Split real and imag part in jacobian
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@ -382,62 +382,81 @@ The enstrophy is defined as
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To compute the Lyapunov exponents, we must first compute the Jacobian of $\hat u^{(n)}\mapsto\hat u^{(n+1)}$.
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This map is always of Runge-Kutta type, that is,
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\begin{equation}
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\hat u^{(n+1)}=\hat u^{(n)}+\delta\sum_{i=1}^q w_i\mathfrak F(\hat u^{(n)})
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\end{equation}
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(see\-~(\ref{ins_k})), so
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\begin{equation}
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D\hat u^{(n+1)}=\mathds 1+\delta\sum_{i=1}^q w_iD\mathfrak F(\hat u^{(n)})
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\hat u(t_{n+1})=\mathfrak F_{t_n}(\hat u(t_n))
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.
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\end{equation}
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We then compute
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Let $D\mathfrak F_{t_{n}}$ be the Jacobian of this map, in which we split the real and imaginary parts: if
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\begin{equation}
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(D\mathfrak F(\hat u))_{k,\ell}
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=
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-\frac{4\pi^2}{L^2}\nu k^2\delta_{k,\ell}
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+\frac{4\pi^2}{L^2|k|}\partial_{\hat u_\ell}T(\hat u,k)
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\hat u_k(t_n)=:\rho_k+i\iota_k
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,\quad
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\mathfrak F_{t_n}(\hat u(t_n))_k=:\phi_k+i\psi_k
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\end{equation}
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and, by\-~(\ref{T}),
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then
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\begin{equation}
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\partial_{\hat u_\ell}T(\hat u,k)
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=
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\sum_{\displaystyle\mathop{\scriptstyle q\in\mathbb Z^2}_{\ell+q=k}}
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\left(
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\frac{(q\cdot \ell^\perp)|q|}{|\ell|}
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+
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\frac{(\ell\cdot q^\perp)|\ell|}{|q|}
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\right)\hat u_q
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=
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(k\cdot \ell^\perp)\left(
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\frac{|k-\ell|}{|\ell|}
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-
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\frac{|\ell|}{|k-\ell|}
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\right)\hat u_{k-\ell}
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(D\mathfrak F_{t_n})_{k,p}:=\left(\begin{array}{cc}
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\partial_{\rho_p}\phi_k&\partial_{\iota_p}\phi_k\\
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\partial_{\rho_p}\psi_k&\partial_{\iota_p}\psi_k
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\end{array}\right)
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\end{equation}
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We compute this Jacobian numerically using a finite difference, by computing
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\begin{equation}
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(D\mathfrak F_{t_n})_{k,p}:=\frac1\epsilon
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\left(\begin{array}{cc}
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\phi_k(\hat u+\epsilon\delta_p)-\phi_k(\hat u)&\phi_k(\hat u+i\epsilon\delta_p)-\phi_k(\hat u)\\
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\psi_k(\hat u+\epsilon\delta_p)-\psi_k(\hat u)&\psi_k(\hat u+i\epsilon\delta_p)-\psi_k(\hat u)
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\end{array}\right)
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.
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\end{equation}
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The parameter $\epsilon$ can be set using the parameter {\tt D\_epsilon}.
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%, so
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%\begin{equation}
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% D\hat u^{(n+1)}=\mathds 1+\delta\sum_{i=1}^q w_iD\mathfrak F(\hat u^{(n)})
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% .
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%\end{equation}
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%We then compute
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%\begin{equation}
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% (D\mathfrak F(\hat u))_{k,\ell}
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% =
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% -\frac{4\pi^2}{L^2}\nu k^2\delta_{k,\ell}
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% +\frac{4\pi^2}{L^2|k|}\partial_{\hat u_\ell}T(\hat u,k)
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%\end{equation}
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%and, by\-~(\ref{T}),
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%\begin{equation}
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% \partial_{\hat u_\ell}T(\hat u,k)
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% =
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% \sum_{\displaystyle\mathop{\scriptstyle q\in\mathbb Z^2}_{\ell+q=k}}
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% \left(
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% \frac{(q\cdot \ell^\perp)|q|}{|\ell|}
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% +
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% \frac{(\ell\cdot q^\perp)|\ell|}{|q|}
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% \right)\hat u_q
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% =
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% (k\cdot \ell^\perp)\left(
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% \frac{|k-\ell|}{|\ell|}
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% -
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% \frac{|\ell|}{|k-\ell|}
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% \right)\hat u_{k-\ell}
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% .
