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On the boundary layer arising in the spin-up of a stratified fluid in a container with sloping walls
Get PDFIn this paper we consider the boundary layer that forms on the sloping walls of a
rotating container (notably a conical container), filled with a stratified fluid, when
flow conditions are changed abruptly from some initial (uniform) state. The structure
of the solution valid away from the cone apex is derived, and it is shown that a
similarity-type solution is appropriate. This system, which is inherently nonlinear in
nature, is solved numerically for several flow regimes, and the results reveal a number
of interesting and diverse features.
In one case, a steady state is attained at large times inside the boundary layer.
In a second case, a finite-time singularity occurs, which is fully analysed. A third
scenario involves a double boundary-layer structure developing at large times, most
significantly including an outer region that grows in thickness as the square-root of
time.
We also consider directly the nonlinear fully steady solutions to the problem, and
map out in parameter space the likely ultimate flow behaviour. Intriguingly, we find
cases where, when the rotation rate of the container is equal to that of the main
body of the fluid, an alternative nonlinear state is preferred, rather than the trivial
(uniform) solution.
Finally, utilizing Laplace transforms, we re-investigate the linear initial-value prob-
lem for small differential spin-up studied by MacCready & Rhines (1991), recovering
the growing-layer solution they found. However, in contrast to earlier work, we
find a critical value of the buoyancy parameter beyond which the solution grows
exponentially in time, consistent with our nonlinear results
Stable iterations for the matrix square root
No full textAny matrix with no nonpositive real eigenvalues has a unique square root for which every eigenvalue lies in the open right half-plane. A link between the matrix sign function and this square root is exploited to derive both old and new iterations for the square root from iterations for the sign function. One new iteration is a quadratically convergent Schulz iteration based entirely on matrix multiplication; it converges only locally, but can be used to compute the square root of any nonsingular M-matrix. A new Padé iteration well suited to parallel implementation is also derived and its properties explained. Iterative methods for the matrix square root are notorious for suffering from numerical instability. It is shown that apparently innocuous algorithmic modifications to the Padé iteration can lead to instability, and a perturbation analysis is given to provide some explanation. Numerical experiments are included and advice is offered on the choice of iterative method for computing the matrix square root
Computing the field of values and pseudospectra using the Lanczos method with continuation
No full textThe field of values and pseudospectra are useful tools for understanding the behaviour of various matrix processes. To compute these subsets of the complex plane it is necessary to estimate one or two eigenvalues of a large number of parametrized Hermitian matrices; these computations are prohibitively expensive for large, possibly sparse, matrices, if done by use of the QR algorithm. We describe an approach based on the Lanczos method with selective reorthogonalization and Chebyshev acceleration that, when combined with continuation and a shift and invert technique, enables efficient and reliable computation of the field of values and pseudospectra for large matrices. The idea of using the Lanczos method with continuation to compute pseudospectra is not new, but in experiments reported here our algorithm is faster and more accurate than existing algorithms of this type
The representation theory of p-adic GL(n) and Deligne-Langlands parameters
No full textLet GL(n) denote the general linear group over a local nonarchimedean field. For the equivalence classes of irreducible smooth representations of GL(n) admitting nonzero Iwahori fixed vectors, we have the classical Deligne-Langlands parameters. We prove that the Deligne-Langlands parameters have a definite geometric structure: the structure of the extended quotient of a complex torus of dimension n by the symmetric group S_n.
Notes:
1. At the time of publication, the extended quotient was called the Brylinski quotient.
2. We say in the Introduction that this chapter is a "re-interpretation" of [8]. Reference [8] was, however, never published, so this eprint is now the primary source: the book itself is out of print.
3. The main result in this chapter is the precursor of a wide-ranging geometric conjecture, in the representation theory of p-adic groups, developed by Anne-Marie Aubert, Paul Baum and myself in a series of papers from 2007 -- 2011
Workshop on perinatally acquired human immuno-deficiency virus infection in long-term surviving children: A collaborative study of factors contributing to slow disease progression
No full textStability of parallel triangular system solvers
Get PDFSeveral parallel algorithms have been proposed for the solution of triangular systems. The stability of four of them is analysed here: a fan-in algorithm, a block elimination method, a method based on a factorized power series expansion of the matrix inverse, and a method based on a divide and conquer matrix inversion technique. New forward error and residual bounds are derived, including an improvement on the bounds of Sameh and Brent for the fan-in algorithm. A forward error bound is identified that holds not only for all the methods described here, but for any triangular equation solver that does not rely on algebraic cancellation; among the implications of the bound is that any such method is extremely accurate for certain special types of triangular systems
Stability of Block LU Factorization
No full textMany of the currently popular ‘block algorithms’ are scalar algorithms in which the operations have been
grouped and reordered into matrix operations. One genuine block algorithm in practical use is block LU
factorization, and this has recently been shown by Demmel and Higham to be unstable in general. It is shown
here that block LU factorization is stable if A is block diagonally dominant by columns. Moreover, for a
general matrix the level of instability in block LU factorization can be bounded in terms of the condition
number K(A) and the growth factor for Gaussian elimination without pivoting. A consequence is that block
LU factorization is stable for a matrix A that is symmetric positive definite or point diagonally dominant by
rows or columns as long as A is well-conditioned
Matrix powers in finite precision arithmetic
Get PDFIf is a square matrix with spectral radius less than 1 then , but the powers computed in finite precision arithmetic may or may not converge. We derive a sufficient condition for and a bound on , both expressed in terms of the Jordan canonical form of . Examples show that the results can be sharp. We show that the sufficient condition can be rephrased in terms of a pseudospectrum of when is diagonalizable, under certain assumptions. Our analysis leads to the rule of thumb that convergence or divergence of the computed powers of can be expected according as the spectral radius computed by any backward stable algorithm is less than or greater than 1
POMPUS: an optimized EIT reconstruction algorithm
Get PDFElectrical impedance tomography (EIT) is a non-invasive imaging technique which aims to image the impedance of material within a test volume from electrical measurements made on the surface. The reconstruction of impedance images is an ill-posed problem which is both extremely sensitive to noise and highly computationally intensive. This paper defines an experimental measurement in EIT and calculates optimal experiments which maximize the distinguishability between the region to be imaged and a best estimate conductivity distribution. These optimal experiments can be derived from measurements made on the boundary. We describe a reconstruction algorithm, known as POMPUS, which is based on the use of optimal experiments. We have shown that, given some mild constraints, if POMPUS converges, it converges to a stationary point of our objective function. It is demonstrated to be many times faster than standard, Newton based, reconstruction algorithms. Results using synthetic data indicate that the images produced by POMPUS are comparable to those produced by these standard algorithms
Chest Impedance Imaging Using Trigonometric Current Patterns
Get PDFThis paper reports the results from the application of the 32 channel Oxford Brookes Adaptve Current Tomograph, OXBACT-III to the chest of a human volunteer. A frame rate of 10 per second was achieved using trigonometric drive currents