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Broad Themes
The focus topics of the workshop reflect three broad themes:
- Emerging applications requiring 3D Finite Element Analysis (FEA), and non-scalar interpolation.
- Essential and evolving techniques from algebraic topology and geometric analysis.
- Geometric inverse problems,
Emerging applications requiring 3D FEA and non-scalar interpolation.
Whitney forms provided the formalism for tackling one of the hardest problems of computational electromagnetics, namely the correct representation of three-dimensional divergence-zero vector fields, or in the language differential forms, the representation of closed 2-forms. For this reason, finite element analysis (FEA) using Whitney forms has transcended the artificial barriers between low frequency applications such as the design of energy conversion devices and analysis of eddy current nondestructive testing schemes, and high frequency applications such as the design of metamaterials and waveguide couplers. For the engineer, it is important to build on the essential role Whitney forms play in the analysis of three dimensional magnetic fields. The workshop will emphasize both fundamental issues and applications to inverse problems, metamaterials, electromagnetic band-gap materials, photonic devices, inductance issues in VLSI microelectronics, and issues in MRI imaging.
It must be stressed that Magnetic Resonance Imaging (MRI) and associated hardware are very interesting from the perspective of this workshop because of the spectacular success of MRI and spin-off fields such as fMRI. From the design of superconducting MRI coils, to the automated construction of anatomically correct finite element models from MRI data, to MREIT (see below), and to the tagging of tumors (via their DNA) with magnetic nanoparticles for the purposes of drug delivery, the implications of the MRI revolution for computational electromagnetics are incredible. Having the workshop in Boston will enable research groups in local hospitals to link up with the computational electromagnetics community on many aspects of MRI, fMRI and MREIT.
Essential and evolving techniques from algebraic topology and geometric analysis.
The analysis of electric circuits, using Kirchhoff’s Laws, brought topology into electrical engineering over 150 years ago. Hermann Weyl’s reformulation Kirchhoff’s laws in terms of homology over 80 years ago, is an abstraction which is proving to be essential in the finite element analysis of three-dimensional electromagnetic fields. It enables computers to be programmed to identify an electrical circuit in an electromagnetic field problem -- a task once considered the domain of the engineer’s intuition. In “control theory” parlance, circuit theory equations are low frequency model reductions of distributed parameter electromagnetic systems, and homology theory yields the key mathematical tools for obtaining robust numerical algorithms. One aspect of the workshop will deal with large scale homology calculations and the realization of cycles representing generators of integral homology groups as embedded manifolds. The underlying homology calculations involve large sparse integer matrices with remarkable structure even when the underlying finite element meshes are “unstructured”. One aim of the workshop is to bring together those performing large scale homology calculations in the context of dynamical systems and point cloud data analysis, with those requiring more geometrical applications of homology groups in electromagnetics.
Over two decades ago, boundary value problems arising in the analysis of quasistatic electromagnetic fields were reinterpreted in terms of Hodge theory on manifolds with boundary. This observation is quite natural when Maxwell’s equations are viewed in the context of differential forms and the problem of defining potentials is phrased in terms of de Rham cohomology. This observation, along with the variational formulation of Hodge theory on manifolds with boundary, created a revolution in the finite element analysis of electromagnetic fields. When phrased this way, the most difficult theoretical problems were actually solved in the 1950’s by Andre Weil and Hassler Whitney who were concerned with problems in algebraic topology. They had an explicit interpolation formula for turning simplicial cochains into piecewise linear differential forms. This formula gives a chain homotopy between the algebraic complexes involved, and an isomorphism of cohomology rings. Although it took 30 years for Whitney forms to impact engineering practice, once the genie was out of the bottle, there was no way to put it back in. In the early 1990s, Whitney form techniques solved the problem of “spurious modes” appearing in electromagnetic cavity resonator calculations and soon after became widely accepted as an essential tool which is only recently being appreciated in the context of nanophotonics.
