13 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
Nonlocal-interaction equation on graphs: gradient flow structure and continuum limit
We consider dynamics driven by interaction energies on graphs. We introduce
graph analogues of the continuum nonlocal-interaction equation and interpret
them as gradient flows with respect to a graph Wasserstein distance. The
particular Wasserstein distance we consider arises from the graph analogue of
the Benamou-Brenier formulation where the graph continuity equation uses an
upwind interpolation to define the density along the edges. While this approach
has both theoretical and computational advantages, the resulting distance is
only a quasi-metric. We investigate this quasi-metric both on graphs and on
more general structures where the set of "vertices" is an arbitrary positive
measure. We call the resulting gradient flow of the nonlocal-interaction energy
the nonlocal nonlocal-interaction equation (NLIE). We develop the existence
theory for the solutions of the NLIE as curves of maximal slope with
respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the
solutions of the NLIE on graphs converge as the empirical measures of the
set of vertices converge weakly, which establishes a valuable
discrete-to-continuum convergence result.Comment: 46 pages. Minor revision with improved presentation and fixed typo
Health care for older people in Italy: The U.L.I.S.S.E. Project (Un link informatico sui servizi sanitari esistenti per l'anziano - a computerized network on health care services for older people).
Objectives: The U.L.I.S.S.E. study is aimed at describing older patients who are cared for in hospitals,
home care or nursing homes in Italy. Design: The U.L.I.S.S.E. study is an observational multicenter prospective
1-year study. Setting: Overall, 23 acute geriatric or internal medicine hospital units, 11 home care services and 31
nursing homes participated in the study. Measurements: The patient\u2019s evaluation was performed using
comprehensive geriatric assessment instruments, i.e. the interRAI Minimum Data Set, while data on service
characteristics were recorded using ad-hoc designed questionnaires. Results: The older subjects who are in need
of acute and long term care in Italy have similar characteristics: their mean age is higher than 80 years, they have
a high level of disability in ADL, an important multimorbidity, and are treated with several drugs. The prevalence
of cognitive impairment is particularly high in nursing homes, where almost 70% of residents suffer from it and
40% have severe cognitive impairment. On the other hand, there is a shortage of health care services, which are
heterogeneous and fragmented. Conclusions: Health care services for older people in Italy are currently
inadequate to manage the complexity of the older patients. An important effort should be undertaken to create a
more integrated health care system