3,766 research outputs found
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Sard kernel theorems on triangular and rectangular domains with extensions and applications to finite element error bounds
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Interpolation remainder theory from taylor expansions with non-rectangular domains of influence
Sobolev norm error bounds are derived for interpolation remainders on triangles using two types of Taylor expansion. These bounds are applied to the finite element analysis of Poisson's equation on a triangulation of a polygonal region
Smooth polynomial interpolation to boundary data on triangles
Boolean sum interpolation theory is used to derive a polynomial interpolant which interpolates a function F ∈ CN(∂T), and its derivatives of order N and less, on the boundary 3T of a triangle T. A triangle with one curved side is also considered
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Compatable smooth interpolation in triangles
Boolean sum smooth interpolation to boundary data on a triangle is described. Sufficient conditions are given so that the functions when pieced together form a CN-1(Ω) function over a triangular subdivision of a polygonal region Ω and the precision sets of the interpolation functions are derived. The interpolants are modified so that the compatability conditions on the function which is interpolated can be removed and a C1 interpolant is used to illustrate the theory. The generation of interpolation schemes for discrete boundary data is also discussed
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Exact and approximate boundary data interpolation in the finite element method
Matching boundary data exactly in an elliptic problem avoids one of Strang's "variational crimes". (Strang and Fix (1973)). Supporting numerical evidence for this procedure is given by Marshall and Mitchell (1973), who considered the solution of Laplace's equation with Dirichlet boundary data by bilinear elements over squares and measured the errors in the L2 norm. Then Marshall and Mitchell (1978) obtained some surprising results: for certain triangular elements, matching the boundary data exactly produced worse results than the usual procedure of interpolating the boundary data
Experimental exploration over a quantum control landscape through nuclear magnetic resonance
The growing successes in performing quantum control experiments motivated the
development of control landscape analysis as a basis to explain these
findings.When a quantum system is controlled by an electromagnetic field, the
observable as a functional of the control field forms a landscape. Theoretical
analyses have revealed many properties of control landscapes, especially
regarding their slopes, curvatures, and topologies. A full experimental
assessment of the landscape predictions is important for future consideration
of controlling quantum phenomena. Nuclear magnetic resonance (NMR) is exploited
here as an ideal laboratory setting for quantitative testing of the landscape
principles. The experiments are performed on a simple two-level proton system
in a HO-DO sample. We report a variety of NMR experiments roving over
the control landscape based on estimation of the gradient and Hessian,
including ascent or descent of the landscape, level set exploration, and an
assessment of the theoretical predictions on the structure of the Hessian. The
experimental results are fully consistent with the theoretical predictions. The
procedures employed in this study provide the basis for future multispin
control landscape exploration where additional features are predicted to exist
Searching for quantum optimal controls in the presence of singular critical points
Quantum optimal control has enjoyed wide success for a variety of theoretical
and experimental objectives. These favorable results have been attributed to
advantageous properties of the corresponding control landscapes, which are free
from local optima if three conditions are met: (1) the quantum system is
controllable, (2) the Jacobian of the map from the control field to the
evolution operator is full rank, and (3) the control field is not constrained.
This paper explores how gradient searches for globally optimal control fields
are affected by deviations from assumption (2). In some quantum control
problems, so-called singular critical points, at which the Jacobian is
rank-deficient, may exist on the landscape. Using optimal control simulations,
we show that search failure is only observed when a singular critical point is
also a second-order trap, which occurs if the control problem meets additional
conditions involving the system Hamiltonian and/or the control objective. All
known second-order traps occur at constant control fields, and we also show
that they only affect searches that originate very close to them. As a result,
even when such traps exist on the control landscape, they are unlikely to
affect well-designed gradient optimizations under realistic searching
conditions.Comment: 14 pages, 2 figure
Searching for quantum optimal controls under severe constraints
The success of quantum optimal control for both experimental and theoretical
objectives is connected to the topology of the corresponding control
landscapes, which are free from local traps if three conditions are met: (1)
the quantum system is controllable, (2) the Jacobian of the map from the
control field to the evolution operator is of full rank, and (3) there are no
constraints on the control field. This paper investigates how the violation of
assumption (3) affects gradient searches for globally optimal control fields.
The satisfaction of assumptions (1) and (2) ensures that the control landscape
lacks fundamental traps, but certain control constraints can still introduce
artificial traps. Proper management of these constraints is an issue of great
practical importance for numerical simulations as well as optimization in the
laboratory. Using optimal control simulations, we show that constraints on
quantities such as the number of control variables, the control duration, and
the field strength are potentially severe enough to prevent successful
optimization of the objective. For each such constraint, we show that exceeding
quantifiable limits can prevent gradient searches from reaching a globally
optimal solution. These results demonstrate that careful choice of relevant
control parameters helps to eliminate artificial traps and facilitate
successful optimization.Comment: 16 pages, 7 figure
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