It is important to re-examine this Whitney form revolution in the context of recent attempts to develop “discrete exterior calculus,” “mimetic discretizations,” “compatible discretizations” etc. For example, in algebraic topology it is well known that simplicial cochains do not admit a graded-commutative, associative product analogous to the wedge product on differential forms. This classical result, known as “the commutative cochain problem,” is surprising and unintuitive in light of the fact that simplicial cochains admit a graded-commutative, associative product on the level of cohomology, analogous to the one induced by the wedge product in the de Rham complex. The bottom line is that these types of classical results are often ignored by newcomers trying to develop a discrete approach to calculus. Obviously, there is still some important technology transfer to be performed between algebraic topology and numerical analysis! Much of the mathematical work was done by Patodi, Dodziuk and Muller in the 1970’s, has been exploited by electrical engineers, but has been largely ignored by applied mathematicians. Although the multiplicative structure on differential forms does not seem to be very important in the context of linear boundary value problems, it seems to play an important role in magnetohydrodynamics. Magnetohydrodynamics, in turn is an essential tool in space physics, in particular, in the growing field of space weather.
The workshop will address other aspects of the Whitney form revolution. There are deep mathematical connections which do not seem to have appeared on the radar screens of applied mathematicians, yet need to be explored. For example, in the 1970s, Whitney forms were a key ingredient in Werner Muller’s proof of the Ray-Singer conjecture (equating Reidemeister and analytic torsions), and this points to a method for discretizing Edward Witten’s topological quantum field theoretic (TQFT) approach to the Jones polynomial where the underlying partition function is expressed in terms of torsion invariants. (From a theoretical point of view, any such discretization raises some very interesting links to computer science since the computational complexity of computing the Jones polynomial of a knot [as a function of the knot’s crossing number], is known to be NP hard.) In a related, but no less surprising way, Reidemeister torsion also pops up in questions related to ambiguities in the definition of the super symmetric functional determinant associated with the Dirichlet to Neumann map of electrical impedance tomography in three dimensions. The pleasant surprise is that Whitney form finite element discretizations of abelian Chern-Simmons functionals (with gauge fixing terms) make the details clear on the cochain level, and in the data structures of finite element analysis!
Geometric inverse problems
Inverse problems involving partial differential equations occur in many types of imaging problems. They are also notoriously ill conditioned, and there is an impulse to “regularize” them in order to improve their conditioning (at the expense of resolving detail). However, if one identifies enough extra structure in the problem, one can both resolve extra detail and avoid using regularization as a “one size fits all” approach to improved conditioning. Unfortunately avoiding regularization usually requires very deep and problem-specific insights. One aim of the workshop is to explore techniques from geometric inverse problems which provide the type of side information which helps avoid regularization. Specific inverse problems the workshop plans to focus on are electrical impedance tomography (EIT), magnetic resonance assisted EIT (or MREIT), and radar. Speakers being lined up for EIT talks are clearly world class mathematicians, who also have very close connections to hardware efforts. Specifically, they are leaders in developing the so-called “d-bar” method based on the Beltrami equation arising in complex analysis, and the workshop will probe possible extensions of this technique to three dimensions.
The time honored problem of interpreting radar returns has, at its root, the fact that numerical methods such as the finite element method, obey the best error estimates in the context of elliptic equations (such as the analog of Helmholtz’ equation associated with the Laplace-Beltrami equation -- along with the constraint of fixed or bounded frequency). However, truly wideband radar signals need to be analyzed in the context of the wave equation -- a hyperbolic equation. For this reason, in the solution of ultra wideband radar problems, numerical computations must be supplemented by asymptotic methods, and geometric analysis provides both a framework and a key for having asymptotically bounded computational cost with increasing crossover frequency between numerical and asymptotic methods.
Geometric and topological aspects also arise in inverse problems involving “near” force-free magnetic fields and superconducting magnet design. Here, an essential ingredient is the topological considerations in choosing isotopy invariant boundary conditions which yield the curl operator self-adjoint on a manifold with boundary (embedded in the three-sphere). Furthermore, the finite element discetization of the curl operator is a prime example of how the metric-free aspects of Chern-Simmons theory have profound implications for finite element analysis, and this connection yields great insights into algorithms for certain inverse problems in three dimensions. Specifically, this connection yields profound insights into both the problems of electrical impedance tomography and force-free coil design.
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