1,211 research outputs found
Computational aspects of helicopter trim analysis and damping levels from Floquet theory
Helicopter trim settings of periodic initial state and control inputs are investigated for convergence of Newton iteration in computing the settings sequentially and in parallel. The trim analysis uses a shooting method and a weak version of two temporal finite element methods with displacement formulation and with mixed formulation of displacements and momenta. These three methods broadly represent two main approaches of trim analysis: adaptation of initial-value and finite element boundary-value codes to periodic boundary conditions, particularly for unstable and marginally stable systems. In each method, both the sequential and in-parallel schemes are used and the resulting nonlinear algebraic equations are solved by damped Newton iteration with an optimally selected damping parameter. The impact of damped Newton iteration, including earlier-observed divergence problems in trim analysis, is demonstrated by the maximum condition number of the Jacobian matrices of the iterative scheme and by virtual elimination of divergence. The advantages of the in-parallel scheme over the conventional sequential scheme are also demonstrated
Algorithm MGB to solve highly nonlinear elliptic PDEs in FLOPS
We introduce Algorithm MGB (Multi Grid Barrier) for solving highly nonlinear
convex Euler-Lagrange equations. This class of problems includes many highly
nonlinear partial differential equations, such as -Laplacians. We prove
that, if certain regularity hypotheses are satisfied, then our algorithm
converges in damped Newton iterations, or FLOPS,
where the tilde indicates that we neglect some polylogarithmic terms. This the
first algorithm whose running time is proven optimal in the big-
sense. Previous algorithms for the -Laplacian required
damped Newton iterations or more
Discretization of the 3D Monge-Ampere operator, between Wide Stencils and Power Diagrams
We introduce a monotone (degenerate elliptic) discretization of the
Monge-Ampere operator, on domains discretized on cartesian grids. The scheme is
consistent provided the solution hessian condition number is uniformly bounded.
Our approach enjoys the simplicity of the Wide Stencil method, but
significantly improves its accuracy using ideas from discretizations of optimal
transport based on power diagrams. We establish the global convergence of a
damped Newton solver for the discrete system of equations. Numerical
experiments, in three dimensions, illustrate the scheme efficiency
Sketch-and-Project Meets Newton Method: Global Convergence with Low-Rank Updates
In this paper, we propose the first sketch-and-project Newton method with
fast global convergence rate for self-concordant
functions. Our method, SGN, can be viewed in three ways: i) as a
sketch-and-project algorithm projecting updates of Newton method, ii) as a
cubically regularized Newton ethod in sketched subspaces, and iii) as a damped
Newton method in sketched subspaces. SGN inherits best of all three worlds:
cheap iteration costs of sketch-and-project methods, state-of-the-art global convergence rate of full-rank Newton-like methods and the
algorithm simplicity of damped Newton methods. Finally, we demonstrate its
comparable empirical performance to baseline algorithms.Comment: 10 page
- …