132 research outputs found
A blob method for diffusion
As a counterpoint to classical stochastic particle methods for diffusion, we
develop a deterministic particle method for linear and nonlinear diffusion. At
first glance, deterministic particle methods are incompatible with diffusive
partial differential equations since initial data given by sums of Dirac masses
would be smoothed instantaneously: particles do not remain particles. Inspired
by classical vortex blob methods, we introduce a nonlocal regularization of our
velocity field that ensures particles do remain particles, and we apply this to
develop a numerical blob method for a range of diffusive partial differential
equations of Wasserstein gradient flow type, including the heat equation, the
porous medium equation, the Fokker-Planck equation, the Keller-Segel equation,
and its variants. Our choice of regularization is guided by the Wasserstein
gradient flow structure, and the corresponding energy has a novel form,
combining aspects of the well-known interaction and potential energies. In the
presence of a confining drift or interaction potential, we prove that
minimizers of the regularized energy exist and, as the regularization is
removed, converge to the minimizers of the unregularized energy. We then
restrict our attention to nonlinear diffusion of porous medium type with at
least quadratic exponent. Under sufficient regularity assumptions, we prove
that gradient flows of the regularized energies converge to solutions of the
porous medium equation. As a corollary, we obtain convergence of our numerical
blob method, again under sufficient regularity assumptions. We conclude by
considering a range of numerical examples to demonstrate our method's rate of
convergence to exact solutions and to illustrate key qualitative properties
preserved by the method, including asymptotic behavior of the Fokker-Planck
equation and critical mass of the two-dimensional Keller-Segel equation
Talking about a Christine Borland sculpture: effective empathy in contemporary anatomy art (and an emerging counterpart in medical training?)
This Introduction and interview discusses the poetical and empathic insights that are a key to the effectiveness of contemporary artist Christine Borland's practice and its relevance to the medical humanities, visual art research and medical studentsâ training. It takes place in a context of intensive interest in reciprocity and conversation as well as expert exchange between the fields of Medicine and Contemporary Arts. The interview develops an understanding of medical research and the application of its historical resources and contemporary practice-based research in contemporary art gallery exhibitions. Artists tend not to follow prescriptive programmes towards new historical knowledge, however, a desire to form productive relationships between history and contemporary art practice does reveal practical advantages. Borland's research also includes investigations in anatomy, medical practices and conservatio
Which Metric on the Space of Collider Events?
Which is the best metric for the space of collider events? Motivated by the
success of the Energy Mover's Distance in characterizing collider events, we
explore the larger space of unbalanced optimal transport distances, of which
the Energy Mover's Distance is a particular case. Geometric and computational
considerations favor an unbalanced optimal transport distance known as the
Hellinger-Kantorovich distance, which possesses a Riemannian structure that
lends itself to efficient linearization. We develop the particle linearized
unbalanced Optimal Transport (pluOT) framework for collider events based on the
linearized Hellinger-Kantorovich distance and demonstrate its efficacy in
boosted jet tagging. This provides a flexible and computationally efficient
optimal transport framework ideally suited for collider physics applications.Comment: 17 pages, 5 figures, 3 table
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