A wavelet approach to adjoint state sensitivity computation for steady state differential equations Abeeb A. Awotunde 1,2 and Roland N. Horne 1 Received 1 February 2010 ; revised 30 October 2010 ; accepted 10 November 2010 ; published 1 March 2011. [1] The computation of the sensitivity matrix is the most time-consuming part of any parameter estimation algorithm that requires sensitivity coefficients. An efficient wavelet approach to adjoint sensitivity analysis is proposed to reduce the computational cost of obtaining sensitivity coefficients. The method exploits a wavelet reduction of the data space to reduce the size of the linear system encountered in steady state adjoint equations. In this regard, wavelet transform is used as a data compression tool. Numerical examples applied to spatial data are used to verify and show the effectiveness of the method. Citation: Awotunde, A. A., and R. N. Horne (2011), A wavelet approach to adjoint state sensitivity computation for steady state differential equations, Water Resour. Res., 47, W03502, doi:10.1029/2010WR009165. 1. Introduction [2] Sensitivity analysis of large-scale systems governed by differential equations has continued to be of importance in groundwater modeling and parameter estimation [ Carter et al., 1974; Sun and Yeh, 1985; Yeh, 1986; Carrera and Neuman, 1986; Yeh and Sun, 1990; Sun and Yeh, 1990a, 1990b]. Applications of such analysis cover a wide spec- trum, including optimization, optimal control, model repar- ameterization, uncertainty analysis, and experimental design. However, the cost of computing sensitivity coeffi- cients often poses a challenge. This cost may be central to determining the choice of optimization method to use in pa- rameter estimation. When the computational overhead of sensitivity calculation becomes prohibitively high, methods such as the conjugate gradient and quasi-Newton, which avoid this computation, are often used in place of gradient- based methods such as Gauss-Newton and Levenberg- Marquardt. [3] Jacquard and Jain [1965] presented a procedure for numerically computing the sensitivity coefficients for his- tory matching and applied the procedure to estimate per- meability in a two-dimensional reservoir from pressure data. Subsequently, Carter et al. [1974] presented a deri- vation of the method to compute the sensitivity coeffi- cients for two-dimensional single-phase flow problems. Chen et al. [1974] and Chavent et al. [1975] independ- ently proposed the optimal control method to calculate the gradient of the objective function with respect to model parameters for single-phase flow. Wasserman et al. [1975] extended the optimal control theory to automatic history matching in a multiphase reservoir but only computed the adjoint variables for the overall pressure equation and used an objective function based on only the pressure mis- match term. Later, Carrera and Neuman [1986] and Sun and Yeh [1990a] used the optimal control theory to solve the parameter identification problem for groundwater flow. A detailed review of the parameter identification proce- dures in groundwater hydrology was done by Yeh [1986]. Wu et al. [1999] later derived the adjoint equations for multiphase flow in a hydrocarbon reservoir, but the com- putational cost is still very high when the number of data is large. [4] Efforts have been vested in finding cheaper methods of computing the sensitivity matrix without compromising the accuracy of the solution. One method, the forward sen- sitivity analysis (also known as the gradient simulator method), is very efficient when the model space is small [Yeh, 1986; Tang et al., 1989; Landa, 1997]. This is because this method requires the solution of a linear system with multiple right-hand-side vectors. The number of right- hand-side vectors is exactly equal to the number of model parameters. Moreover, in the forward sensitivity method, the sensitivities of all grid block variables are computed. This is inefficient because only the sensitivities of variables at measurement locations are required. For high-dimen- sional problems, this method becomes prohibitively expen- sive. Model space reduction via reparameterization [Oldenburg et al., 1993; Reynolds et al., 1996; Lu and Horne, 2000; Sahni and Horne, 2006; Sarma et al., 2008] is often coupled with the forward sensitivity method to sta- bilize the algorithm and speed up the computation of sensi- tivity coefficients. [5] Another method, the adjoint approach [ Shah et al., 1978; Anterion et al., 1989; Plessix, 2006; Michalak and Kitanidis, 2004; Li and Petzold, 2004], is commonly used to compute the sensitivity coefficients and the gradient of the objective function. The adjoint method of sensitivity computation is particularly useful when the number of data is relatively small. This method is also based on solving a linear system with multiple right-hand-side vectors. How- ever, the number of right-hand-side vectors in this case is equal to the number of data for which sensitivities are to be calculated. The number is therefore independent of the number of parameters. This approach is preferred to the 1 Department of Energy Resources Engineering, Stanford University, Stanford, California, USA. 2 Now at Department of Petroleum Engineering, KFUPM, Dhahran, Saudi Arabia. Copyright 2011 by the American Geophysical Union. 0043-1397/11/2010WR009165 W03502 1 of 21 WATER RESOURCES RESEARCH, VOL. 47, W03502, doi :10.1029/2010WR009165, 2011