193 research outputs found
Eigenfunction scarring and improvements in bounds
We study the relationship between growth of eigenfunctions and
their concentration as measured by defect measures. In particular, we
show that scarring in the sense of concentration of defect measure on certain
submanifolds is incompatible with maximal growth. In addition, we
show that a defect measure which is too diffuse, such as the Liouville measure,
is also incompatible with maximal eigenfunction growth.Comment: 10 page
Averages of eigenfunctions over hypersurfaces
Let be a compact, smooth, Riemannian manifold and an
-normalized sequence of Laplace eigenfunctions with defect measure .
Let be a smooth hypersurface. Our main result says that when is
concentrated conormally to , the eigenfunction restrictions
to and the restrictions of their normal derivatives to have integrals
converging to 0 as .Comment: 18 pages, 1 figur
Strong practical stability based robust stabilization of uncertain discrete linear repetitive processes
Repetitive processes are a distinct class of 2D systems of both theoretical and practical interest whose dynamics evolve over a subset of the positive quadrant in the 2D plane. The stability theory for these processes originally consisted of two distinct concepts termed asymptotic stability and stability along the pass respectively where the former is a necessary condition for the latter. Stability along the pass demands a bounded-input bounded-output property over the complete positive quadrant of the 2D plane and this is a very strong requirement, especially in terms of control law design. A more feasible alternative for some cases is strong practical stability, where previous work has formulated this property and obtained necessary and sufficient conditions for its existence together with Linear Matrix Inequality (LMI) based tests, which then extend to allow control law design. This paper develops considerably simpler, and hence computationally more efficient, stability tests that extend to allow control law design in the presence of uncertainty in process model
Sharp preasymptotic error bounds for the Helmholtz -FEM
In the analysis of the -version of the finite-element method (FEM), with
fixed polynomial degree , applied to the Helmholtz equation with wavenumber
, the is when is
sufficiently small and the sequence of Galerkin solutions are quasioptimal;
here is the norm of the Helmholtz solution operator, normalised
so that for nontrapping problems. In the
, one expects that if is
sufficiently small, then (for physical data) the relative error of the Galerkin
solution is controllably small. In this paper, we prove the natural error
bounds in the preasymptotic regime for the variable-coefficient Helmholtz
equation in the exterior of a Dirichlet, or Neumann, or penetrable obstacle (or
combinations of these) and with the radiation condition
realised exactly using the Dirichlet-to-Neumann map on the boundary of a ball
approximated either by a radial perfectly-matched layer (PML) or
an impedance boundary condition. Previously, such bounds for were only
available for Dirichlet obstacles with the radiation condition approximated by
an impedance boundary condition. Our result is obtained via a novel
generalisation of the "elliptic-projection" argument (the argument used to
obtain the result for ) which can be applied to a wide variety of abstract
Helmholtz-type problems
High-frequency estimates on boundary integral operators for the Helmholtz exterior Neumann problem
We study a commonly-used second-kind boundary-integral equation for solving
the Helmholtz exterior Neumann problem at high frequency, where, writing
for the boundary of the obstacle, the relevant integral operators map
to itself. We prove new frequency-explicit bounds on the norms of
both the integral operator and its inverse. The bounds on the norm are valid
for piecewise-smooth and are sharp up to factors of (where
is the wavenumber), and the bounds on the norm of the inverse are valid for
smooth and are observed to be sharp at least when is smooth
with strictly-positive curvature. Together, these results give bounds on the
condition number of the operator on ; this is the first time
condition-number bounds have been proved for this operator for
obstacles other than balls
Eigenvalues of the truncated Helmholtz solution operator under strong trapping
For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we
prove that if there exists a family of quasimodes (as is the case when the
exterior of the obstacle has stable trapped rays), then there exist near-zero
eigenvalues of the standard variational formulation of the exterior Dirichlet
problem (recall that this formulation involves truncating the exterior domain
and applying the exterior Dirichlet-to-Neumann map on the truncation boundary).
Our motivation for proving this result is that a) the finite-element method
for computing approximations to solutions of the Helmholtz equation is based on
the standard variational formulation, and b) the location of eigenvalues, and
especially near-zero ones, plays a key role in understanding how iterative
solvers such as the generalised minimum residual method (GMRES) behave when
used to solve linear systems, in particular those arising from the
finite-element method. The result proved in this paper is thus the first step
towards rigorously understanding how GMRES behaves when applied to
discretisations of high-frequency Helmholtz problems under strong trapping (the
subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021])
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