Past Numerical Analysis Group Internal Seminar

Flow thought a porous media is usually described by assuming the superficial velocity can be expressed in terms of a constant permeability and a pressure gradient. In poroelastic flows the underlying elastic matrix responds to changes in the fluid pressure. When the elastic deformation is allowed to influence the permeability through the elastic strain, it becomes possible for increased fluid pressure gradient not to result in increased flow, but to decrease the permeability and potentially this may close off or choke the flow. I will talk about a simple model problem for a number of different elastic constitutive models and a number of different permeability-strain models and examine whether there is a general criterion that can be derived to show when, or indeed if, choking can occur for different elasticity-permeability combinations.

D-finite functions are solutions of linear differential equations with polynomial coefficients.  They have drawn a lot of attention, both in Computer Algebra--because of their numerous (algorithmic) closure properties--but also in Numerical Analysis, because their defining ODEs can be numerically solved very efficiently.  In this talk, I will show how a mix of symbolic and numerical methods yields fast and well-conditioned spectral methods on various domains and using different bases of functions.

It is well-known that a matrix $A$ is Hurwitz stable if and only if there exists a positive definite solution to the Lyapunov matrix equation $A X + X A^* = B$, where $B$ is Hermitian negative definite. We present a verified numerical algorithm to rigorously prove the stability of a given matrix $A$ in the presence of rounding errors.  The computational cost of the algorithm is cubic and it is fast since we can cast almost all operations in level 3 BLAS for which interval arithmetic can be implemented very efficiently.  This is a joint work with Andreas Frommer and the results are already published in ETNA in 2013.

This talk concerns the numerical solution of elliptic partial differential equations posed on general smooth surfaces by the Closest Point Method. Based on the closest point representation of the surface, we formulate an embedding equation in a narrow band surrounding the surface, then discretize it using standard finite differences and interpolation schemes. Numerical convergence of the method will be discussed. In order to solve the resulting large sparse linear systems, we propose a specific geometric multigrid method which makes use of the closest point representation of the surface.
 

We discuss the iterative solution of a finite element discretisation of the magma dynamics equations.  These equations share features of the Stokes equations, however, Elman-Silvester-Wathen (ESW) preconditioners for the magma dynamics equations are not optimal. By introducing a new field, the compaction pressure, into the magma dynamics equations, we have developed a new three-field preconditioner which is optimal in terms of problem size and less sensitive to physical parameters compared to the ESW preconditioners.

There is a well established body of research on quadratic optimization problems based on reformulations of the original problem as a conic program over the cone of completely positive matrices, or its conic dual, the cone of copositive matrices. As a result of this reformulation approach, novel solution schemes for quadratic polynomial optimization problems have been designed by drawing on conic programming tools, and the extensively studied cones of completely positive and of copositive matrices. In particular, this approach has been applied to address key combinatorial optimization problems. Along this line of research, we consider quadratically constrained quadratic programs and provide sufficient and necessary conditions for
this type of problems to be reformulated as a conic program over the cone of completely positive matrices. Thus, recent related results for quadratic problems can be further strengthened. Moreover, these results can be generalized to optimization problems involving higher order polynomias.

A key question to develop our understanding of turbulence in shear flows is: what is the smallest perturbation to the laminar flow that causes a transition to turbulence, and how does this change with the Reynolds number, R?  Finding this so-called ``minimal seed'' is as yet unachievable in direct numerical simulations of the Navier-Stokes equations. We search for the minimal seed in a low-dimensional model analogue to the full Navier-Stokes in plane sinusoidal flow, developed by Waleffe (1997). A previous such calculation found the minimal seed as the least distance (energy norm) from the origin (laminar flow) to the basin of attraction of another fixed point (turbulent attractor).  However, using a non-linear optimization technique, we found an internal boundary of the basin of attraction of the origin that separates flows which directly relaminarize from flows which undergo transient turbulence. It is this boundary which contains the minimal seed, and we find it to be smaller than the previously calculated minimal seed. We present results over a range of Reynolds numbers up to 2000 and find an R^{-1} scaling law fits reasonably well. We propose a new scaling law which asymptotes to R^{-1} for large R but, using some additional information, matches the minimal seed scaling better at low R.

