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Study Notes

• Note on Linear System Solver:In general, we can model a PDE problem and then discretize it into a linear/nonlinear system to be solved. Here, we will focus on the solver to the linear system of equations that has a sparse structure. If not sparse systems, say dense matrix. we can recall the BLAS, Lapack or ScaLapack. Here we take the Landau-Lifshitz equation as an application example and study its linear system of equation for the discrete equation.

• Note on Ubermag Tutorial: Ubermag is a python package for micromagnetics.

• Note on OOMMF Tutorial: OOMMF is a package for micromagnetics.

Research Notes

• A Machine-learning Method for Time-dependent Wave Equations Over Unbounded Domains.

• A note on a high order accurate linear numerical scheme for LL equation.

• A note on a new model on the soft substrate with MCLs.

• A note on a second order GSPM for micromagnetics simulations.

• A note on Absorbing-net An absorbing boundary learning to high dimensional PDEs.

• A note on ALE for two-phase flow using dealII.

• A note on boundary-net boundary learning to high dimensional PDEs.

• A note on constraint comes in the neural network A unified framework.

• A note on dealII for TO with surface elasticity for reducing the inner singularity.

• A note on dealII.

• A note on deep Ritz method for micromagnetics.

• A note on deep Ritz method for topology optimization.

• A note on deriving the electron temperature from schordinger equation.

• A note on diffuse interface model with MCL.

• A note on diffuse interface model with MCLs by dealII.

• A note on Dune-FEM.

• A note on elasticity-nets based on Cauchy-Born rule.

• A note on enforcing the ABC with high order accuracy and the absorbing net.

• A note on FSI using dealII.

• A note on hyperelastic materials.

• A note on hysteresis on soft substrate.

• A note on learning nonlinear autonomous systems with stiffness.

• A note on learning the LL equation.

• A note on learning the multiphase model.

• A note on least action principle and Hamiton's princinple.

• A note on MCL of soft substrate.

• A note on MCL with surface effect.

• A note on MCLs on rigid solid substrates.

• A note on MCLs with ShuttleWorth effect.

• A note on ML solver faster than FFT and beyond.

• A note on molecular dynamics and spin dynamics modeling of magnetic nanoparticles.

• A note on nonlinear basis selection of NN.

• A note on numerical methods and deep nets for PDEs.

• A note on numerical methods for LL equation.

• A note on python FEM lib Nutils tutorial.

• A note on restudy the neural network.

• A note on revisit FSI by dealII.

• A note on RFM for inverse problems.

• A note on several implementations for PINNs for PDEs.

• A note on some research proposals on scientifc computing.

• A note on splitting iterative strategy on semi-implicit and fully implicit methods for LLG equation.

• A note on stress recovery.

• A note on surface elasticity.

• A note on the CB approxiamtion under magnetic field at finite temperature.

• A note on the Redbkit.

• A note on Theano and symbolic ML for solving PDE.

• A note on variational integrator learning by bilevel least action principle.

• Efficient random feature method for topology optimization.

• least action principle for unitary dynamics.

• LL model.

• Note on a ML method with onsager variational principle for LLG equation.

• Note on adaptivity for ELM.

• Note on ELM.

• Note on MCLs of the soft substrate.

• Note on nonlinear euqations and optimization.

• Note on numerical methods and deep nets.

• Note on phase field model on the soft substrate.

• Note on topology optimization using the SIMP method.

• Note on some open problems.

Lecture Notes

• Talk Topic: Physics of Open Systems:The fundamental physics of open quantum systems, including topics of nonequilibrium dynamics, non-Markovianity memory phenomena, entanglement decoherence and decoherence of topological states, quantum thermalization, quantum phase transition, and quantum transport in mesoscopic physics are explored. Applications to various open systems are also presented.

• Talk Topic: FEM for option price: Explore the FEM for possible Black-Sholes equation. Explore the industrial and engineering collabration

Course Projects Report

• Numerical PDE Project1:This Report contains finite difference approximation to solve second-order ODE (BVP), verify the accuracy.

• Numerical PDE Project2:This Report contains Jacobi Method, Gauss-Seidel Method to solve linear systems of equation which's corresponding to second-order ODE.

• Numerical PDE Project3:This Report contains approximation of the solution to the 2nd-order ODE equation numerically using the Fast Fourier Transform (FFT) and verify 2nd-order accuracy.

• Numerical PDE Project4:This Report contains Newton's method to solve the nonlinear system of equations from the nonlinear time-dependent PDEs.

• Numerical PDE Project5:This Report contains Three schemes (Forward Time, Centered Space; Leapfrog; Lax-Friedrichs) to solve transport equation with periodic boundary condition; Von Neumann Analysis for FTCS; Find the order of accuracy of each method).

Take-home Notes

• Homogenization hw1: Consider the propoties of weak convergence, as a consequence of the principle of uniform boundedness, every weakly convergent sequence is bounded. The norm is weakly lower-semicontinuous.

• Homogenization hw2: Prove C_0^{\infty}(\Omega) is dense in L^p(\Omega);

• Homogenization hw3: The energy minimization to deduce the Landau-Lifhiz equation;

• Homogenization hw4: Asymptotics analysis on boundary layer;

• Homogenization hw5: The multi-dimensional case for periodic composite materials;

• Homogenization hw6: The Optimized problem of homogenization coefficients;

• Homogenization hw7: The linear response representation.