This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

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Discrete mechanics and variational integrators

Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

高等学校計算数学学報 = Numerical mathematics

Many differential equations of practical interest evolve on Lie groups or on manifolds acted upon by Lie groups. The retention of Lie-group structure under discretization is often vital in the recovery of qualitatively correct geometry and dynamics and in the minimization of numerical error. Having introduced requisite elements of differential geometry, this paper surveys the novel theory of numerical integrators that respect Lie-group structure, highlighting theory, algorithmic issues and a number of applications.

Lie-group methods

The first ever energy-preserving B-series numerical integration method for (ordinary) differential equations is presented and applied to several Hamiltonian systems. Related novel Lie algebraic results are also discussed.

https://iopscience.iop.org/article/10.1088/1751-8113/41/4/045206/pdf

A new class of energy-preserving numerical integration methods

This paper discusses the discrete analogue of the gradient of a function and shows how discrete gradients can be used in the numerical integration of ordinary differential equations (ODEs). Given a...

Geometric integration using discrete gradients

Systems with a first integral (i.e., constant of motion) or a Lyapunov function can be written as ``linear-gradient systems'' $\dot x= L(x)\nabla V(x)$ for an appropriate matrix function $L$, with a generalization to several integrals or Lyapunov functions. The discrete-time analogue, $\Delta x/\Delta t = L \bar\nabla V$ where $\bar\nabla$ is a ``discrete gradient,'' preserves $V$ as an integral or Lyapunov function, respectively.

https://arxiv.org/pdf/math-ph/9805021.pdf

Unified approach to hamiltonian systems, poisson systems, gradient systems, and systems with lyapunov functions or first integrals

Functionals for variational grid generation, in which the Jacobian matrix and matrix norms play a central role, are presented. All first-order geometric qualities of a structured grid (length, area, volume, angle between sides, aspect ratio, grid alignment) are determined by the Jacobian matrix of the induced mapping between the logical and physical regions. Grids with desirable quality can be generated by requiring the Jacobian matrix or the corresponding metric tensor to have certain properties. Functionals can be obtained by integrating over the logical or physical domain a power of the norm of a matrix derived from the Jacobian matrix. A judicious choice of this derived matrix results in a functional whose minima are mappings with desired features. Many well-known grid generators fit in this framework; for example, the harmonic generator arises when one drives the metric tensor toward the identity. The framework provides insight into what grid qualities these functionals describe.
Three new directional control functionals are proposed and tested.

A Framework for Variational Grid Generation: Conditioning the Jacobian Matrix with Matrix Norms

Mimetic discretization methods for the numerical solution of continuum mechanics problems directly use vector calculus and differential forms identities for their derivation and analysis. Fully mimetic discretizations satisfy discrete analogs of the continuum theory results used to derive energy inequalities. Consequently, continuum arguments carry over and can be used to show that discrete problems are well-posed and discrete solutions converge. A fully mimetic discrete vector calculus on three dimensional tensor product grids is derived and its key properties proven. Opinions regarding the future of the field are stated. 2000 Mathematical subject classification: 65N12; 65N06.

A Discrete Vector Calculus in Tensor Grids

Commodity imaging systems rely on hardware image signal processing (ISP) pipelines. These low-level pipelines consist of a sequence of processing blocks that, depending on their hyperparameters, reconstruct a color image from RAW sensor measurements. Hardware ISP hyperparameters have a complex interaction with the output image, and therefore with the downstream application ingesting these images. Traditionally, ISPs are manually tuned in isolation by imaging experts without an end-to-end objective. Very recently, ISPs have been optimized with 1st-order methods that require differentiable approximations of the hardware ISP. Departing from such approximations, we present a hardware-in-the-loop method that directly optimizes hardware image processing pipelines for end-to-end domain-specific losses by solving a nonlinear multi-objective optimization problem with a novel 0th-order stochastic solver directly interfaced with the hardware ISP. We validate the proposed method with recent hardware ISPs and 2D object detection, segmentation, and human viewing as end-to-end downstream tasks. For automotive 2D object detection, the proposed method outperforms manual expert tuning by 30% mean average precision (mAP) and recent methods using ISP approximations by 18% mAP.

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Nicolas Robidoux

Papers

Geometric integration using discrete gradients

Unified approach to hamiltonian systems, poisson systems, gradient systems, and systems with lyapunov functions or first integrals

A Framework for Variational Grid Generation: Conditioning the Jacobian Matrix with Matrix Norms

A Discrete Vector Calculus in Tensor Grids

Hardware-in-the-Loop End-to-End Optimization of Camera Image Processing Pipelines