Notes On Micro-Methods

Notes on Multigrid Methods

1 The model problem: 1D Possion equation. On the unit 1D domain x ∈ [0, 1], we numerically solve Poisson equation with homogeneous boundary condition ∆u = f, x(0) = x(1) = 0. (1) Discretize the domain with n cells and locate the knowns fj and unknowns uj at nodes xj = j/n = jh, j = 0, 1, . . . , n. We would like to approximate the second derivative of u using the discrete values at the nodes. U...

Notes on Decomposition Methods

4 More general decomposition structures 17 4.1 General examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2 Framework for decomposition structures . . . . . . . . . . . . . . . . . . . . 19 4.3 Primal decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.4 Dual decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2...

Notes on Matrix and Micro Strings ∗

We review some recent developments in the study of M-theory compactifications via Matrix theory. In particular we highlight the appearance of IIA strings and their interactions, and explain the unifying role of the M-theory five-brane for describing the spectrum of the T 5 compactification and its duality symmetries. The 5+1-dimensional micro-string theory that lives on the fivebrane world-volu...

Intermediate Micro Lecture Notes

5 A Brief Review of Optimization 27 5.1 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Lagrange . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.3 Constrained Optimization . . . . . . . . . . . . . . . . . 29 5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.5 Understanding Lagrangians . . . . . . . . . . . . . . . . 31 5.6 Concavity and Quas...

Lecture Notes on Mathematical Methods