Abstract
The paper develops a new algorithm for parameter identification in a partial differential equation associated with an inhomogeneous aquifer system. The parameters chosen for identification are the storage coefficient, a constant, and transmissivities, functions of the space variable. An implicit finite-difference scheme is used to approximate the solutions of the governing equation. A least-squares criterion is then established. Using distributed observations on the dependent variable within the system, parameters are identified directly by solving a sequence of quadratic programming problems such that the final solution converges to the original problem. The advantages of this new algorithm problem. The advantages of this new algorithm include rapid rate of convergence, ability to handle any inequality constraints, and easy computer implementation. The numerical example presented demonstrates the simultaneous identification of 12 parameters in only seconds of computer time. parameters in only seconds of computer time
Introduction
Simulation and mathematical models are often used in analyzing aquifer systems. Physically based mathematical models are implemented by high-speed computers. Most models are of a parametric type, in which parameters used in parametric type, in which parameters used in deriving the governing equation are not measurable directly from the physical point of view and, therefore, must be determined from historical records. The literature dealing with parameter identification in partial differential equations has become available only within the last decade. The approaches include gradient searching procedures, the balanced error-weighted gradient method, the classical Gauss-Newton least-squares procedure, optimal control and gradient optimization, quasi linearization, influence coefficient algorithm, linear programming, maximum principle, and regression analysis allied with the steepest-descent algorithm. Yeh analyzed a typical parameter identification problem governed by a second-order, nonlinear, parabolic partial differential equation using five different methods (the gradient method, quasilinearization, maximum principle, influence coefficient method, and linear programming) and then compared these methods.
The problem under consideration is that of an unsteady radial flow in a confined aquifer system. The governing equation is(1)1 h h--- --- rT(r) ----- = S ------ + Q, r r r t
subject to the following initial and boundary conditions:0t = 0, h = h, 0 < r < re(2)r= 0, h = h (t), t >00Bhr = r ----- = 0, t >0e r
Eq. 1 represents a distributed system in which parameters are functions of the space variable. The parameters are functions of the space variable. The assumptions used in deriving Eq. 1 include (1) the aquifer is confined with a constant depth, b; (2) the aquifer overlays on an infinite horizontal impermeable bed; (3) the Dupuit-Forchheimer assumptions are valid; and (4) water is released instantaneously because of the change of the flow potential.
SPEJ
P. 371
A new regularization method for a parameter identification problem in a non-linear partial differential equation