The variational formulation of the Fokker-Planck equation
SIAM Journal on Mathematical Analysis
An Adaptive Grid Method and Its Application to Steady Euler Flow Calculations
SIAM Journal on Scientific Computing
Modeling the aggregative behavior of ants of the species Polyergus rufescens
Nonlinear Analysis: Real World Applications - Special issue on Spatial Heterogeneity in ecological models
Optimal Mass Transport and Image Registration
VLSM '01 Proceedings of the IEEE Workshop on Variational and Level Set Methods (VLSM'01)
Optimal Mass Transport for Registration and Warping
International Journal of Computer Vision
Precise computations of chemotactic collapse using moving mesh methods
Journal of Computational Physics
Identification of Asymptotic Decay to Self-Similarity for One-Dimensional Filtration Equations
SIAM Journal on Numerical Analysis
Lagrangian Numerical Approximations to One-Dimensional Convolution-Diffusion Equations
SIAM Journal on Scientific Computing
A finite volume scheme for the Patlak–Keller–Segel chemotaxis model
Numerische Mathematik
SIAM Journal on Numerical Analysis
Moving Mesh Generation Using the Parabolic Monge-Ampère Equation
SIAM Journal on Scientific Computing
One-Dimensional Numerical Algorithms for Gradient Flows in the p-Wasserstein Spaces
Acta Applicandae Mathematicae: an international survey journal on applying mathematics and mathematical applications
Numerical solution of the Optimal Transportation problem using the Monge-Ampère equation
Journal of Computational Physics
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We propose a numerical algorithm for solving nonlinear continuity equations written in Lagrangian coordinates. This transformation is intimately related to variational approaches for the well-posedness of gradient flows of energy functionals with respect to the quadratic transportation distance in optimal transport theory. These schemes allow the numerical approximation of both diffusive and aggregation regimes of different models. Positivity, energy decreasing, and mesh adaptation are built-in in the numerical scheme, and thus we are capable of capturing blow-up densities and of dealing with vacuum regions and merging of mass patches in a natural way.