520 research outputs found
The Polynomial CarathΓ©odoryβFejΓ©r Approximation Method for Jordan Regions
We propose a method for the approximation of analytic functions on Jordan regions that is based on a CarathΓ©odoryβFejΓ©r type of economization of the Faber series. The method turns out to be very effective if the boundary of the region is analytic. It often still works when the region degenerates to a Jordan arc. We also derive related lower and upper bounds for the error of the best approximatio
Conformal Mapping on Rough Boundaries II: Applications to bi-harmonic problems
We use a conformal mapping method introduced in a companion paper to study
the properties of bi-harmonic fields in the vicinity of rough boundaries. We
focus our analysis on two different situations where such bi-harmonic problems
are encountered: a Stokes flow near a rough wall and the stress distribution on
the rough interface of a material in uni-axial tension. We perform a complete
numerical solution of these two-dimensional problems for any univalued rough
surfaces. We present results for sinusoidal and self-affine surface whose slope
can locally reach 2.5. Beyond the numerical solution we present perturbative
solutions of these problems. We show in particular that at first order in
roughness amplitude, the surface stress of a material in uni-axial tension can
be directly obtained from the Hilbert transform of the local slope. In case of
self-affine surfaces, we show that the stress distribution presents, for large
stresses, a power law tail whose exponent continuously depends on the roughness
amplitude
ΠΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠ°Ρ Π±Π΅Π·ΠΎΠΏΠ°ΡΠ½ΠΎΡΡΡ ΡΡΠ½ΠΊΡΠΈΠΎΠ½ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΏΡΠ΅Π΄ΠΏΡΠΈΡΡΠΈΡ Π² ΡΡΠ»ΠΎΠ²ΠΈΡΡ ΡΠ΅ΡΠ΅Π²ΠΎΠΉ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΠΊΠΈ
Π Π½Π°ΡΠ΅ Π²ΡΠ΅ΠΌΡ Π²ΠΎΠ·Π½ΠΈΠΊΠ½ΠΎΠ²Π΅Π½ΠΈΠ΅ ΡΠ΅ΡΠ΅Π²ΡΡ
ΠΎΡΠΎΠ±Π΅Π½Π½ΠΎΡΡΠ΅ΠΉ Π² ΡΠΊΠΎΠ½ΠΎΠΌΠΈΠΊΠ΅ ΡΠ²ΡΠ·ΡΠ²Π°ΡΡ Ρ ΡΠ°Π·Π²ΠΈΡΠΈΠ΅ΠΌ ΠΈΠ½ΡΠΎΡΠΌΠ°ΡΠΈΠΎΠ½Π½ΡΡ
ΡΠ΅Ρ
Π½ΠΎΠ»ΠΎΠ³ΠΈΠΉ, ΡΡΠΎ ΠΏΡΠΈΠ²ΠΎΠ΄ΠΈΡ ΠΊ ΡΠ²ΠΎΠ»ΡΡΠΈΠΈ ΡΠΎΠ²ΡΠ΅ΠΌΠ΅Π½Π½ΡΡ
ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠΈΡ
ΡΠΈΡΡΠ΅ΠΌ, ΡΠ°Π·Π²ΠΈΡΠΈΡ Π½Π΅ΡΡΠ½ΠΎΡΠ½ΡΡ
ΠΌΠ΅Ρ
Π°Π½ΠΈΠ·ΠΌΠΎΠ² ΡΠ΅Π³ΡΠ»ΠΈΡΠΎΠ²Π°Π½ΠΈΡ ΠΈ ΡΠ΅ΡΠ΅Π²ΡΡ
ΠΎΡΠ³Π°Π½ΠΈΠ·Π°ΡΠΈΠΎΠ½Π½ΡΡ
ΡΡΡΡΠΊΡΡΡ. ΠΡΡΠ³ΠΈΠΌΠΈ ΡΠ»ΠΎΠ²Π°ΠΌΠΈ, ΡΠ΅ΡΠ΅Π²ΡΠ΅ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΎΡΠ½ΠΎΡΠ΅Π½ΠΈΡ ΠΈΠ³ΡΠ°ΡΡ ΠΎΡΠΎΠ±ΡΡ ΡΠΎΠ»Ρ Π² ΠΏΡΠΎΡΠ΅ΡΡΠ΅ ΠΊΠΎΠΎΡΠ΄ΠΈΠ½Π°ΡΠΈΠΈ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠΈΡ
Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡΠ²ΠΈΠΉ. ΠΠ°Π½Π½ΡΠ΅ ΠΈΠ·ΠΌΠ΅Π½Π΅Π½ΠΈΡ ΠΎΠ±ΠΎΡΡΡΡΡΡ ΠΏΡΠΎΠ±Π»Π΅ΠΌΡ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΡΠ΅ΡΠΊΠΎΠΉ Π±Π΅Π·ΠΎΠΏΠ°ΡΠ½ΠΎΡΡΠΈ ΠΏΡΠ΅Π΄ΠΏΡΠΈΡΡΠΈΡ Π² ΡΡΠ»ΠΎΠ²ΠΈΡΡ
ΡΠ°Π·Π²ΠΈΡΠΈΡ ΠΌΠ΅ΠΆΠΎΡΠ³Π°Π½ΠΈΠ·Π°ΡΠΈΠΎΠ½Π½ΡΡ
Π²Π·Π°ΠΈΠΌΠΎΠ΄Π΅ΠΉΡΡΠ²ΠΈΠΉ ΡΠΎΡΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ ΠΈ Π½Π΅ΡΠΎΡΠΌΠ°Π»ΡΠ½ΠΎΠ³ΠΎ Ρ
Π°ΡΠ°ΠΊΡΠ΅ΡΠ° Ρ ΠΏΠΎΠ·ΠΈΡΠΈΠΈ ΡΠ΅ΡΠ΅Π²ΠΎΠΉ ΡΠΊΠΎΠ½ΠΎΠΌΠΈΠΊΠΈ
Conformal Mapping on Rough Boundaries I: Applications to harmonic problems
The aim of this study is to analyze the properties of harmonic fields in the
vicinity of rough boundaries where either a constant potential or a zero flux
is imposed, while a constant field is prescribed at an infinite distance from
this boundary. We introduce a conformal mapping technique that is tailored to
this problem in two dimensions. An efficient algorithm is introduced to compute
the conformal map for arbitrarily chosen boundaries. Harmonic fields can then
simply be read from the conformal map. We discuss applications to "equivalent"
smooth interfaces. We study the correlations between the topography and the
field at the surface. Finally we apply the conformal map to the computation of
inhomogeneous harmonic fields such as the derivation of Green function for
localized flux on the surface of a rough boundary
Order reduction approaches for the algebraic Riccati equation and the LQR problem
We explore order reduction techniques for solving the algebraic Riccati
equation (ARE), and investigating the numerical solution of the
linear-quadratic regulator problem (LQR). A classical approach is to build a
surrogate low dimensional model of the dynamical system, for instance by means
of balanced truncation, and then solve the corresponding ARE. Alternatively,
iterative methods can be used to directly solve the ARE and use its approximate
solution to estimate quantities associated with the LQR. We propose a class of
Petrov-Galerkin strategies that simultaneously reduce the dynamical system
while approximately solving the ARE by projection. This methodology
significantly generalizes a recently developed Galerkin method by using a pair
of projection spaces, as it is often done in model order reduction of dynamical
systems. Numerical experiments illustrate the advantages of the new class of
methods over classical approaches when dealing with large matrices
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
A bootstrap method for sum-of-poles approximations
A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138β1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples
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