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Fermionic quantum criticality and the fractal nodal surface
http://arxiv.org/pdf/0804.2161v1.pdf
Solution of the Schrödinger equation by a spectral method ☆
http://www.sciencedirect.com/science/article/pii/0021999182900912
An Efficient Numerical Spectral Method for Solving the Schrödinger Equation
http://dl.acm.org/citation.cfm?id=1098532
http://www.quantum-espresso.org/?page_id=16
matlab spectral method
http://www.mathworks.ir/downloads/Spectral%20Methods%20in%20MATLAB%5Bwww.mathworks.ir%5D.pdf
An Efficient Chebyshev–Lanczos Method for Obtaining Eigensolutions of the Schrödinger Equation on a Grid
http://www.sciencedirect.com/science/article/pii/S0021999196901400
Numerical Solutions of the Schr¨odinger Equation
http://physics.bu.edu/~py502/lectures4/schrod.pdf
Solving the discretized time‐independent Schrödinger equation with the Lanczos procedure
http://jcp.aip.org/resource/1/jcpsa6/v92/i7/p4374_s1
IMPORTANT
http://www.itp.phys.ethz.ch/education/lectures_fs08/cqp/cqp
K2D
http://math.mit.edu/classes/18.086/2006/project1-Dominguez.pdf
http://ocw.mit.edu/courses/mathematics/18-086-mathematical-methods-for-engineers-ii-spring-2006/assignments/
##Early results
#atomic, precise steps. to take action whenever I know how to do so.
##'useful' code to learn from
https://github.com/NNemec/NNlib
##Done
**Dec 2**
* seed() and then st = get_state() to store seed to repeat a random simulation
* cffi to call C function
**Nov 21**
* in vals, vecs = eigs(matrix), separate the vectors based on v[:,i], (i.e. scipy's eigs works to give us the proper eigenvector, of course) so, there is no need to implement the diagonalization algorithm manually (but I learned the lanczos algorithm other diagonalization method (another e.g. power iteration) along the way)
* use scipy.spy to display matrix
* finfo(float)
**Nov 6**
* https://code.google.com/p/glumpy/ for rapid GPU-based numpy array visualization. see example in toy/ folder.
* alternative to matplotlib, 'ggplot', https://github.com/ContinuumIO/Bokeh
**Oct 31**
* considering using hdf5 for efficient storing
**Jul 30**
6 electron 100 meshsize improves from 2.12s to 1.68s
**Apr 6**
* Plot the slater-det gs wavefunction when N-1 electrons are fixed
* Identify the nodes
* as expected, all the 'position' of the other electrons lies along the node
**Apr 7**
* switch to use symmetric PBC instead of box boundary
* python code cprofile-ing. slow
* comment: "the nodal surface for a real wavefunction is always closed loops. (Notice that the apparent ends at the edges of the box always have partners on the other side which is related by the periodic boundary conditions.) This is so because the nodes separate the regions of positive from regions of negative wavefunction and therefore can't end."
* as of April 7, 24 electrons require 5mins of simulation
* as of April 8, 24 electrons require 2mins of simulation (remove duplication in calculation)
full performance report on April 8, 24e, 143s:
* psi 5.882s ncalls 10000
* result 114.826 ncalls 5760000
6e require 13.37s
**Apr 16**
* implement reduced slater determinant. The old way is to blindly antisymmetrize wf from scratch each time
* performance report on April 16: (new since the implementation of reduced slater det), 6e require 2.73s
* beware http://stackoverflow.com/questions/5956783/numpy-float-10x-slower-than-builtin-in-arithmetic-operations
* implement http://stackoverflow.com/questions/6876377/numpy-arbitrary-precision-linear-algebra
* starting to implement slater determinant update. It should become O(N) as advertised by the paper here http://arxiv.org/pdf/0906.4354.pdf
A fast and efficient algorithm for Slater determinant updates in quantum Monte Carlo simulations http://jcp.aip.org/resource/1/jcpsa6/v130/i20/p204105_s1
##Todo
* implement diffusion monte carlo
* summarize the review paper
* really explore http://code.google.com/p/qutip/
* urgent: how the zeros of slater determinant is efficiently calculated. **Consult http://arxiv.org/abs/0804.2161 for optimization**. Alternatively, check out http://code.google.com/p/pi-qmc/. But then, scipy's det also uses LU decomposition
* wtf is 'bosonization' in ceperley's path integral? Pauli surface?
* How different is the usual antisymmetrization vs superposition? (and also entanglement). check out Neumaier's ans in physics.SE
* ground state and "overdamping" is a bad analogy, but I need to know how meaningless it really is.
