Meshless methods for PDEs

Direct discretizations with applications to meshless methods for PDEs

A central problem of numerical analysis is the approximate evaluation of integrals or derivatives of functions. In more generality, this is the approximate evaluation of a linear functional defined on a space of functions. Users often just have values of a function u at scattered points x1, . . . ,xN in the domain Ω of u, and then the value λ (u) of a linear functional λ must be approximated vi...

Meshless Methods for Volume Visualization

A revolution in sensor technology is introducing new problems that must be addressed by visualization research. These technologies will enable earthquake engineers to monitor tectonic plate activity, park rangers to respond more effectively to wildfires, and marine biologists to uncover mysteries from the ocean depths using a network of small, lightweight sensors collecting data in real-time. V...

Numerical Methods for PDEs

Numerical Methods for PDEs

Kernel–Based Meshless Methods

2 Kernels 1 2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Positive Definiteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 General Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.4 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.5 Dualit...