Motivations

2D Euler equations

Ergodic hypotesis: for $T$ large and a functional $\phi$

for some invariant measure $\mu$ on $H$ Hilbert space

Motivations

Invariant measures

• Enstrophy measure: $\nu(d\omega)\approx e^{-E(\omega)}d\omega$
  More precisely, centered Gaussian measure on $H^{-1-}=\cap_{\varepsilon>0}H^{-1-\varepsilon}$
   
• We would like to take $\mu_{\alpha,k}(d\omega)\approx e^{-\alpha C_k(\omega)}\nu(d\omega)$, for some Casimir $C_k(\omega)=\int_S\omega^kdS$
   
• Higher Casimir invariant measures: open problem!
   
• Can we study an analogue problem for a finite dimensional system? YES!

Zeitlin model

• $W\in\mathfrak{su}(N)$
• Enstrophy measure $\nu(dW)=e^{-Tr(W^*W)}dW$
• Higher Casimir measures
  $\mu_{\alpha,\beta,k}(dW)=e^{-\alpha Tr(W^*W)-\beta Tr((W^*W)^k)}dW$
• $\mu$ well defined for any $k\geq 1$

Given some 

$\mu_{\alpha,\beta,k}(dW)=e^{-\alpha Tr(W^*W)-\beta Tr((W^*W)^k)}dW$

we would like to generate $W_0$ random matrix whose Law is $\mu$

A possible technique to generate such $W_0$ is Metropolis-Hastings algorithm

Metropolis-Hastings

1. Let $W_0\in\mathfrak{su}(N)$ be given
2. Pick $\Delta W\in\mathfrak{su}(N)$ generated by a distribution $p$ on $\mathfrak{su}(N)$
3. Let define $\delta:=V(W_0+\Delta W)-V(W_0)$
4. Define $W_0:=W_0+\Delta W$ with probability $e^{-(\delta\wedge 0)}$
5. Repeat from 1 until $V$ becomes stationary

Generate $W$ distributed according to
$\mu(dW)=e^{-V(W)}dW$, $V\geq 0$

Metropolis-Hastings

• This approach has not be fruitful
   
• Apparently not much changed varying the parameters $\alpha,\beta,k$
   
• Not clear the influence of the higher Casimirs

 $V(W)=\alpha Tr(W^*W)-\beta Tr((W^*W)^k)$,
for some $\alpha,\beta,k$

Alternative approach

• Let us consider $W=W(t)$ solution of Euler-Zeitlin equations
   
• $W(t) = U(t)^*W_0 U(t)$, $U(t)$ unitary for all $t\geq 0$
   
• Can we say something of $U(\infty):=\lim_{t\rightarrow\infty}U(t)$?
   
• It turns out that typically $\sigma(U(\infty))\sim Unif(\mathbb{S}^1)$ and possibly $U(\infty))\sim Haar(SU(N))$

Haar generated $U$

$Re(U)$

$eig(U)$

 $U(\infty)$

$eig(U)$

$Re(U)$

Haar measure on $SU(N)$

• Unique probability measure on the Borel sets of $SU(N)$ invariant with respect to left and right multiplication of $U\in SU(N)$
• Thm[Eaton, 1983]. If $Z\in GL(N,\mathbb{C})$ has entries i.i.d. standard complex normal random variables, then applying the Gram-Schmidt orthonormalization to the columns of $Z$, gives a unitary matrix $Q$ is distributed with Haar measure.

Haar measure on $SU(N)$

• Gram-Schmidt algorithm is numerically unstable
• QR decomposition does not provide a unique Q and in general not distributed as Haar (Edelman and Rao, 2005)
   
• Mezzadri, 2007 provides a stable algorithm:
  - QR decomposition of $Z=QR$ with entries i.i.d. normally distributed on $\mathbb{C}$
  - Define $Q':=Q\Lambda$, where $\Lambda:=diag(R)/|diag(R)|$, which is distributed as Haar

A first test

• Take some non-zero $W_0\in\mathfrak{su}(N)$
   
• Generate $U\sim Haar(SU(N))$ with the Mezzadri algorithm
   
• Define $W_{end}:=U^*W_0U$
   

What do we get?

A first test

Enstrophy spectrum $\approx l^1$

A first test

• Conjecture: $U\sim Haar(SU(N))$ if and only if $Ad^*_U(W_0)\sim \nu(SU(N))$, for almost any $W_0\in\mathfrak{su}(N)$, where $\nu$ is the enstrophy measure.
• The unitary matrix $U(\infty)$ not clearly distributed as Haar, in particular $Ad^*_{U(\infty)}(W_0)$ is not distributed as $\nu$
• We need to add some constraint to the transformation $U$. In particular, we want to retain energy and angular momentum

Metropolis-Hastings on $SU(N)$

1. Let $W_0\in\mathfrak{su}(N)$ be given
2. Generate $U\sim Haar(SU(N)$
3. Resize $U$: $\hat U:=Cay(r CayInv(U))$, for some $r<<1$
4. $W_{new}:=\hat{U}^*W_0\hat U$
5. Let define $\delta:=V(W_0+\Delta W)-V(W_0)$,
   for $V(W)=\alpha(H(W)-H_0)^2+\beta(M(W)-M_0)^2$
6. Define $W_0:=W_0+\Delta W$ with probability $e^{-(\delta\wedge 0)}$
7. Repeat from 1 until $V$ becomes stationary

Generate $W$ distributed according to
$\mu(dW)=e^{-\alpha(H(W)-H_0)^2-\beta(M(W)-M_0)^2}dW$,

via a random walk on $SU(N)$

Metropolis-Hastings on $SU(N)$

1. Let $W_0\in\mathfrak{su}(N)$ be given
2. Generate $\Xi\sim Unif(\mathbb{S}_{SU(N)})$
3. Generate $U:=Cay(r \Xi$), for some $r<<1$
4. $W_{new}:=U^*W_0 U$
5. Let define $\delta:=V(W_0+\Delta W)-V(W_0)$,
   for $V(W)=\alpha(H(W)-H_0)^2+\beta(M(W)-M_0)^2$
6. Define $W_0:=W_0+\Delta W$ with probability $e^{-(\delta\wedge 0)}$
7. Repeat from 1.

Generate $W$ distributed according to
$\mu(dW)=e^{-\alpha(H(W)-H_0)^2-\beta(M(W)-M_0)^2}dW$,

via a random walk on $SU(N)$

Numerical test

Enstrophy spectrum: red vorticity transformed with Haar unitary matrix, green final spectrum, black evolving spectrum of $W$ according to Metropolis

Numerical test

$eig(U(\infty))$

• The enstrophy distributes according to the power law $l^1$ up to the smallest possible wavenumber $l=2$
   
• Therefore, either the random walk does not replicate correctly the Euler-Zeitlin statistics or it is much faster
   
• We propose to add information on the scale separation adding to $V$ the term
  $-\alpha_s (H(W_s)-H(W_0))^2$

Numerical test

Enstrophy spectrum: red vorticity transformed with Haar unitary matrix, green final spectrum, black spectrum of $W(\infty)$ generated with Metropolis

Numerical test

$eig(U(\infty))$

Conclusions

• We have shown that some statistics of Euler-Zeitlin can be recover via simulating a random walk on $SU(N)$
   
• Different interpretations of the results
   
• Need for more theoretical insight
   
• ...

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