Given a subset E of R^n with empty interior, what is the maximum regularity exponent s for which there exist non-zero distributions in the Bessel potential Sobolev space H^s_p(R^n) that are supported entirely inside E? This question has arisen many times in my recent investigations into boundary integral equation formulations of linear wave scattering by fractal screens, and it is closely related to other fundamental questions concerning Sobolev spaces defined on ``rough'' (i.e. non-Lipschitz) domains. Roughly speaking, one expects that the ``fatter'' the set, the higher the maximum regularity that can be supported. For sets of zero Lebesgue measure one can show, using results on certain set capacities from classical potential theory, that the maximum regularity (if it exists) is negative, and is (almost) characterised by the fractal (Hausdorff) dimension of E. For sets with positive measure the maximum regularity (if it exists) is non-negative,but appears more difficult to characterise in terms of geometrical properties of E.  I will present some partial results in this direction, which have recently been obtained by studying the asymptotic behaviour of the Fourier transform of the characteristic functions of certain fat Cantor sets.

It is well known that piecewise smooth signals are approximately sparse in a wavelet basis. However, other sparse representations are possible, such as the discrete gradient basis. It turns out that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in Haar wavelets (with high probability). I will talk about this result and its implications, and also show some numerical experiments in which the use of the gradient basis improves compressive signal reconstruction.

One general approach to random number generation is to take a uniformly distributed (0,1) random variable and then invert the cumulative distribution function (CDF) to generate samples from another distribution.  This talk follows this approach, approximating the inverse CDF for the Poisson distribution in a way which is particularly efficient for vector execution on NVIDIA GPUs.

Rational Krylov subspaces have been proven to be useful for many applications, like the approximation of matrix functions or the solution of matrix equations. It will be shown that extended and rational Krylov subspaces —under some assumptions— can be retrieved without any explicit inversion or system solves involved. Instead we do the necessary computations of $A^{-1} v$ in an implicit way using the information from an enlarged standard Krylov subspace.

It is well-known that both for classical and extended Krylov spaces, direct unitary similarity transformations exist providing us the matrix of recurrences. In practice, however, for large dimensions computing time is saved by making use of iterative procedures to gradually gather the recurrences in a matrix. Unfortunately, for extended Krylov spaces one is required to frequently solve, in some way or another a system of equations. In this talk both techniques will be integrated. We start with an orthogonal basis of a standard Krylov subspace of dimension $m+m+p$. Then we will apply a unitary similarity built by rotations compressing thereby significantly the initial subspace and resulting in an orthogonal basis approximately spanning an extended or rational Krylov subspace.

Numerical experiments support our claims that this approximation is very good if the large Krylov subspace contains $A^{-(m+1)} v$, …, $A^{-1} v$ and thus can culminate in nonneglectable dimensionality reduction and as such also can lead to time savings when approximating, e.g., matrix functions.

A stable algorithm to compute the roots of polynomials is presented. The roots are found by computing the eigenvalues of the associated companion matrix by Francis's implicitly-shifted $QR$ algorithm.  A companion matrix is an upper Hessenberg matrix that is unitary-plus-rank-one, that is, it is the sum of a unitary matrix and a rank-one matrix.  These properties are preserved by iterations of Francis's algorithm, and it is these properties that are exploited here. The matrix is represented as a product of $3n-1$ Givens rotators plus the rank-one part, so only $O(n)$ storage space is required.  In fact, the information about the rank-one part is also encoded in the rotators, so it is not necessary to store the rank-one part explicitly.  Francis's algorithm implemented on this representation requires only $O(n)$ flops per iteration and thus $O(n^{2})$ flops overall.  The algorithm is described, backward stability is proved under certain conditions on the polynomial coefficients, and an extensive set of numerical experiments is presented.  The algorithm is shown to be about as accurate as the (slow) Francis $QR$ algorithm applied to the companion matrix without exploiting the structure.  It is faster than other fast methods that have been proposed, and its accuracy is comparable or better.