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%\end{equation}
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\bigskip
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\indent
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The Lyapunov exponents are then defined as
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\begin{equation}
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\frac1n\log\mathrm{spec}(\pi_i)
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\frac1{t_n}\log\mathrm{spec}(\pi_{t_n})
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,\quad
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\pi_i:=\prod_{i=1}^nD\hat u^{(i)}
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\pi_{t_n}:=\prod_{i=1}^nD\hat u(t_i)
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.
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\end{equation}
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However, the product of $D\hat u$ may become quite large or quite small (if the exponents are not all 1).
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To avoid this, we periodically rescale the product.
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We set $\mathfrak L_r\in\mathbb N_*$ (set by adjusting the {\tt lyanpunov\_reset} parameter), and, when $n$ is a multiple of $\mathfrak L_r$, we rescale the eigenvalues of $\pi_i$ to 1.
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We set $\mathfrak L_r>0$ (set by adjusting the {\tt lyanpunov\_reset} parameter), and, when $t_n$ crosses a multiple of $\mathfrak L_r$, we rescale the eigenvalues of $\pi_i$ to 1.
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To do so, we perform a $QR$ decomposition:
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\begin{equation}
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\pi_{\alpha\mathfrak L_r}
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=Q^{(\alpha)}R^{(\alpha)}
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\end{equation}
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where $Q^{(\alpha)}$ is diagonal and $R^{(\alpha)}$ is an orthogonal matrix, and we divide by $Q^{(\alpha)}$ (thus only keeping $R^{(\alpha)}$).
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Thus, we replace
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\begin{equation}
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\pi_{\alpha\mathfrak L_r+i}\mapsto R^{(\alpha)}\prod_{j=1}^iD\hat u^{(j)}
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.
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=R^{(\alpha)}Q^{(\alpha)}
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\end{equation}
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where $Q^{(\alpha)}$ is orthogonal and $R^{(\alpha)}$ is a diagonal matrix, and we divide by $R^{(\alpha)}$ (thus only keeping $Q^{(\alpha)}$).
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The Lyapunov exponents at time $\alpha\mathfrak L_r$ are then
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\begin{equation}
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\frac1{\alpha\mathfrak L_r}\sum_{\beta=1}^\alpha\log\mathrm{spec}(Q^{(\beta)})
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@ -21,7 +21,8 @@ limitations under the License.
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#include <math.h>
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#include <stdlib.h>
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#define MATSIZE (K1*(2*K2+1)+K2)
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#define MATSIZE (2*(K1*(2*K2+1)+K2))
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#define USIZE (K1*(2*K2+1)+K2)
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// compute Lyapunov exponents
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int lyapunov(
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@ -50,8 +51,8 @@ int lyapunov(
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){
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double* lyapunov;
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double* lyapunov_avg;
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_Complex double* Du_prod;
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_Complex double* Du;
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double* Du_prod;
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double* Du;
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_Complex double* u;
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_Complex double* prevu;
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_Complex double* tmp1;
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@ -63,7 +64,7 @@ int lyapunov(
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_Complex double* tmp7;
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_Complex double* tmp8;
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_Complex double* tmp9;
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_Complex double* tmp10;
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double* tmp10;
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double time;
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fft_vect fft1;
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fft_vect fft2;
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@ -82,9 +83,6 @@ int lyapunov(
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double step=delta;
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double next_step=step;
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const _Complex double one=1;
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const _Complex double zero=0;
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int i;
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// init average
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for (i=0; i<MATSIZE; i++){
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@ -109,9 +107,9 @@ int lyapunov(
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// multiply Jacobian
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// switch to column major order, so transpose Du
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cblas_zgemm(CblasColMajor, CblasNoTrans, CblasTrans, MATSIZE, MATSIZE, MATSIZE, &one, Du_prod, MATSIZE, Du, MATSIZE, &zero, tmp10, MATSIZE);
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cblas_dgemm(CblasColMajor, CblasNoTrans, CblasTrans, MATSIZE, MATSIZE, MATSIZE, 1., Du_prod, MATSIZE, Du, MATSIZE, 0., tmp10, MATSIZE);
