This site contains: mathematics courses and book; covers: image analysis, data analysis, and discrete modelling; provides: image analysis software. Created and run by Peter Saveliev.

From Intelligent Perception

We deal with partial differential equations, i.e., equations with respect to derivatives. However, we don't "sample" or "discretize" these PDEs via finite differences. Instead, we look at the derivations of these PDEs and, based on the physics, represent all of the processes as equations with respect to differential forms or, better, discrete differential forms whenever possible. The discrete versions of these equation are ready-made simulations (similar to cellular automata). The advantage of this approach is that the laws of physics (conservation of energy, conservation of mass, etc, similar to Finite volume method) are satisfied exactly rather than approximately, as is the case with discretization of PDEs.

Algebraic topology

• The algebra of chains
• Cell complexes and simplicial complexes
• Manifolds
• Homology as a vector space
• Boundary operator
• Properties of homology groups
• Cell maps, simplicial maps and their homology maps

Discrete exterior calculus

• Calculus of discrete differential forms
• Discretization of calculus
• Calculus is topology
• Cubical complexes
• Isotropy in numerical PDEs
• Discrete Calculus: Applied Analysis on Graphs for Computational Science by Grady and Polimeni

Simulations of PDEs based on the physics

• wave equation,
• fluid flow produced by a vector field and/or particle modeling,
• lattice gas cellular automata,
• heat transfer and diffusion, heat transfer;
• Maxwell equations.

Tools of discrete exterior calculus:

• cubical complexes, simplicial complexes, cell complexes, and their duals,
• discrete differential forms,
• exterior derivative,
• Hodge duality operator.