اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدGreedy kernel methods for accelerating implicit integrators for parametric ODEs
60
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
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 methods features.
قيم البحث
اقرأ أيضاً
We study ways to accelerate greedy coordinate descent in theory and in practice, where accelerate refers either to $O(1/k^2)$ convergence in theory, in practice, or both. We introduce and study two algorithms: Accelerated Semi-Greedy Coordinate Desce
nt (ASCD) and Accelerated Greedy Coordinate Descent (AGCD). While ASCD takes greedy steps in the $x$-updates and randomized steps in the $z$-updates, AGCD is a straightforward extension of standard greedy coordinate descent that only takes greedy steps. On the theory side, our main results are for ASCD: we show that ASCD achieves $O(1/k^2)$ convergence, and it also achieves accelerated linear convergence for strongly convex functions. On the empirical side, we observe that both AGCD and ASCD outperform Accelerated Randomized Coordinate Descent on a variety of instances. In particular, we note that AGCD significantly outperforms the other accelerated coordinate descent methods in numerical tests, in spite of a lack of theoretical guarantees for this method. To complement the empirical study of AGCD, we present a Lyapunov energy function argument that points to an explanation for why a direct extension of the acceleration proof for AGCD does not work; and we also introduce a technical condition under which AGCD is guaranteed to have accelerated convergence. Last of all, we confirm that this technical condition holds in our empirical study.
The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy rando
mized Motzkin-Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices, and the blo
Time integration methods for solving initial value problems are an important component of many scientific and engineering simulations. Implicit time integrators are desirable for their stability properties, significantly relaxing restrictions on time
step size. However, implicit methods require solutions to one or more systems of nonlinear equations at each timestep, which for large simulations can be prohibitively expensive. This paper introduces a new family of linearly implicit multistep methods (LIMM), which only requires the solution of one linear system per timestep. Order conditions and stability theory for these methods are presented, as well as design and implementation considerations. Practical methods of order up to five are developed that have similar error coefficients, but improved stability regions, when compared to the widely used BDF methods. Numerical testing of a self-starting variable stepsize and variable order implementation of the new LIMM methods shows measurable performance improvement over a similar BDF implementation.
A novel class of high-order linearly implicit energy-preserving exponential integrators are proposed for the nonlinear Schrodinger equation. We firstly done that the original equation is reformulated into a new form with a modified quadratic energy b
y the scalar auxiliary variable approach. The spatial derivatives of the system are then approximated with the standard Fourier pseudo-spectral method. Subsequently, we apply the extrapolation technique to the nonlinear term of the semi-discretized system and a linearized system is obtained. Based on the Lawson transformation, the linearized system is rewritten as an equivalent one and we further apply the symplectic Runge-Kutta method to the resulting system to gain a fully discrete scheme. We show that the proposed scheme can produce numerical solutions along which the modified energy is precisely conserved, as is the case with the analytical solution and is extremely efficient in the sense that only linear equations with constant coefficients need to be solved at every time step. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other high-order structure-preserving method.
This paper studies fixed-step convergence of implicit-explicit general linear methods. We focus on a subclass of schemes that is internally consistent, has high stage order, and favorable stability properties. Classical, index-1 differential algebrai
c equation, and singular perturbation convergence analyses results are given. For all these problems IMEX GLMs from the class of interest converge with the full theoretical orders under general assumptions. The convergence results require the time steps to be sufficiently small, with upper bounds that are independent on the stiffness of the problem.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
