46,693 research outputs found
H-Infinity Optimal Interconnections
In this paper, a general ℋ∞ problem for continuous time, linear time invariant systems is formulated and solved in a behavioral framework. This general formulation, which includes standard ℋ∞ optimization as a special case, provides added freedom in the design of sub-optimal compensators, and can in fact be viewed as a means of designing optimal systems. In particular, the formulation presented allows for singular interconnections, which naturally occur when interconnecting first principles models
On the decidability and complexity of Metric Temporal Logic over finite words
Metric Temporal Logic (MTL) is a prominent specification formalism for
real-time systems. In this paper, we show that the satisfiability problem for
MTL over finite timed words is decidable, with non-primitive recursive
complexity. We also consider the model-checking problem for MTL: whether all
words accepted by a given Alur-Dill timed automaton satisfy a given MTL
formula. We show that this problem is decidable over finite words. Over
infinite words, we show that model checking the safety fragment of MTL--which
includes invariance and time-bounded response properties--is also decidable.
These results are quite surprising in that they contradict various claims to
the contrary that have appeared in the literature
Curse of dimensionality reduction in max-plus based approximation methods: theoretical estimates and improved pruning algorithms
Max-plus based methods have been recently developed to approximate the value
function of possibly high dimensional optimal control problems. A critical step
of these methods consists in approximating a function by a supremum of a small
number of functions (max-plus "basis functions") taken from a prescribed
dictionary. We study several variants of this approximation problem, which we
show to be continuous versions of the facility location and -center
combinatorial optimization problems, in which the connection costs arise from a
Bregman distance. We give theoretical error estimates, quantifying the number
of basis functions needed to reach a prescribed accuracy. We derive from our
approach a refinement of the curse of dimensionality free method introduced
previously by McEneaney, with a higher accuracy for a comparable computational
cost.Comment: 8pages 5 figure
Recovering edges in ill-posed inverse problems: optimality of curvelet frames
We consider a model problem of recovering a function from noisy Radon data. The function to be recovered is assumed smooth apart from a discontinuity along a curve, that is, an edge. We use the continuum white-noise model, with noise level .
Traditional linear methods for solving such inverse problems behave poorly in the presence of edges. Qualitatively, the reconstructions are blurred near the edges; quantitatively, they give in our model mean squared errors (MSEs) that tend to zero with noise level only as as . A recent innovation--nonlinear shrinkage in the wavelet domain--visually improves edge sharpness and improves MSE convergence to . However, as we show here, this rate is not optimal.
In fact, essentially optimal performance is obtained by deploying the recently-introduced tight frames of curvelets in this setting. Curvelets are smooth, highly anisotropic elements ideally suited for detecting and synthesizing curved edges. To deploy them in the Radon setting, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build "curvelet shrinkage" estimators based on thresholding of the noisy curvelet coefficients. In effect, the estimator detects edges at certain locations and orientations in the Radon domain and automatically synthesizes edges at corresponding locations and directions in the original domain.
We prove that the curvelet shrinkage can be tuned so that the estimator will attain, within logarithmic factors, the MSE as noise level . This rate of convergence holds uniformly over a class of functions which are except for discontinuities along curves, and (except for log terms) is the minimax rate for that class. Our approach is an instance of a general strategy which should apply in other inverse problems; we sketch a deconvolution example
Parabolic Molecules
Anisotropic decompositions using representation systems based on parabolic
scaling such as curvelets or shearlets have recently attracted significantly
increased attention due to the fact that they were shown to provide optimally
sparse approximations of functions exhibiting singularities on lower
dimensional embedded manifolds. The literature now contains various direct
proofs of this fact and of related sparse approximation results. However, it
seems quite cumbersome to prove such a canon of results for each system
separately, while many of the systems exhibit certain similarities.
In this paper, with the introduction of the notion of {\em parabolic
molecules}, we aim to provide a comprehensive framework which includes
customarily employed representation systems based on parabolic scaling such as
curvelets and shearlets. It is shown that pairs of parabolic molecules have the
fundamental property to be almost orthogonal in a particular sense. This result
is then applied to analyze parabolic molecules with respect to their ability to
sparsely approximate data governed by anisotropic features. For this, the
concept of {\em sparsity equivalence} is introduced which is shown to allow the
identification of a large class of parabolic molecules providing the same
sparse approximation results as curvelets and shearlets. Finally, as another
application, smoothness spaces associated with parabolic molecules are
introduced providing a general theoretical approach which even leads to novel
results for, for instance, compactly supported shearlets
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