16:15
16:15
Jacobians and Hessians are scarcely matrices!!
Abstract
To numerical analysts and other applied mathematicians Jacobians and Hessians
are matrices, i.e. rectangular arrays of numbers or algebraic expressions.
Possibly taking account of their sparsity such arrays are frequently passed
into library routines for performing various computational tasks.
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A central goal of an activity called automatic differentiation has been the
accumulation of all nonzero entries from elementary partial derivatives
according to some variant of the chainrule. The elementary partials arise
in the user-supplied procedure for evaluating the underlying vector- or
scalar-valued function at a given argument.
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We observe here that in this process a certain kind of structure that we
call "Jacobian scarcity" might be lost. This loss will make the subsequent
calculation of Jacobian vector-products unnecessarily expensive.
Instead we advocate the representation of the Jacobian as a linear computational
graph of minimal complexity. Many theoretical and practical questions remain unresolved.
17:00
17:00
15:00
12:00
17:00
Geometry and physics of packing and unpacking, DNA to origami
(Alan Tayler Lecture)
16:30
14:15
16:30
16:30
16:15
Conditioning in optimization and variational analysis
Abstract
Condition numbers are a central concept in numerical analysis.
They provide a natural parameter for studying the behavior of
algorithms, as well as sensitivity and geometric properties of a problem.
The condition number of a problem instance is usually a measure
of the distance to the set of ill-posed instances. For instance, the
classical Eckart and Young identity characterizes the condition
number of a square matrix as the reciprocal of its relative distance
to singularity.
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We present concepts of conditioning for optimization problems and
for more general variational problems. We show that the Eckart and
Young identity has natural extension to much wider contexts. We also
discuss conditioning under the presence of block-structure, such as
that determined by a sparsity pattern. The latter has interesting
connections with the mu-number in robust control and with the sign-real
spectral radius.
12:00
Moduli Kahler Potential for M-theory on a G_2 Manifold
Abstract
I present a calculation of the moduli Kahler potential for M-theory
on a G_2 manifold in a large radius approximation. The result is used to
analyze moduli dynamics and moduli stabilization in the context of the
associated four-dimensional effective theory.
12:00
17:00
The Aviles Giga functional
Abstract
Take any region omega and let function u defined inside omega be the
distance from the boundary, u solves the iconal equation \lt|Du\rt|=1 with
boundary condition zero. Functional u is also conjectured (in some cases
proved) to be the "limiting minimiser" of various functionals that
arise models of blistering and micro magnetics. The precise formulation of
these problems involves the notion of gamma convergence. The Aviles Giga
functional is a natural "second order" generalisation of the Cahn
Hilliard model which was one of the early success of the theory of gamma
convergence. These problems turn out to be surprisingly rich with connections
to a number of areas of pdes. We will survey some of the more elementary
results, describe in detail of one main problems in field and state some
partial results.