21 research outputs found
Asymptotics for optimal design problems for the Schr\"odinger equation with a potential
We study the problem of optimal observability and prove time asymptotic
observability estimates for the Schr\"odinger equation with a potential in
, with , using spectral theory.
An elegant way to model the problem using a time asymptotic observability
constant is presented. For certain small potentials, we demonstrate the
existence of a nonzero asymptotic observability constant under given conditions
and describe its explicit properties and optimal values. Moreover, we give a
precise description of numerical models to analyze the properties of important
examples of potentials wells, including that of the modified harmonic
oscillator
Multiclass Data Segmentation using Diffuse Interface Methods on Graphs
We present two graph-based algorithms for multiclass segmentation of
high-dimensional data. The algorithms use a diffuse interface model based on
the Ginzburg-Landau functional, related to total variation compressed sensing
and image processing. A multiclass extension is introduced using the Gibbs
simplex, with the functional's double-well potential modified to handle the
multiclass case. The first algorithm minimizes the functional using a convex
splitting numerical scheme. The second algorithm is a uses a graph adaptation
of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates
between diffusion and thresholding. We demonstrate the performance of both
algorithms experimentally on synthetic data, grayscale and color images, and
several benchmark data sets such as MNIST, COIL and WebKB. We also make use of
fast numerical solvers for finding the eigenvectors and eigenvalues of the
graph Laplacian, and take advantage of the sparsity of the matrix. Experiments
indicate that the results are competitive with or better than the current
state-of-the-art multiclass segmentation algorithms.Comment: 14 page
Asymptotics for Optimal Design Problems for the Schrodinger Equation with a Potential
We study the problem of optimal observability and prove time asymptotic observability estimates for the Schrodinger equation with a potential in L-infinity(Omega), with Omega subset of R-d, using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator
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