2,473 research outputs found
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
This work develops a nonlinear multigrid method for diffusion problems
discretized by cell-centered finite volume methods on general unstructured
grids. The multigrid hierarchy is constructed algebraically using aggregation
of degrees of freedom and spectral decomposition of reference linear operators
associated with the aggregates. For rapid convergence, it is important that the
resulting coarse spaces have good approximation properties. In our approach,
the approximation quality can be directly improved by including more spectral
degrees of freedom in the coarsening process. Further, by exploiting local
coarsening and a piecewise-constant approximation when evaluating the nonlinear
component, the coarse level problems are assembled and solved without ever
re-visiting the fine level, an essential element for multigrid algorithms to
achieve optimal scalability. Numerical examples comparing relative performance
of the proposed nonlinear multigrid solvers with standard single-level
approaches -- Picard's and Newton's methods -- are presented. Results show that
the proposed solver consistently outperforms the single-level methods, both in
efficiency and robustness
Numerical Verification of Affine Systems with up to a Billion Dimensions
Affine systems reachability is the basis of many verification methods. With
further computation, methods exist to reason about richer models with inputs,
nonlinear differential equations, and hybrid dynamics. As such, the scalability
of affine systems verification is a prerequisite to scalable analysis for more
complex systems. In this paper, we improve the scalability of affine systems
verification, in terms of the number of dimensions (variables) in the system.
The reachable states of affine systems can be written in terms of the matrix
exponential, and safety checking can be performed at specific time steps with
linear programming. Unfortunately, for large systems with many state variables,
this direct approach requires an intractable amount of memory while using an
intractable amount of computation time. We overcome these challenges by
combining several methods that leverage common problem structure. Memory is
reduced by exploiting initial states that are not full-dimensional and safety
properties (outputs) over a few linear projections of the state variables.
Computation time is saved by using numerical simulations to compute only
projections of the matrix exponential relevant for the verification problem.
Since large systems often have sparse dynamics, we use Krylov-subspace
simulation approaches based on the Arnoldi or Lanczos iterations. Our method
produces accurate counter-examples when properties are violated and, in the
extreme case with sufficient problem structure, can analyze a system with one
billion real-valued state variables
Algorithmic Verification of Continuous and Hybrid Systems
We provide a tutorial introduction to reachability computation, a class of
computational techniques that exports verification technology toward continuous
and hybrid systems. For open under-determined systems, this technique can
sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661
Analysis of parametric biological models with non-linear dynamics
In this paper we present recent results on parametric analysis of biological
models. The underlying method is based on the algorithms for computing
trajectory sets of hybrid systems with polynomial dynamics. The method is then
applied to two case studies of biological systems: one is a cardiac cell model
for studying the conditions for cardiac abnormalities, and the second is a
model of insect nest-site choice.Comment: In Proceedings HSB 2012, arXiv:1208.315
Design of defect spins in piezoelectric aluminum nitride for solid-state hybrid quantum technologies
Spin defects in wide-band gap semiconductors are promising systems for the
realization of quantum bits, or qubits, in solid-state environments. To date,
defect qubits have only been realized in materials with strong covalent bonds.
Here, we introduce a strain-driven scheme to rationally design defect spins in
functional ionic crystals, which may operate as potential qubits. In
particular, using a combination of state-of-the-art ab-initio calculations
based on hybrid density functional and many-body perturbation theory, we
predicted that the negatively charged nitrogen vacancy center in piezoelectric
aluminum nitride exhibits spin-triplet ground states under realistic uni- and
bi-axial strain conditions; such states may be harnessed for the realization of
qubits. The strain-driven strategy adopted here can be readily extended to a
wide range of point defects in other wide-band gap semiconductors, paving the
way to controlling the spin properties of defects in ionic systems for
potential spintronic technologies.Comment: In press. 32 pages, 4 figures, 3 tables, Scientific Reports 201
Fluid dynamics: an emerging route for the scalable production of graphene in the last five years
Bulk applications of graphene in fields such as advanced composites,
conductive ink, and energy storage require cheap and scalable graphene.
Fortunately, in the last decade, liquid-phase exfoliation of graphite to give
pristine graphene has been thought as a promising way to massive production of
graphene at high efficiency and low cost, in terms of the cheap and abundant
graphite source and a variety of cost-effective exfoliation techniques. Though
many exfoliation techniques are available so far, this article will highlight
the recent progress of fluid dynamics route which emerges as a promising
scalable and efficient way for graphene production in the last five years. The
emphasis is set on vortex fluidic devices and pressure- and mixer-driven fluid
dynamics, with our perspectives on the latest progress, exfoliation mechanism,
and some key issues that require further study in order to realize industrial
applications.Comment: 18 figure
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