Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For a $s$-stage IRK scheme used to integrate a $d$-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same $sd \times sd$ real coefficient matrix. We propose rewriting such $sd$-dimensional linear systems as an equivalent $(s+1)d$-dimensional systems that can be solved by performing the LU decompositions of $[s/2] +1$ real matrices of size $d \times d$. We present a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation

Inexact Newton based Lifted Implicit Integrators for fast Nonlinear MPC

Nonlinear Model Predictive Control (NMPC) requires the online solution of an Optimal Control Problem (OCP) at every sampling instant. In the context of multiple shooting, a numerical integration is needed to discretize the continuous time dynamics. For stiff, implicitly defined or differential-algebraic systems, implicit schemes are preferred to carry out the integration. The Newton-type optimization method and the implicit integrator then form a nested Newton scheme, solving the optimization and integration problem on two different levels. In recent research, an exact lifting technique was proposed to improve the computational efficiency of the latter framework. Inspired by that work, this paper presents a novel class of lifted implicit integrators, using an inexact Newton method. An additional iterative scheme for computing the sensitivities is proposed, which provides similar properties as the exact lifted integrator at considerably reduced computational costs. Using the example of an industrial robot, computational speedups of up to factor 8 are reported. The proposed methods have been implemented in the open-source ACADO code generation software

Inexact Newton based Lifted Implicit Integrators for fast Nonlinear MPC

Nonlinear Model Predictive Control (NMPC) requires the online solution of an Optimal Control Problem (OCP) at every sampling instant. In the context of multiple shooting, a numerical integration is needed to discretize the continuous time dynamics. For stiff, implicitly defined or differential-algebraic systems, implicit schemes are preferred to carry out the integration. The Newton-type optimization method and the implicit integrator then form a nested Newton scheme, solving the optimization and integration problem on two different levels. In recent research, an exact lifting technique was proposed to improve the computational efficiency of the latter framework. Inspired by that work, this paper presents a novel class of lifted implicit integrators, using an inexact Newton method. An additional iterative scheme for computing the sensitivities is proposed, which provides similar properties as the exact lifted integrator at considerably reduced computational costs. Using the example of an industrial robot, computational speedups of up to factor 8 are reported. The proposed methods have been implemented in the open-source ACADO code generation software

Efficient implementation of symplectic implicit Runge-Kutta schemes with simplified Newton iterations

We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For a s-stage IRK scheme used to integrate a d-dimensional system of ordinary differential equations, the application of simplified versions of Newton iterations requires solving at each step several linear systems (one per iteration) with the same sd × sd real coefficient matrix. We propose rewriting such sd-dimensional linear systems as an equivalent (s + 1)d-dimensional systems that can be solved by performing the LU decompositions of [s/2] + 1 real matrices of size d × d. We present a C implementation (based on Newton-like iterations) of Runge-Kutta collocation methods with Gaussian nodes that make use of such a rewriting of the linear system and that takes special care in reducing the effect of round-off errors. We report some numerical experiments that demonstrate the reduced round-off error propagation of our implementation.Project of the Spanish Ministry of Economy and Competitiveness with reference MTM2016-76329-R (AEI/FEDER, EU). Project MTM2013-46553-C3-2-P from Spanish Ministry of Economy and Trade. Consolidated Research Group IT649-13 from the Basque Government

Resolving Entropy Growth from Iterative Methods

We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton's method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers' equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.Comment: 25 pages, 6 figure

A semi-implicit version of the MPAS-atmosphere dynamical core

An important question for atmospheric modeling is the viability of semi-implicit time integration schemes on massively parallel computing architectures. Semi-implicit schemes can provide increased stability and accuracy. However, they require the solution of an elliptic problem at each time step, creating concerns about their parallel efficiency and scalability. Here, a semi-implicit (SI) version of the Model for Prediction Across Scales (MPAS) is developed and compared with the original model version, which uses a split Runge-Kutta (SRK3) time integration scheme. The SI scheme is based on a quasi-Newton iteration toward a Crank-Nicolson scheme. Each Newton iteration requires the solution of a Helmholtz problem; here, the Helmholtz problem is derived, and its solution using a geometric multigrid method is described. On two standard test cases, a midlatitude baroclinic wave and a small-planet nonhydrostatic gravity wave, the SI and SRK3 versions produce almost identical results. On the baroclinic wave test, the SI version can use somewhat larger time steps (about 60%) than the SRK3 version before losing stability. The SI version costs 10%-20% more per step than the SRK3 version, and the weak and strong scalability characteristics of the two versions are very similar for the processor configurations the authors have been able to test (up to 1920 processors). Because of the spatial discretization of the pressure gradient in the lowest model layer, the SI version becomes unstable in the presence of realistic orography. Some further work will be needed to demonstrate the viability of the SI scheme in this case.UK Natural Environment Research Council as part of the G8 ICOMEX projec

Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes

We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.Ministerio de Economía y Comercio: proyecto MTM2013-46553-C3-2-P, Spanish Ministry of Economy and Competitiveness: project MTM2016-76329-R “IMAGEARTH”, Basque Government: Consolidated Research Group IT649-1