Traditionally threshold partial pivoting is used to ensure stability of sparse LDL^T factorizations of symmetric matrices. This involves comparing a candidate pivot with all entries in its row/column to ensure that growth in the size of the factors is limited by a threshold at each stage of the factorization. It is capabale of delivering a scaled backwards error on the level of machine precision for practically all real world matrices. However it has significant flaws when used in a massively parallel setting, such as on a GPU or modern supercomputer. It requires all entries of the column to be up-to-date and requires an all-to-all communication for every column. The latter requirement can be performance limiting as the factorization cannot proceed faster than k*(communication latency), where k is the length of the longest path in the sparse elimination tree.

We introduce a new family of communication-avoiding pivoting techniques that reduce the number of messages required by a constant factor allowing the communication cost to be more effectively hidden by computation. We exhibit two members of this family. The first deliver equivalent stability to threshold partial pivoting, but is more pessimistic, leading to additional fill in the factors. The second provides similar fill levels as traditional techniques and, whilst demonstrably unstable for pathological cases, is able to deliver machine accuracy on even the worst real world examples.

The relationship of diagonal dominance ideas to the convergence of stationary iterations is well known. There are a multitude of situations in which such considerations can be used to guarantee convergence when the splitting matrix (the preconditioner) is positive definite. In this talk we will describe and prove sufficient conditions for convergence of a stationary iteration based on a splitting with an indefinite preconditioner. Simple examples covered by this theory coming from Optimization and Economics will be described.

This is joint work with Michael Ferris and Tom Rutherford

This brief lecture will highlight several best-practice observations and
research for writing software for mathematical research, drawn from a
number of sources, including; Best Practices for Scientific Computing
[BestPractices], Code Complete [CodeComplete], and personal observation
from the presenter.  Specific focus will be given to providing the
definition of important concepts, then describing how to apply them
successfully in day-to-day research settings.  Following the outline from
Best Practices, we will cover the following topics:

* Write Programs for People, Not Computers
* Let the Computer Do the Work
* Make Incremental Changes
* Don't Repeat Yourself (or Others)
* Plan for Mistakes
* Optimize Software Only after It Works Correctly
* Document Design and Purpose, Not Mechanics
* Collaborate

[BestPractices]
http://www.plosbiology.org/article/info%3Adoi%2F10.1371%2Fjournal.pbio.1001745
[CodeComplete] http://www.cc2e.com/Default.aspx

We present advances in several fundamental fields of computer vision: image segmentation, object tracking, stereo reconstruction for depth map estimation and full 3D multi-view reconstruction. The basic method applied to these fields is convex relaxation. Convex relaxation methods allow for global optimization of numerous energy functionals and provide a step towards less user input and more automation. We will show how the respective computer vision problems can be formulated in this convex optimization framework. Efficient parallel implementations of the arising numerical schemes using graphics processing units allow for interactive applications.

We investigate an X-ray imaging system that fires multiple point sources of radiation simultaneously from close proximity to a probe. Radiation traverses the probe in a non-parallel fashion, which makes it necessary to use tomosynthesis as a preliminary step to calculating a 2D shadowgraph. The system geometry requires imaging techniques that differ substantially from planar X-rays or CT tomography. We present a proof of concept of such an imaging system, along with relevant artefact removal techniques.  This work is joint with Kishan Patel.

Economic dispatch is a critical part of electricity planning and 
operation. Enhancing the dispatch problem to improve its robustness 
in the face of equipment failures or other contingencies is standard 
practice, but extremely time intensive, leading to restrictions on 
the richness of scenarios considered. We model post-contingency 
corrective actions in the security-constrained economic dispatch 
and consider multiple stages of rescheduling to meet different 
security constraints. The resulting linear program is not solvable
by traditional LP methods due to its large size. We devise and 
implement a series of algorithmic enhancements based on the Benders'
decomposition method to ameliorate the computational difficulty.
In addition, we propose a set of online measures to diagnose
and correct infeasibility issues encountered in the solution process.

The overall solution approach is able to process the ``N-1'' 
contingency list in ten minutes for all large network cases
available for experiments. Extensions to the nonlinear setting will 
be discussed via a semidefinite relaxation.

We present our recent results  on the fluctuation of Optimal Alignments of random sequences and Longest Common Subsequences (LCS). We show how OA and LCS are special cases of certain Last Passage Percolation models which can also be viewed as growth models. this is joint work with Saba Amsalu, Raphael Hauser and Ionel Popescu.