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// copy back to Du_prod
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_Complex double* move=tmp10;
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double* move=tmp10;
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tmp10=Du_prod;
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Du_prod=move;
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@ -124,10 +122,10 @@ int lyapunov(
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// QR decomposition
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// do not touch tmp1, it might be used in the next step
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LAPACKE_zgerqf(LAPACK_COL_MAJOR, MATSIZE, MATSIZE, Du_prod, MATSIZE, tmp2);
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LAPACKE_dgerqf(LAPACK_COL_MAJOR, MATSIZE, MATSIZE, Du_prod, MATSIZE, tmp10);
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// extract eigenvalues (diagonal elements of Du_prod)
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for(i=0; i<MATSIZE; i++){
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lyapunov[i]=log(cabs(Du_prod[i*MATSIZE+i]))/(time-prevtime);
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lyapunov[i]=log(fabs(Du_prod[i*MATSIZE+i]))/(time-prevtime);
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}
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// sort lyapunov exponents
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@ -153,7 +151,7 @@ int lyapunov(
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}
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// set Du_prod to Q:
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LAPACKE_zungrq(LAPACK_COL_MAJOR, MATSIZE, MATSIZE, MATSIZE, Du_prod, MATSIZE, tmp2);
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LAPACKE_dorgrq(LAPACK_COL_MAJOR, MATSIZE, MATSIZE, MATSIZE, Du_prod, MATSIZE, tmp10);
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// reset prevtime
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prevtime=time;
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@ -177,7 +175,7 @@ int compare_double(const void* x, const void* y) {
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// Jacobian of u_n -> u_{n+1}
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int ns_D_step(
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_Complex double* out,
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double* out,
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_Complex double* un,
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_Complex double* un_next,
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int K1,
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@ -218,7 +216,7 @@ int ns_D_step(
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for(ly=(lx>0 ? -K2 : 1);ly<=K2;ly++){
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// derivative in the real direction
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// finite difference vector
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for (i=0;i<MATSIZE;i++){
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for (i=0;i<USIZE;i++){
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if(i==klookup_sym(lx,ly,K2)){
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tmp1[i]=un[i]+epsilon;
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}else{
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@ -230,15 +228,16 @@ int ns_D_step(
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step=delta;
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next_step=delta;
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ns_step(algorithm, tmp1, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, time, fft1, fft2, ifft, &tmp2, &tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, irreversible, keep_en_cst, target_en);
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// compute derivative
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for (i=0;i<MATSIZE;i++){
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// derivative
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for (i=0;i<USIZE;i++){
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// use row major order
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out[klookup_sym(lx,ly,K2)*MATSIZE+i]=(tmp1[i]-un_next[i])/epsilon;
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out[2*klookup_sym(lx,ly,K2)*USIZE+2*i]=(__real__ (tmp1[i]-un_next[i]))/epsilon;
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out[(2*klookup_sym(lx,ly,K2)+1)*USIZE+2*i]=(__imag__ (tmp1[i]-un_next[i]))/epsilon;
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}
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// derivative in the imaginary direction
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// finite difference vector
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for (i=0;i<MATSIZE;i++){
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for (i=0;i<USIZE;i++){
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if(i==klookup_sym(lx,ly,K2)){
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tmp1[i]=un[i]+epsilon*I;
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}else{
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@ -251,9 +250,10 @@ int ns_D_step(
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next_step=delta;
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ns_step(algorithm, tmp1, K1, K2, N1, N2, nu, &step, &next_step, adaptive_tolerance, adaptive_factor, max_delta, adaptive_norm, L, g, time, time, fft1, fft2, ifft, &tmp2, &tmp3, tmp4, tmp5, tmp6, tmp7, tmp8, irreversible, keep_en_cst, target_en);
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// compute derivative
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for (i=0;i<MATSIZE;i++){
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for (i=0;i<USIZE;i++){
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// use row major order
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out[klookup_sym(lx,ly,K2)*MATSIZE+i]=-(tmp1[i]-un_next[i])/epsilon*I;
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out[2*klookup_sym(lx,ly,K2)*USIZE+2*i+1]=(__real__ (tmp1[i]-un_next[i]))/epsilon;
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out[(2*klookup_sym(lx,ly,K2)+1)*USIZE+2*i+1]=(__imag__ (tmp1[i]-un_next[i]))/epsilon;
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}