* read http://prl.aps.org/abstract/PRL/v72/i15/p2442_1
* read http://altair.physics.ncsu.edu/articles.htm
* read Path integrals in the theory of condensed helium http://rmp.aps.org/abstract/RMP/v67/i2/p279_1
* read Quantum Monte Carlo simulations of solics http://rmp.aps.org/abstract/RMP/v73/i1/p33_1
* See Ceperley publications http://people.physics.illinois.edu/Ceperley/papers/index.htm
* read http://prl.aps.org/pdf/PRL/v72/i15/p2442_1
* read http://prb.aps.org/pdf/PRB/v44/i17/p9410_1
* see http://code.google.com/p/pi-qmc/source/browse/trunk/src/fixednode/FreeParticleNodes.cc -> not useful
* read http://physics.stackexchange.com/questions/11605/when-is-the-minus-sign-problem-in-quantum-simulations-an-obstacle
* http://stackoverflow.com/questions/1053928/python-numpy-very-large-matrices
* http://www.physik.uni-muenchen.de/lehre/vorlesungen/sose_11/comp_phys_sose11/exercises/notes_private.pdf
* http://stackoverflow.com/questions/599619/how-to-do-numerical-integration-with-quantum-harmonic-oscillator-wavefunction
* you're encouraged to implement the code from scratch, should be ..
##Abandoned
* use scipy.weave.blitz() to improve the slater determinant calculation. Meh, doesn't help that much
* http://stackoverflow.com/questions/9337188/animating-a-contour-plot-in-matplotlib-using-funcanimation not flexible. That's it
```python
fig = figure()
ims = []
for i in range(2):
otherelectronsplot += array([(0,.05)]).T
zerox,zeroy = bruteforcezerofinder(wfgrid,length,meshsize,X,Y)
wfgrid = wfgridcreator(wavefunction, X, Y, list(otherelectronsplot.T), meshsize)
ims.append(plot(zerox,zeroy,'bo'))
ani = animation.ArtistAnimation(fig, ims, interval=50, blit = True, repeat_delay = 1000)
ani.save('blah.mp4')
```
* cythonize
##Papers and references
* MAIN review paper Fermion nodes http://www.springerlink.com/content/m344615283325722/fulltext.pdf
* fermion monte carlo without fixed nodes http://jcp.aip.org/resource/1/jcpsa6/v131/i5/p054106_s1
* Structure of fermion nodes http://arxiv.org/pdf/cond-mat/0601485v1.pdf
* H. J. M. van Bemmel et al, "Fixed-node quantum Monte Carlo method for lattice fermions", [Phys. Rev. Lett. 72, 2442-2445 (1994)](http://prl.aps.org/abstract/PRL/v72/i15/p2442_1)
* Numerical sign problem http://en.wikipedia.org/wiki/Numerical_sign_problem
Basically, integration of highly oscillatory functions give you headache (this sentence doesn't say anything).
current strategies: meron-cluster algorithm, stochastic quantization(doesn't work for fermions), fixed node method
* more articles on fixed node method http://www.lorentz.leidenuniv.nl/~saarloos/Correlateds/fixednode.html
* [Antisymmetrizer](http://en.wikipedia.org/wiki/Antisymmetrizer)
* Cython: http://docs.cython.org/src/userguide/numpy_tutorial.html
###Electronic Wavefunction Calculations
* Electronic Wave Functions. I. A General Method of Calculation for the Stationary States of Any Molecular System http://rspa.royalsocietypublishing.org/content/200/1063/542.short
##Google keywords
* fermion node
* fixed node quantum monte carlo
* free fermion node
* slater determinant monte carlo
##Node searching strategy
1. expensive: the most rudimentary one is by using cutoff method abs(wavefunction) < cutoff
2. http://www.mathworks.com/matlabcentral/answers/15082-multiple-solutions-using-fsolve basically, square the wavefunction, and find the minimum using fmin
3. bisection method. http://stackoverflow.com/questions/4326667/solving-equation-using-bisection-method -> but this example is only for 1D. more: http://stackoverflow.com/questions/3513660/multivariate-bisection-method
4. random walk bisection method. will be implemented soon
##sparse matrix
[references](http://docs.scipy.org/doc/scipy/reference/sparse.html)
[sparse linalg references](http://docs.scipy.org/doc/scipy/reference/sparse.linalg.html)
http://csc.media.mit.edu/divisi efficient SVD
http://stackoverflow.com/questions/2540059/scipy-sparse-arrays sparse benchmark
###Readlater
O(n) quantum monte carlo using maximally-localized wannier function
http://prb.aps.org/abstract/PRB/v71/i12/e121105
http://prl.aps.org/pdf/PRL/v87/i24/e246406
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