Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExtending Representation Formulae for Boundary Voltage Perturbations of Low Volume Fraction to Very Contrasted Conductivity Inhomogeneities
68
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Imposing either Dirichlet or Neumann boundary conditions on the boundary of a smooth bounded domain $Omega$, we study the perturbation incurred by the voltage potential when the conductivity is modified in a set of small measure. We consider $left(gamma_{n}right)_{ninmathbb{N}}$, a sequence of perturbed conductivity matrices differing from a smooth $gamma_{0}$ background conductivity matrix on a measurable set well within the domain, and we assume $left(gamma_{n}-gamma_{0}right)gamma_{n}^{-1}left(gamma_{n}-gamma_{0}right)to0$ in $L^{1}(Omega)$. Adapting the limit measure, we show that the general representation formula introduced for bounded contrasts in citep{capdeboscq-vogelius-03a} can be extended to unbounded sequencesof matrix valued conductivities.
rate research
Read More
We consider a singularly-perturbed nonconvex energy functional which arises in the study of microstructures in shape memory alloys. The scaling law for the minimal energy predicts a transition from a parameter regime in which uniform structures are favored, to a regime in which the formation of fine patterns is expected. We focus on the transition regime and derive the reduced model in the sense of $Gamma$-convergence. The limit functional turns out to be similar to the Mumford-Shah functional with additional constraints on the jump set of admissible functions. One key ingredient in the proof is an approximation result for $SBV^p$ functions whose jump sets have a prescribed orientation.
We study microstructure formation in two nonconvex singularly-perturbed variational problems from materials science, one modeling austenite-martensite interfaces in shape-memory alloys, the other one slip structures in the plastic deformation of crystals. For both functionals we determine the scaling of the optimal energy in terms of the parameters of the problem, leading to a characterization of the mesoscopic phase diagram. Our results identify the presence of a new phase, which is intermediate between the classical laminar microstructures and branching patterns. The new phase, characterized by partial branching, appears for both problems in the limit of small volume fraction, that is, if one of the variants (or of the slip systems) dominates the picture and the volume fraction of the other one is small.
In this short note we prove two elegant generalized continued fraction formulae $$e= 2+cfrac{1}{1+cfrac{1}{2+cfrac{2}{3+cfrac{3}{4+ddots}}}}$$ and $$e= 3+cfrac{-1}{4+cfrac{-2}{5+cfrac{-3}{6+cfrac{-4}{7+ddots}}}}$$ using elementary methods. The first formula is well-known, and the second one is newly-discovered in arXiv:1907.00205 [cs.LG]. We then explore the possibility of automatic verification of such formulae using computer algebra systems (CASs).
In this work, we present a novel 3D-Convolutional Neural Network (CNN) architecture called I2I-3D that predicts boundary location in volumetric data. Our fine-to-fine, deeply supervised framework addresses three critical issues to 3D boundary detection: (1) efficient, holistic, end-to-end volumetric label training and prediction (2) precise voxel-level prediction to capture fine scale structures prevalent in medical data and (3) directed multi-scale, multi-level feature learning. We evaluate our approach on a dataset consisting of 93 medical image volumes with a wide variety of anatomical regions and vascular structures. In the process, we also introduce HED-3D, a 3D extension of the state-of-the-art 2D edge detector (HED). We show that our deep learning approach out-performs, the current state-of-the-art in 3D vascular boundary detection (structured forests 3D), by a large margin, as well as HED applied to slices, and HED-3D while successfully localizing fine structures. With our approach, boundary detection takes about one minute on a typical 512x512x512 volume.
We consider the inverse problem of recovering an isotropic electrical conductivity from interior knowledge of the magnitude of one current density field generated by applying current on a set of electrodes. The required interior data can be obtained by means of MRI measurements. On the boundary we only require knowledge of the electrodes, their impedances, and the corresponding average input currents. From the mathematical point of view, this practical question leads us to consider a new weighted minimum gradient problem for functions satisfying the boundary conditions coming from the Complete Electrode Model of Somersalo, Cheney and Isaacson. This variational problem has non-unique solutions. The surprising discovery is that the physical data is still sufficient to determine the geometry of the level sets of the minimizers. In particular, we obtain an interesting phase retrieval result: knowledge of the input current at the boundary allows determination of the full current vector field from its magnitude. We characterize the non-uniqueness in the variational problem. We also show that additional measurements of the voltage potential along one curve joining the electrodes yield unique determination of the conductivity. A nonlinear algorithm is proposed and implemented to illustrate the theoretical results.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
