A Successive Linear Programming Approach for Initialization and Reinitialization after Discontinuities of Differential-Algebraic Equations

Vipin Gopal, Lorenz T. Biegler
1998 SIAM Journal on Scientific Computing  
Determination of consistent initial conditions is an important aspect of the solution of differential algebraic equations (DAEs). Specification of inconsistent initial conditions, even if they are slightly inconsistent, often leads to a failure in the initialization problem. In this paper, we present a Successive Linear Programming (SLP) approach for the solution of the DAB derivative array equations for the initialization problem. The SLP formulation handles roundoff errors and inconsistent
more » ... r specifications among others and allows for reliable convergence strategies that incorporate variable bounds and trust region concepts. A new consistent set of initial conditions is obtained by minimizing the deviation of the variable values from the specified ones. For problems with discontinuities caused by a step change in the input functions, a new criterion is presented for identifying the subset of variables which are continuous across the discontinuity. The LP formulation is then applied to determine a consistent set of initial conditions for further solution of the problem in the domain after the discontinuity. Numerous example problems are solved to illustrate these concepts. Pantelides (1988) utilizes the property that the hidden or latent equations are generated from those subsets of equations in whidi the number of variables (x\y) present in the subset is less than the number of equations themselves. He uses a graph-theoretical algorithm to locate the minimally structurally singular subsets and differentiates them. However, the algorithm is structural and subsets of equations which are numerically singular might escape detection. Unger et al. (1995) also present an algorithm based on structural analysis. They determine the structural rank and the structural index of the problem by using a structural version of the symbolic algorithm for index reduction by Gear (1988). Gear's algorithm is an extension of the algorithm by Bachmann et al. (1990) for linear systems. Here the structural rank and index of the DAE (1) is solely determined by the patterns of dF/dv 1 and dF/dv. However the structurally determined index and degrees of freedom will only provide a lower bound and upper bound respectively for their corresponding differential quantities. They also proposed a combination of structural methods with numerical/symbolic methods to overcome these difficulties as a direction for future research. INITIALIZATION AND REINITIALIZATION OF DAES Campbell and Moore (1994) proposed a method for solving nonlinear, high index problems for which BDF and Implicit RK methods might be inappropriate. They define the derivative array equations F"(a:,z,z',...,*V) = 0 (4) as the set of equations derived by differentiating the original DAE (1) v number of times where v is the index of (1) and * = {*',y} (5) They solve F ¥ = 0 by a least squares iterative solution. The method was found to be computationally expensive especially for computing the singular value decomposition of JF» (the Jacobian of the derivative array equations) used for the least squares solution. Leimkuhler et al. (1991) characterize the consistency requirement by posing a set of equations which comprise the derivative array equations (4) (defined at time to) and the set of user specified information on initial conditions (6). £(*<>, z o ,y o )=0 (6) The higher derivatives are approximated by forward finite differences. The resulting approximate system, however, might not have a solution and hence is solved in a least squares sense. This is complicated as well because of the rank deficiency of the Jacobian. The method is illustrated for index one and semi explicit index 2 problems in the triangular form. Chung and Westerberg (1990) and Chung (1991) proposed a numerical algorithm for consistent initialization by identifying singular subsets of equations from the Jacobian and symbolically differentiating them. Majer et al. (1995) considers the problem of reinitialization of DAEs after discontinuities which we also discuss in detail in section 3. Leimkuhler et al. (1991) defines the consistent initialization problem as: given specified information about the initial state of the problem Uiat is sufficient to specify a unique solution to a DAE, determine the complete vector (t/(t<>), v'(to)). However, for a particular problem, the user may not know which variables to specify to sufficiently determine an exact solution for the initial condition vector. In other words, the user does not know the degrees of freedom a priori. On the other hand, the user may know a certain set of specifications and we would like our initial conditions to be as dose to those given by the user (whether it is over or exactly specified). Many approaches analyze the set of equations and ask the user for the initial conditions for a particular set of variables, which he might not have. Hence, it would make sense to work the other way. Given the V. GOPAL AND L. T. BIEGLER known set of initial conditions we would like to find a consistent initial vector (x',x,y) closest to the known set. In many practical applications, the exact numerical value of the initial conditions may not be known for satisfying the relevant set of equations within the specified error criteria. The minimization formulation which we present in section 2 has its significance in this context. The method is also relevant when there are roundoff errors. In die next section we develop a successive linear programming (SLP) formulation for the consistent initialization of DAE systems. The motivation for this approach is that the derivative array equations require solution of an underdetermined system, but roundoff errors and incomplete user specifications may render this system inconsistent. An SLP formulation handles this underdetermined nature efficiently and allows for reliable convergence strategies that incorporate variable bounds and trust region concepts. Section 3 deals with the related problem of reinitialization after input discontinuities are encountered for the DAE system. Here an analysis on the continuity of the state profiles is presented and compatible SLP formulations are derived for this problem. The SLP formulations in both sections are illustrated with the successful treatment of numerous high index examples. Section 4 concludes the paper and outlines directions for future work.
doi:10.1137/s1064827596307725 fatcat:om3lwxtejzcwhbajmvl2vlxqja