380 research outputs found
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
We study linear parabolic initial-value problems in a space-time variational
formulation based on fractional calculus. This formulation uses "time
derivatives of order one half" on the bi-infinite time axis. We show that for
linear, parabolic initial-boundary value problems on , the
corresponding bilinear form admits an inf-sup condition with sparse tensor
product trial and test function spaces. We deduce optimality of compressive,
space-time Galerkin discretizations, where stability of Galerkin approximations
is implied by the well-posedness of the parabolic operator equation. The
variational setting adopted here admits more general Riesz bases than previous
work; in particular, no stability in negative order Sobolev spaces on the
spatial or temporal domains is required of the Riesz bases accommodated by the
present formulation. The trial and test spaces are based on Sobolev spaces of
equal order with respect to the temporal variable. Sparse tensor products
of multi-level decompositions of the spatial and temporal spaces in Galerkin
discretizations lead to large, non-symmetric linear systems of equations. We
prove that their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of freedom, the
convergence orders equal, up to logarithmic terms, those of best -term
approximations of solutions of the corresponding elliptic problems.Comment: 26 page
Exact resolution method for general 1D polynomial Schr\"odinger equation
The stationary 1D Schr\"odinger equation with a polynomial potential
of degree N is reduced to a system of exact quantization conditions of
Bohr-Sommerfeld form. They arise from bilinear (Wronskian) functional relations
pairing spectral determinants of (N+2) generically distinct operators, all the
transforms of one quantum Hamiltonian under a cyclic group of complex scalings.
The determinants' zeros define (N+2) semi-infinite chains of points in the
complex spectral plane, and they encode the original quantum problem. Each
chain can now be described by an exact quantization condition which constrains
it in terms of its neighbors, resulting in closed equilibrium conditions for
the global chain system; these are supplemented by the standard
(Bohr-Sommerfeld) quantization conditions, which bind the infinite tail of each
chain asymptotically. This reduced problem is then probed numerically for
effective solvability upon test cases (mostly, symmetric quartic oscillators):
we find that the iterative enforcement of all the quantization conditions
generates discrete chain dynamics which appear to converge geometrically
towards the correct eigenvalues/eigenfunctions. We conjecture that the exact
quantization then acts by specifying reduced chain dynamics which can be stable
(contractive) and thus determine the exact quantum data as their fixed point.
(To date, this statement is verified only empirically and in a vicinity of
purely quartic or sextic potentials .)Comment: flatex text.tex, 4 files Submitted to: J. Phys. A: Math. Ge
On the computation of geometric features of spectra of linear operators on Hilbert spaces
Computing spectra is a central problem in computational mathematics with an
abundance of applications throughout the sciences. However, in many
applications gaining an approximation of the spectrum is not enough. Often it
is vital to determine geometric features of spectra such as Lebesgue measure,
capacity or fractal dimensions, different types of spectral radii and numerical
ranges, or to detect essential spectral gaps and the corresponding failure of
the finite section method. Despite new results on computing spectra and the
substantial interest in these geometric problems, there remain no general
methods able to compute such geometric features of spectra of
infinite-dimensional operators. We provide the first algorithms for the
computation of many of these longstanding problems (including the above). As
demonstrated with computational examples, the new algorithms yield a library of
new methods. Recent progress in computational spectral problems in infinite
dimensions has led to the Solvability Complexity Index (SCI) hierarchy, which
classifies the difficulty of computational problems. These results reveal that
infinite-dimensional spectral problems yield an intricate infinite
classification theory determining which spectral problems can be solved and
with which type of algorithm. This is very much related to S. Smale's
comprehensive program on the foundations of computational mathematics initiated
in the 1980s. We classify the computation of geometric features of spectra in
the SCI hierarchy, allowing us to precisely determine the boundaries of what
computers can achieve and prove that our algorithms are optimal. We also
provide a new universal technique for establishing lower bounds in the SCI
hierarchy, which both greatly simplifies previous SCI arguments and allows new,
formerly unattainable, classifications
Global Hopf bifurcation in the ZIP regulatory system
Regulation of zinc uptake in roots of Arabidopsis thaliana has recently been
modeled by a system of ordinary differential equations based on the uptake of
zinc, expression of a transporter protein and the interaction between an
activator and inhibitor. For certain parameter choices the steady state of this
model becomes unstable upon variation in the external zinc concentration.
Numerical results show periodic orbits emerging between two critical values of
the external zinc concentration. Here we show the existence of a global Hopf
bifurcation with a continuous family of stable periodic orbits between two Hopf
bifurcation points. The stability of the orbits in a neighborhood of the
bifurcation points is analyzed by deriving the normal form, while the stability
of the orbits in the global continuation is shown by calculation of the Floquet
multipliers. From a biological point of view, stable periodic orbits lead to
potentially toxic zinc peaks in plant cells. Buffering is believed to be an
efficient way to deal with strong transient variations in zinc supply. We
extend the model by a buffer reaction and analyze the stability of the steady
state in dependence of the properties of this reaction. We find that a large
enough equilibrium constant of the buffering reaction stabilizes the steady
state and prevents the development of oscillations. Hence, our results suggest
that buffering has a key role in the dynamics of zinc homeostasis in plant
cells.Comment: 22 pages, 5 figures, uses svjour3.cl
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