123 research outputs found
A derivative-free bracketing scheme for univariate minimization
AbstractA derivative-free scheme for univariate minimization is developed. This scheme has a quadratic convergence rate and requires two function evaluations each iteration
Higher Order Variational Integrators: a polynomial approach
We reconsider the variational derivation of symplectic partitioned
Runge-Kutta schemes. Such type of variational integrators are of great
importance since they integrate mechanical systems with high order accuracy
while preserving the structural properties of these systems, like the
symplectic form, the evolution of the momentum maps or the energy behaviour.
Also they are easily applicable to optimal control problems based on mechanical
systems as proposed in Ober-Bl\"obaum et al. [2011].
Following the same approach, we develop a family of variational integrators
to which we refer as symplectic Galerkin schemes in contrast to symplectic
partitioned Runge-Kutta. These two families of integrators are, in principle
and by construction, different one from the other. Furthermore, the symplectic
Galerkin family can as easily be applied in optimal control problems, for which
Campos et al. [2012b] is a particular case.Comment: 12 pages, 1 table, 23rd Congress on Differential Equations and
Applications, CEDYA 201
Smoothing a program soundly and robustly
We study the foundations of smooth interpretation, a recently-proposed program approximation scheme that facilitates the use of local numerical search techniques (e.g., gradient descent) in program analysis and synthesis. While the popular techniques for local optimization works well only on relatively smooth functions, functions encoded by real-world programs are infested with discontinuities and local irregular features. Smooth interpretation attenuates such features by taking the convolution of the program with a Gaussian function, effectively replacing discontinuous switches in the program by continuous transitions. In doing so, it extends to programs the notion of Gaussian smoothing, a popular signal-processing technique used to filter noise and discontinuities from signals.
Exact Gaussian smoothing of programs is undecidable, so algorithmic implementations of smooth interpretation must necessarily be approximate. In this paper, we characterize the approximations carried out by such algorithms. First, we identify three correctness properties—soundness, robustness, and β-robustness—that an approximate smooth interpreter should satisfy. In particular, a smooth interpreter is sound if it computes an abstraction of a program’s “smoothed” semantics, and robust if it has arbitrary-order derivatives in the input variables at every point in its input space. Second, we describe the design of an approximate smooth interpreter that provably satisfies these properties. The interpreter combines program abstraction using a new domain with symbolic calculation of convolution.National Science Foundation (U.S.) (Grant CCF-0953507)Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laborator
Continuum description of profile scaling in nanostructure decay
The relaxation of axisymmetric crystal surfaces with a single facet below the
roughening transition is studied via a continuum approach that accounts for
step energy g_1 and step-step interaction energy g_3>0. For diffusion-limited
kinetics, free-boundary and boundary-layer theories are used for self-similar
shapes close to the growing facet. For long times and g_3/g_1 < 1, (a) a
universal equation is derived for the shape profile, (b) the layer thickness
varies as (g_3/g_1)^{1/3}, (c) distinct solutions are found for different
g_3/_1, and (d) for conical shapes, the profile peak scales as
(g_3/g_1)^{-1/6}. These results compare favorably with kinetic simulations.Comment: 4 pages including 3 figure
Accurate evaluation of the interstitial KKR-Green function
It is shown that the Brillouin zone integral for the interstitial KKR-Green
function can be evaluated accurately by taking proper care of the free-electron
singularities in the integrand. The proposed method combines two recently
developed methods, a supermatrix method and a subtraction method. This
combination appears to provide a major improvement compared with an earlier
proposal based on the subtraction method only. By this the barrier preventing
the study of important interstitial-like defects, such as an electromigrating
atom halfway along its jump path, can be considered as being razed.Comment: 23 pages, RevTe
Constrained dogleg methods for nonlinear systems with simple bounds
We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problem
Efficient Nonlinear Programming Algorithms for Chemical Process Control and Operations
Nonlinear programming (NLP) has been a key enabling tool for model-based decision-making in the chemical industry for over 50 years. Opti-mization is frequently applied in numerous ar-eas of chemical engineering including the de-velopment of process models from experimen-tal data, design of process flowsheets and equip-ment, planning and scheduling of chemical pro-cess operations, and the analysis of chemical pro-cesses under uncertainty and adverse conditions. These off-line tasks frequently require the solu-tion of NLPs formulated with detailed, lareg-scale process models. More recently, these tasks are complemented by time-critical, on-line optimization problem
- …