759,405 research outputs found
Necessary conditions for variational regularization schemes
We study variational regularization methods in a general framework, more
precisely those methods that use a discrepancy and a regularization functional.
While several sets of sufficient conditions are known to obtain a
regularization method, we start with an investigation of the converse question:
How could necessary conditions for a variational method to provide a
regularization method look like? To this end, we formalize the notion of a
variational scheme and start with comparison of three different instances of
variational methods. Then we focus on the data space model and investigate the
role and interplay of the topological structure, the convergence notion and the
discrepancy functional. Especially, we deduce necessary conditions for the
discrepancy functional to fulfill usual continuity assumptions. The results are
applied to discrepancy functionals given by Bregman distances and especially to
the Kullback-Leibler divergence.Comment: To appear in Inverse Problem
On lower bounds for the L_2-discrepancy
The L_2-discrepancy measures the irregularity of the distribution of a finite
point set. In this note we prove lower bounds for the L_2 discrepancy of
arbitrary N-point sets. Our main focus is on the two-dimensional case.
Asymptotic upper and lower estimates of the L_2-discrepancy in dimension 2 are
well-known and are of the sharp order sqrt(log N). Nevertheless the gap in the
constants between the best known lower and upper bounds is unsatisfactory large
for a two-dimensional problem. Our lower bound improves upon this situation
considerably. The main method is an adaption of the method of K. F. Roth using
the Fourier coefficients of the discrepancy function with respect to the Haar
basis
Bounded Verification with On-the-Fly Discrepancy Computation
Simulation-based verification algorithms can provide formal safety guarantees
for nonlinear and hybrid systems. The previous algorithms rely on user provided
model annotations called discrepancy function, which are crucial for computing
reachtubes from simulations. In this paper, we eliminate this requirement by
presenting an algorithm for computing piece-wise exponential discrepancy
functions. The algorithm relies on computing local convergence or divergence
rates of trajectories along a simulation using a coarse over-approximation of
the reach set and bounding the maximal eigenvalue of the Jacobian over this
over-approximation. The resulting discrepancy function preserves the soundness
and the relative completeness of the verification algorithm. We also provide a
coordinate transformation method to improve the local estimates for the
convergence or divergence rates in practical examples. We extend the method to
get the input-to-state discrepancy of nonlinear dynamical systems which can be
used for compositional analysis. Our experiments show that the approach is
effective in terms of running time for several benchmark problems, scales
reasonably to larger dimensional systems, and compares favorably with respect
to available tools for nonlinear models.Comment: 24 page
A theoretical approach to thermal noise caused by an inhomogeneously distributed loss -- Physical insight by the advanced modal expansion
We modified the modal expansion, which is the traditional method used to
calculate thermal noise. This advanced modal expansion provides physical
insight about the discrepancy between the actual thermal noise caused by
inhomogeneously distributed loss and the traditional modal expansion. This
discrepancy comes from correlations between the thermal fluctuations of the
resonant modes. The thermal noise spectra estimated by the advanced modal
expansion are consistent with the results of measurements of thermal
fluctuations caused by inhomogeneous losses.Comment: 10 pages, 4 figure
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