Stochastic Differential Equations and Random Matrices

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Presentation transcript:

Stochastic Differential Equations and Random Matrices Alan Edelman MIT: Dept of Mathematics, Computer Science Artificial Intelligence Laboratory SIAM Applied Linear Algebra July 18, 2003 11/8/2018

Subject + Random = Wow! Quantum Mechanics Statistical Mechanics Randomized Algorithms Random Variation & Natural Selection Option Pricing Model Why not applied linear algebra???? 11/8/2018

Everyone’s Favorite Tridiagonal -2 1 1 n2 … … … … … d2 dx2 11/8/2018

Everyone’s Favorite Tridiagonal -2 1 G 1 (βn)1/2 1 n2 … … … + … … dW β1/2 d2 dx2 + 11/8/2018

G 11/8/2018

G 11/8/2018

7 G O 11/8/2018

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7 G O 11/8/2018

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7 G O 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 11/8/2018

7 G O 6 5 4 3 11/8/2018

7 G O 6 5 4 3 11/8/2018

7 G O 6 5 4 3 11/8/2018

7 G O 6 5 4 3 11/8/2018

7 G O 6 5 4 3 11/8/2018

7 G O 6 5 4 3 11/8/2018

7 G O 6 5 4 3 2 11/8/2018

7 G O 6 5 4 3 2 11/8/2018

7 G O 6 5 4 3 2 11/8/2018

7 G O 6 5 4 3 2 11/8/2018

7 G O 6 5 4 3 2 11/8/2018

7 G O 6 5 4 3 2 1 11/8/2018

7 G O 6 5 4 3 2 1 11/8/2018

G 6 5 4 3 2 1 Same idea: sym matrix to tridiagonal form 11/8/2018

G 6 5 4 3 2  Same idea: General beta beta: 1: reals 2: complexes 4: quaternions 11/8/2018

Stochastic Operator Limit , dW β 2 x dx d + - , N(0,2) χ n β 2 1 ~ H 2) (n 1) ÷ ø ö ç è æ - … … … , G β 2 H n + » ¥ 11/8/2018

Largest Eigenvalue Plots 11/8/2018

MATLAB beta=1; n=1e9; opts.disp=0;opts.issym=1; alpha=10; k=round(alpha*n^(1/3)); % cutoff parameters d=sqrt(chi2rnd( beta*(n:-1:(n-k-1))))'; H=spdiags( d,1,k,k)+spdiags(randn(k,1),0,k,k); H=(H+H')/sqrt(4*n*beta); eigs(H,1,1,opts) 11/8/2018

Tricks to get O(n9) speedup Sparse matrix storage (Only O(n) storage is used) Tridiagonal Ensemble Formulas (Any beta is available due to the tridiagonal ensemble) The Lanczos Algorithm for Eigenvalue Computation ( This allows the computation of the extreme eigenvalue faster than typical general purpose eigensolvers.) The shift-and-invert accelerator to Lanczos and Arnoldi (Since we know the eigenvalues are near 1, we can accelerate the convergence of the largest eigenvalue) The ARPACK software package as made available seamlessly in MATLAB (The Arnoldi package contains state of the art data structures and numerical choices.) The observation that if k = 10n1/3 , then the largest eigenvalue is determined numerically by the top k × k segment of n. (This is an interesting mathematical statement related to the decay of the Airy function.) 11/8/2018

Open Problems The distribution for general beta Seems to be governed by a convection-diffusion equation 11/8/2018

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