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 (NL$^2$IE). We develop the existence theory for the solutions of the NL$^2$IE as curves of maximal slope with respect to the upwind Wasserstein quasi-metric. Furthermore, we show that the solutions of the NL$^2$IE 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