394 research outputs found
Convex Combinatorial Optimization
We introduce the convex combinatorial optimization problem, a far reaching
generalization of the standard linear combinatorial optimization problem. We
show that it is strongly polynomial time solvable over any edge-guaranteed
family, and discuss several applications
Geometry of Log-Concave Density Estimation
Shape-constrained density estimation is an important topic in mathematical
statistics. We focus on densities on that are log-concave, and
we study geometric properties of the maximum likelihood estimator (MLE) for
weighted samples. Cule, Samworth, and Stewart showed that the logarithm of the
optimal log-concave density is piecewise linear and supported on a regular
subdivision of the samples. This defines a map from the space of weights to the
set of regular subdivisions of the samples, i.e. the face poset of their
secondary polytope. We prove that this map is surjective. In fact, every
regular subdivision arises in the MLE for some set of weights with positive
probability, but coarser subdivisions appear to be more likely to arise than
finer ones. To quantify these results, we introduce a continuous version of the
secondary polytope, whose dual we name the Samworth body. This article
establishes a new link between geometric combinatorics and nonparametric
statistics, and it suggests numerous open problems.Comment: 22 pages, 3 figure
Robust Region-of-Attraction Estimation
We propose a method to compute invariant subsets of the region-of-attraction for asymptotically stable equilibrium points of polynomial dynamical systems with bounded parametric uncertainty. Parameter-independent Lyapunov functions are used to characterize invariant subsets of the robust region-of-attraction. A branch-and-bound type refinement procedure reduces the conservatism. We demonstrate the method on an example from the literature and uncertain controlled short-period aircraft dynamics
Modeling and Control of Robot-Structure Coupling During In-Space Structure Assembly
This paper considers the problem of robot-structure coupling dynamics during in-space robotic assembly of large flexible structures. A two-legged walking robot is used as a construction agent, whose primary goal is to stably walking on the flexible structure while carrying a substructure component to a designated location. The reaction forces inserted by the structure to the walking robot are treated as bounded disturbance inputs, and a trajectory tracking robotic controller is proposed that combines the standard full state feedback motion controller and an adaptive controller to account for the disturbance inputs. In this study, a reduced-order Euler-Bernoulli beam structure model is adapted, and a finite number of co-located sensors and actuators are distributed along the span of the beam structure. The robot-structure coupling forces are treated as a bounded external forcing function to the structure, and hence an output covariance constraint problem can be formulated, in terms of linear matrix inequality, for optimal structure control by utilizing the direct output feedback controllers. The numerical simulations show the effectiveness of the proposed robot-structure modeling and control methodology
Accuracy controlled data assimilation for parabolic problems
This paper is concerned with the recovery of (approximate) solutions to
parabolic problems from incomplete and possibly inconsistent observational
data, given on a time-space cylinder that is a strict subset of the
computational domain under consideration. Unlike previous approaches to this
and related problems our starting point is a regularized least squares
formulation in a continuous infinite-dimensional setting that is based on
stable variational time-space formulations of the parabolic PDE. This allows us
to derive a priori as well as a posteriori error bounds for the recovered
states with respect to a certain reference solution. In these bounds the
regularization parameter is disentangled from the underlying discretization. An
important ingredient for the derivation of a posteriori bounds is the
construction of suitable Fortin operators which allow us to control oscillation
errors stemming from the discretization of dual norms. Moreover, the
variational framework allows us to contrive preconditioners for the discrete
problems whose application can be performed in linear time, and for which the
condition numbers of the preconditioned systems are uniformly proportional to
that of the regularized continuous problem.
In particular, we provide suitable stopping criteria for the iterative
solvers based on the a posteriori error bounds. The presented numerical
experiments quantify the theoretical findings and demonstrate the performance
of the numerical scheme in relation with the underlying discretization and
regularization
Monogamy of highly symmetric states
We study the question of how highly entangled two particles can be when also
entangled in a similar way with other particles on the complete graph for the
case of Werner, isotropic and Brauer states. In order to do so we solve
optimization problems motivated by many-body physics, computational complexity
and quantum cryptography. We formalize our question as a semi-definite program
and then solve this optimization problem analytically, using tools from
representation theory. In particular, we determine the exact maximum values of
the projection to the maximally entangled state and antisymmetric Werner state
possible, solving long-standing open problems. We find these optimal values by
use of SDP duality and representation theory of the symmetric and orthogonal
groups, and the Brauer algebra.Comment: Submitted to QIP202
Space-time residual minimization for parabolic partial differential equations
Many processes in nature and engineering are governed by partial differential equations (PDEs). We focus on parabolic PDEs, that describe time-dependent phenomena like heat conduction, chemical concentration, and fluid flow. Even if we know that a unique solution exists, we can express it in closed form only under very strict circumstances. So, to understand what it looks like, we turn to numerical approximation. Historically, parabolic PDEs are solved using time-stepping. One first discretizes the PDE in space as to obtain a system of coupled ordinary differential equations in time. This system is then solved using the vast theory for ODEs. While efficient in terms of memory and computational cost, time-stepping schemes take global time steps, which are independent of spatial position. As a result, these methods cannot efficiently resolve details in localized regions of space and time. Moreover, being inherently sequential, they have limited possibilities for parallel computation. In this thesis, we take a different approach and reformulate the parabolic evolution equation as an equation posed in space and time simultaneously. Space-time methods mitigate the aforementioned issues, and moreover produce approximations to the unknown solution that are uniformly quasi-optimal. The focal point of this thesis is the space-time minimal residual (MR) method introduced by R. Andreev, that finds the approximation that minimizes both PDE- and initial error. We discuss its theoretical properties, provide numerical algorithms for its computation, and discuss its applicability in data assimilation (the problem of fusing measured data to its underlying PDE)
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