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 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.

Cubature is the business of describing a probability measure in terms of an empirical measure sharing its support with the original measure, of small support, and with identical integrals for a class of functions (eg polynomials with degree less than k). 

Applying cubature to already discrete sets of scenarios provides a powerful tool for scenario management and summarising data.  We refer to this process as recombination. It is a feasible operation in real time and has lead to high accuracy pde solvers.

The practical complexity of this operation has changed! By a factor corresponding to the dimension of the space of polynomials. 

We discuss the algorithm and give home computed examples of nested sparse grids with only positive weights in moderate dimensions (eg degree 1-8 in dimension 7).  Positive weights have significant advantage over signed ones when available.
 

 Recently several conjectures about l2-invariants of
CW-complexes have been disproved. At the heart of the counterexamples
is a method of computing the spectral measure of an element of the
complex group ring. We show that the same method can be used to
compute the finite field analog of the l2-Betti numbers, the homology
gradient. As an application we point out that (i) the homology
gradient over any field of characteristic different than 2 can be an
irrational number, and (ii) there exists a CW-complex whose homology
gradients over different fields have infinitely many different values.
 

Lawler, Schramm and Werner studied 2-point chordal restriction measures and gave several constructions using SLE tools.

It is possible to characterize 3-point chordal restriction measures in a similar manner. Their boundaries are SLE(8/3)-like curves with a slightly different drift term.

@email

   Stéphane Mallat

   Ecole Normale Superieure

Learning functionals in high dimension requires to find sources of regularity and invariants, to reduce dimensionality. Stability to actions of diffeomorphisms is a strong property satisfied by many physical functionals and most signal classification problems. We introduce a scattering operator in a path space, calculated with iterated multiscale wavelet transforms, which is invariant to rigid movements and stable to diffeomorphism actions. It provides a Euclidean embedding of geometric distances and a representation of stationary random processes. Applications will be shown for image classification and to learn quantum chemistry energy functionals.