1,686 research outputs found
Shape reconstructions by using plasmon resonances
We study the shape reconstruction of a dielectric inclusion from the faraway
measurement of the associated electric field. This is an inverse problem of
practical importance in biomedical imaging and is known to be notoriously
ill-posed. By incorporating Drude's model of the dielectric parameter, we
propose a novel reconstruction scheme by using the plasmon resonance with a
significantly enhanced resonant field. We conduct a delicate sensitivity
analysis to establish a sharp relationship between the sensitivity of the
reconstruction and the plasmon resonance. It is shown that when plasmon
resonance occurs, the sensitivity functional blows up and hence ensures a more
robust and effective construction. Then we combine the Tikhonov regularization
with the Laplace approximation to solve the inverse problem, which is an
organic hybridization of the deterministic and stochastic methods and can
quickly calculate the minimizer while capture the uncertainty of the solution.
We conduct extensive numerical experiments to illustrate the promising features
of the proposed reconstruction scheme
Determining a stationary mean field game system from full/partial boundary measurement
In this paper, we propose and study the utilization of the
Dirichlet-to-Neumann (DN) map to uniquely identify the discount functions and cost function in a stationary mean field game (MFG) system. This
study features several technical novelties that make it highly intriguing and
challenging. Firstly, it involves a coupling of two nonlinear elliptic partial
differential equations. Secondly, the simultaneous recovery of multiple
parameters poses a significant implementation challenge. Thirdly, there is the
probability measure constraint of the coupled equations to consider. Finally,
the limited information available from partial boundary measurements adds
another layer of complexity to the problem. Considering these challenges and
problems, we present an enhanced higher-order linearization method to tackle
the inverse problem related to the MFG system. Our proposed approach involves
linearizing around a pair of zero solutions and fulfilling the probability
measurement constraint by adjusting the positive input at the boundary. It is
worth emphasizing that this technique is not only applicable for uniquely
identifying multiple parameters using full-boundary measurements but also
highly effective for utilizing partial-boundary measurements
Sampling reduced density matrix to extract fine levels of entanglement spectrum
Low-lying entanglement spectrum provides the quintessential fingerprint to
identify the highly entangled quantum matter with topological and conformal
field-theoretical properties. However, when the entangling region acquires long
boundary with the environment, such as that between long coupled chains or in
two or higher dimensions, there unfortunately exists no universal yet practical
method to compute the entanglement spectra with affordable computational cost.
Here we propose a new scheme to overcome such difficulty and successfully
extract the low-lying fine entanglement spectrum (ES). We trace out the
environment via quantum Monte Carlo simulation and diagonalize the reduced
density matrix to gain the ES. We demonstrate the strength and reliability of
our method through long coupled spin chains and answer its long-standing
controversy. Our simulation results, with unprecedentedly large system sizes,
establish the practical computation scheme of the entanglement spectrum with a
huge freedom degree of environment
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