12,928 research outputs found
Three embedded techniques for finite element heat flow problem with embedded discontinuities
The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-017-1382-7The present paper explores the solution of a heat conduction problem considering discontinuities embedded within the mesh and aligned at arbitrary angles with respect to the mesh edges. Three alternative approaches are proposed as solutions to the problem. The difference between these approaches compared to alternatives, such as the eXtended Finite Element Method (X-FEM), is that the current proposal attempts to preserve the global matrix graph in order to improve performance. The first two alternatives comprise an enrichment of the Finite Element (FE) space obtained through the addition of some new local degrees of freedom to allow capturing discontinuities within the element. The new degrees of freedom are statically condensed prior to assembly, so that the graph of the final system is not changed. The third approach is based on the use of modified FE-shape functions that substitute the standard ones on the cut elements. The imposition of both Neumann and Dirichlet boundary conditions is considered at the embedded interface. The results of all the proposed methods are then compared with a reference solution obtained using the standard FE on a mesh containing the actual discontinuity.Peer ReviewedPostprint (author's final draft
Discontinuities in the Maximum-Entropy Inference
We revisit the maximum-entropy inference of the state of a finite-level
quantum system under linear constraints. The constraints are specified by the
expected values of a set of fixed observables. We point out the existence of
discontinuities in this inference method. This is a pure quantum phenomenon
since the maximum-entropy inference is continuous for mutually commuting
observables. The question arises why some sets of observables are distinguished
by a discontinuity in an inference method which is still discussed as a
universal inference method. In this paper we make an example of a discontinuity
and we explain a characterization of the discontinuities in terms of the
openness of the (restricted) linear map that assigns expected values to states.Comment: 8 pages, 3 figures, 32nd International Workshop on Bayesian Inference
and Maximum Entropy Methods in Science and Engineering, Garching, Germany,
15-20 July 201
Continuity of the maximum-entropy inference: Convex geometry and numerical ranges approach
We study the continuity of an abstract generalization of the maximum-entropy
inference - a maximizer. It is defined as a right-inverse of a linear map
restricted to a convex body which uniquely maximizes on each fiber of the
linear map a continuous function on the convex body. Using convex geometry we
prove, amongst others, the existence of discontinuities of the maximizer at
limits of extremal points not being extremal points themselves and apply the
result to quantum correlations. Further, we use numerical range methods in the
case of quantum inference which refers to two observables. One result is a
complete characterization of points of discontinuity for matrices.Comment: 27 page
A course space construction based on local Dirichlet-to-Neumann maps
Coarse-grid correction is a key ingredient of scalable domain decomposition methods. In this work we construct coarse-grid space using the low-frequency modes of the subdomain Dirichlet-to-Neumann maps and apply the obtained two-level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition. Our method is suitable for parallel implementation, and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings
Entanglement in fermionic chains with finite range coupling and broken symmetries
We obtain a formula for the determinant of a block Toeplitz matrix associated
with a quadratic fermionic chain with complex coupling. Such couplings break
reflection symmetry and/or charge conjugation symmetry. We then apply this
formula to compute the Renyi entropy of a partial observation to a subsystem
consisting of contiguous sites in the limit of large . The present work
generalizes similar results due to Its, Jin, Korepin and Its, Mezzadri, Mo. A
striking new feature of our formula for the entanglement entropy is the
appearance of a term scaling with the logarithm of the size of . This
logarithmic behaviour originates from certain discontinuities in the symbol of
the block Toeplitz matrix. Equipped with this formula we analyse the
entanglement entropy of a Dzyaloshinski-Moriya spin chain and a Kitaev
fermionic chain with long range pairing.Comment: 27 pages, 5 figure
Reconstruction of cracks and material losses by perimeter-like penalizations and phase-field methods: numerical results
We numerically implement the variational approach for reconstruction in the
inverse crack and cavity problems developed by one of the authors. The method
is based on a suitably adapted free-discontinuity problem. Its main features
are the use of phase-field functions to describe the defects to be
reconstructed and the use of perimeter-like penalizations to regularize the
ill-posed problem.
The numerical implementation is based on the solution of the corresponding
optimality system by a gradient method. Numerical simulations are presented to
show the validity of the method.Comment: 15 pages, 12 figure
Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric Finite Element Method
Let \Omega \subset \RR^d, , be a bounded domain with
piecewise smooth boundary and let be an open subset of a
Banach space . Motivated by questions in "Uncertainty Quantification," we
consider a parametric family of uniformly strongly
elliptic, second order partial differential operators on . We
allow jump discontinuities in the coefficients. We establish a regularity
result for the solution u: \Omega \times U \to \RR of the parametric,
elliptic boundary value/transmission problem , , with
mixed Dirichlet-Neumann boundary conditions in the case when the boundary and
the interface are smooth and in the general case for . Our regularity and
well-posedness results are formulated in a scale of broken weighted Sobolev
spaces \hat\maK^{m+1}_{a+1}(\Omega) of Babu\v{s}ka-Kondrat'ev type in
, possibly augmented by some locally constant functions. This implies
that the parametric, elliptic PDEs admit a shift theorem that
is uniform in the parameter . In turn, this then leads to
-quasi-optimal rates of convergence (i.e. algebraic orders of convergence)
for the Galerkin approximations of the solution , where the approximation
spaces are defined using the "polynomial chaos expansion" of with respect
to a suitable family of tensorized Lagrange polynomials, following the method
developed by Cohen, Devore, and Schwab (2010)
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