Greedy kernel methods for accelerating implicit integrators for parametric ODEs
This work addresses the computational bottleneck of solving parametric ODEs with implicit integrators, offering a general acceleration framework that preserves accuracy.
The paper proposes a greedy kernel-based surrogate model to accelerate implicit integrators for parametric ODEs by providing a good initial guess for the nonlinear solver, reducing iterations and achieving speedup while maintaining solution accuracy. For the Burgers equation with Implicit Euler, the method reduces solver iterations and yields overall simulation speedup.
We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method's features.