2,647 research outputs found
A recursively feasible and convergent Sequential Convex Programming procedure to solve non-convex problems with linear equality constraints
A computationally efficient method to solve non-convex programming problems
with linear equality constraints is presented. The proposed method is based on
a recursively feasible and descending sequential convex programming procedure
proven to converge to a locally optimal solution. Assuming that the first
convex problem in the sequence is feasible, these properties are obtained by
convexifying the non-convex cost and inequality constraints with inner-convex
approximations. Additionally, a computationally efficient method is introduced
to obtain inner-convex approximations based on Taylor series expansions. These
Taylor-based inner-convex approximations provide the overall algorithm with a
quadratic rate of convergence. The proposed method is capable of solving
problems of practical interest in real-time. This is illustrated with a
numerical simulation of an aerial vehicle trajectory optimization problem on
commercial-of-the-shelf embedded computers
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many
statistics and machine learning applications ranging from support vector
machines to Gaussian processes and kernel embeddings of distributions.
Operators acting on such spaces are, for instance, required to embed
conditional probability distributions in order to implement the kernel Bayes
rule and build sequential data models. It was recently shown that transfer
operators such as the Perron-Frobenius or Koopman operator can also be
approximated in a similar fashion using covariance and cross-covariance
operators and that eigenfunctions of these operators can be obtained by solving
associated matrix eigenvalue problems. The goal of this paper is to provide a
solid functional analytic foundation for the eigenvalue decomposition of RKHS
operators and to extend the approach to the singular value decomposition. The
results are illustrated with simple guiding examples
Minkowski and Galilei/Newton Fluid Dynamics: A Geometric 3+1 Spacetime Perspective
A kinetic theory of classical particles serves as a unified basis for
developing a geometric spacetime perspective on fluid dynamics capable of
embracing both Minkowski and Galilei/Newton spacetimes. Parallel treatment of
these cases on as common a footing as possible reveals that the particle
four-momentum is better regarded as comprising momentum and \textit{inertia}
rather than momentum and energy; and consequently, that the object now known as
the stress-energy or energy-momentum tensor is more properly understood as a
stress-\textit{inertia} or \textit{inertia}-momentum tensor. In dealing with
both fiducial and comoving frames as fluid dynamics requires, tensor
decompositions in terms of the four-velocities of observers associated with
these frames render use of coordinate-free geometric notation not only fully
viable, but conceptually simplifying. A particle number four-vector,
three-momentum tensor, and kinetic energy four-vector characterize a
simple fluid and satisfy balance equations involving spacetime divergences on
both Minkowski and Galilei/Newton spacetimes. Reduced to a fully form,
these equations yield the familiar conservative formulations of special
relativistic and non-relativistic hydrodynamics as partial differential
equations in inertial coordinates, and in geometric form will provide a useful
conceptual bridge to arbitrary-Lagrange-Euler and general relativistic
formulations.Comment: Belated upload of version accepted by MDPI Fluids. Additional
material in the Introduction; added several tables and an additional appendi
Dissipative hydrodynamics for viscous relativistic fluids
Explicit equations are given for describing the space-time evolution of
non-ideal (viscous) relativistic fluids undergoing boost-invariant longitudinal
and arbitrary transverse expansion. The equations are derived from the
second-order Israel-Stewart approach which ensures causal evolution. Both
azimuthally symmetric (1+1)-dimensional and non-symmetric (2+1)-dimensional
transverse expansion are discussed. The latter provides the formal basis for
the hydrodynamic computation of elliptic flow in relativistic heavy-ion
collisions including dissipative effects.Comment: 12 pages, no figures. Submitted to Physical Review
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