Abstract.A fully variational approach is developed for solving nonlinear elliptic equations that enables accurate discretization and fast solution methods. The equations are converted to a first-order system that is then linearized via Newton's method. First-order system least squares (FOSLS) is used to formulate and discretize the Newton step, and the resulting matrix equation is solved using algebraic multigrid (AMG). The approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse-level computation. A general theory is developed that confirms the usual full multigrid efficiency: accuracy comparable to the finest-level discretization is achieved at a cost proportional to the number of finest-level degrees of freedom. In a companion paper, the theory is applied to elliptic grid generation (EGG) and supported by numerical results.
Abstract.A new fully variational approach is studied for elliptic grid generation (EGG). It is based on a general algorithm developed in a companion paper [A. L. Codd, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 41 (2003), pp. 2197-2209 that involves using Newton's method to linearize an appropriate equivalent first-order system, first-order system least squares (FOSLS) to formulate and discretize the Newton step, and algebraic multigrid (AMG) to solve the resulting matrix equation. The approach is coupled with nested iteration to provide an accurate initial guess for finer levels using coarse-level computation. The present paper verifies the assumptions of the companion work and confirms the overall efficiency of the scheme with numerical experiments.Key words. least-squares discretization, multigrid, nonlinear elliptic boundary value problems AMS subject classifications. 35J65, 65N15, 65N30, 65N50, 65F10 DOI. 10.1137/S00361429024044181. Introduction. A companion paper [10] develops an algorithm using Newton's method, first-order system least squares (FOSLS), and algebraic multigrid (AMG) for efficient solution of general nonlinear elliptic equations. The equations are first converted to an appropriate first-order system, and an approximate solution to the coarsest-grid problem is then computed (by any suitable method such as Newton iteration coupled perhaps with direct solvers, damping, or continuation). The approximation is then interpolated to the next finer level, where it is used as an initial guess for one Newton linearization of the nonlinear problem, with a few AMG cycles applied to the resulting matrix equation. This algorithm repeats itself until the finest grid is processed, again by one Newton/AMG step. At each Newton step, FOSLS is applied to the linearized system, and the resulting matrix equation is solved using just a few V-cycles of AMG.In the present paper, we apply this algorithm to elliptic grid generation (EGG) equations. Grid generation is usually based on a map between a relatively simple computational region and a possibly complicated physical region. It can be used numerically to create a mesh for a discretization method to solve a given system of equations posed on the physical domain. Alternatively, it can be used to transform equations posed on the physical region into ones posed on the computational region, where the transformed equations are then solved. If the Jacobian of the transformation is positive throughout the computational region, the equation type is unchanged [12]. Actually, the relative minimum value of the Jacobian is important in practice because relatively small values signal small angles between the grid lines and large errors in approximating the equations [20].
We present a new inversion method for Electrical Resistivity Tomography which, in contrast to established approaches, minimizes the cost function prior to finite element discretization for the unknown electric conductivity and electric potential. Minimization is performed with the Broyden-Fletcher-Goldfarb-Shanno method (BFGS) in an appropriate function space. BFGS is self-preconditioning and avoids construction of the dense Hessian which is the major obstacle to solving large 3-D problems using parallel computers. In addition to the forward problem predicting the measurement from the injected current, the so-called adjoint problem also needs to be solved. For this problem a virtual current is injected through the measurement electrodes and an adjoint electric potential is obtained. The magnitude of the injected virtual current is equal to the misfit at the measurement electrodes. This new approach has the advantage that the solution process of the optimization problem remains independent to the meshes used for discretization and allows for mesh adaptation during inversion. Computation time is reduced by using superposition of pole loads for the forward and adjoint problems. A smoothed aggregation algebraic multigrid (AMG) preconditioned conjugate gradient is applied to construct the potentials for a given electric conductivity estimate and for constructing a first level BFGS preconditioner. Through the additional reuse of AMG operators and coarse grid solvers inversion time for large 3-D problems can be reduced further. We apply our new inversion method to synthetic survey data created by the resistivity profile representing the characteristics of subsurface fluid injection. We further test it on data obtained from a 2-D surface electrode survey on Heron Island, a small tropical island off the east coast of central Queensland, Australia.
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