponding binary tree structure provides a natural parallelism allowing concurrent evaluation of subproblems using parallel computing technology. Of special interest here are techniques derived from interval analysis, in particular an interval-Newton/generalized-bisection procedure. In this context, we discuss issues of load balancing and work scheduling that arise in the implementation of parallel interval-Newton on a cluster of workstations using message passing, and describe and analyze techniques for this purpose. Results using an asynchronous diffusive load balancing strategy show that a consistently high efficiency can be achieved in solving nonlinear equations, providing excellent scalability, especially with the use of a two-dimensional torus virtual network. The effectiveness of the approach used, especially in connection with a novel stack management scheme, is also demonstrated in the consistent superlinear speedups observed in performing global optimization.

Parallel Branch-and-Bound for Chemical Engineering Applications: Load Balancing and Scheduling Issues

Branch-and-prune (BP) and branch-and-bound (BB) techniques are commonly used for intelligent search in finding all solutions, or the optimal solution, within a space of interest. The corresponding binary tree structure provides a natural parallelism allowing concurrent evaluation of subproblems using parallel computing technology. Of special interest here are techniques derived from interval analysis, in particular an interval-Newton/generalized-bisection procedure. In this context, we discuss issues of load balancing and work scheduling that arise in the implementation of parallel BB and BP, and describe and analyze techniques for this purpose. These techniques are applied to solve problems appearing in chemical process engineering using a distributed parallel computing system. Results show that a consistently high efficiency can be achieved in solving nonlinear equations, providing excellent scalability. The effectiveness of the approach used is also demonstrated in the consistent superlinear speedup observed in performing global optimization.

Parallel Interval-Newton Using Message Passing: Dynamic Load Balancing Strategies

Branch-and-prune and branch-and-bound techniques are commonly used for intelligent search in finding all solutions, or the optimal solution, within a space of interest. The corresponding binary tree structure provides a natural parallelism allowing concurrent evaluation of subproblems using parallel computing technology. Of special interest here are techniques derived from interval analysis, in particular an interval-Newton/generalized-bisection procedure. In this context, we discuss issues of load balancing and work scheduling that arise in the implementation of parallel interval-Newton on a cluster of workstations using message passing, and describe and analyze techniques for this purpose. Results using an asynchronous diffusive load balancing strategy show that a consistently high efficiency can be achieved in solving non-linear equations, providing excellent scalability, especially with the use of a two-dimensional torus virtual network. The effectiveness of the approach used, especially in connection with a novel stack management scheme, is also demonstrated in the consistent superlinear speedups observed in performing global optimization.