5:15 PM - 7:15 PM
[MGI30-P04] Attempt to estimate initial guesses for iterative solvers using polynomial extrapolation: Toward improving convergence of 3-D simulations of mantle convection
Keywords:iterative methods, Lagrange interpolation, mantle convection
In this study, we attempt to reduce the computational costs of the iterative methods, which are widely used to solve nonlinear or linear simultaneous equations. Our ultimate goal is to help speed-up 3-D numerical simulations of mantle convection in a simple "data-driven" manner, most of whose computational time (typically more than 90 %) is spent in the iterative solutions for (large-scale) linear simultaneous equations describing the flow fields of highly viscous incompressible (or anelastic) fluids at each time instance.
Let us consider an iterative solution method to find the data f1 at a future time instance t=t1, assuming that the data f0 at present (t=t0) as well as those of several past time instances (f-1, f-2, and so on) are known. If a value of F1 close to the true solution f1 can be successfully estimated, the iterative solvers starting with F1 are expected to reach f1 with small computational costs. Our strategy is to estimate the initial guess F1 (≃f1) by constructing the Lagrange polynomial spanning from t=t0 to some pasts; we estimate the value of F1 by extrapolating the polynomial to the time instance t=t1.
In order to verify the usefulness of the method proposed here, we conducted numerical experiments of mantle convection in 3-D Cartesian domain. The results showed that, by estimating the initial guesses of flow fields from extrapolations either with linear or quadratic polynomials, the number of iterations to convergence can be reduced to about half those of the cases without any extrapolation. A detailed comparison of the convergence of the two methods shows that for the cases with linear extrapolations the numbers of iterations increase for the time instances where the flow fields substantially change with time, whereas the iterative solvers with quadratic extrapolations arrived at convergent solutions at almost same computational costs throughout the calculations. We also found that extrapolations with third-order polynomials or even higher ones deteriorate the convergence of iterative solvers. This is probably because the higher-order polynomials have coefficients with larger absolute values, which in turn degrades the predictions by extrapolations.
Let us consider an iterative solution method to find the data f1 at a future time instance t=t1, assuming that the data f0 at present (t=t0) as well as those of several past time instances (f-1, f-2, and so on) are known. If a value of F1 close to the true solution f1 can be successfully estimated, the iterative solvers starting with F1 are expected to reach f1 with small computational costs. Our strategy is to estimate the initial guess F1 (≃f1) by constructing the Lagrange polynomial spanning from t=t0 to some pasts; we estimate the value of F1 by extrapolating the polynomial to the time instance t=t1.
In order to verify the usefulness of the method proposed here, we conducted numerical experiments of mantle convection in 3-D Cartesian domain. The results showed that, by estimating the initial guesses of flow fields from extrapolations either with linear or quadratic polynomials, the number of iterations to convergence can be reduced to about half those of the cases without any extrapolation. A detailed comparison of the convergence of the two methods shows that for the cases with linear extrapolations the numbers of iterations increase for the time instances where the flow fields substantially change with time, whereas the iterative solvers with quadratic extrapolations arrived at convergent solutions at almost same computational costs throughout the calculations. We also found that extrapolations with third-order polynomials or even higher ones deteriorate the convergence of iterative solvers. This is probably because the higher-order polynomials have coefficients with larger absolute values, which in turn degrades the predictions by extrapolations.