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}
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}
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@ -265,8 +265,8 @@ int ns_D_step(
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int lyapunov_init_tmps(
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double ** lyapunov,
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double ** lyapunov_avg,
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_Complex double ** Du_prod,
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_Complex double ** Du,
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double ** Du_prod,
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double ** Du,
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_Complex double ** u,
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_Complex double ** prevu,
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_Complex double ** tmp1,
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@ -278,7 +278,7 @@ int lyapunov_init_tmps(
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_Complex double ** tmp7,
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_Complex double ** tmp8,
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_Complex double ** tmp9,
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_Complex double ** tmp10,
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double ** tmp10,
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fft_vect* fft1,
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fft_vect* fft2,
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fft_vect* ifft,
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@ -293,12 +293,12 @@ int lyapunov_init_tmps(
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*lyapunov=calloc(sizeof(double),MATSIZE);
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*lyapunov_avg=calloc(sizeof(double),MATSIZE);
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*Du_prod=calloc(sizeof(_Complex double),MATSIZE*MATSIZE);
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*Du=calloc(sizeof(_Complex double),MATSIZE*MATSIZE);
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*prevu=calloc(sizeof(_Complex double),MATSIZE);
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*tmp1=calloc(sizeof(_Complex double),MATSIZE);
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*tmp2=calloc(sizeof(_Complex double),MATSIZE);
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*tmp10=calloc(sizeof(_Complex double),MATSIZE*MATSIZE);
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*Du_prod=calloc(sizeof(double),MATSIZE*MATSIZE);
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*Du=calloc(sizeof(double),MATSIZE*MATSIZE);
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*prevu=calloc(sizeof(_Complex double),USIZE);
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*tmp1=calloc(sizeof(_Complex double),USIZE);
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*tmp2=calloc(sizeof(_Complex double),USIZE);
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*tmp10=calloc(sizeof(double),MATSIZE*MATSIZE);
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return(0);
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}
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@ -307,8 +307,8 @@ int lyapunov_init_tmps(
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int lyapunov_free_tmps(
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double* lyapunov,
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double* lyapunov_avg,
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_Complex double* Du_prod,
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_Complex double* Du,
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double* Du_prod,
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double* Du,
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_Complex double* u,
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_Complex double* prevu,
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_Complex double* tmp1,
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@ -320,7 +320,7 @@ int lyapunov_free_tmps(
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_Complex double* tmp7,
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_Complex double* tmp8,
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_Complex double* tmp9,
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_Complex double* tmp10,
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double* tmp10,
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fft_vect fft1,
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fft_vect fft2,
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fft_vect ifft,
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@ -26,12 +26,12 @@ int lyapunov( int K1, int K2, int N1, int N2, double final_time, double lyapunov
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int compare_double(const void* x, const void* y);
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// Jacobian of step
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int ns_D_step( _Complex double* out, _Complex double* un, _Complex double* un_next, int K1, int K2, int N1, int N2, double nu, double epsilon, double delta, unsigned int algorithm, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, double L, _Complex double* g, double time, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* tmp1, _Complex double* tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, _Complex double* tmp8, bool irreversible, bool keep_en_cst, double target_en);
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int ns_D_step( double* out, _Complex double* un, _Complex double* un_next, int K1, int K2, int N1, int N2, double nu, double epsilon, double delta, unsigned int algorithm, double adaptive_tolerance, double adaptive_factor, double max_delta, unsigned int adaptive_norm, double L, _Complex double* g, double time, fft_vect fft1, fft_vect fft2, fft_vect ifft, _Complex double* tmp1, _Complex double* tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, _Complex double* tmp8, bool irreversible, bool keep_en_cst, double target_en);
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// init vectors
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int lyapunov_init_tmps(double** lyapunov, double** lyapunov_avg, _Complex double ** Du_prod, _Complex double ** Du, _Complex double ** u, _Complex double ** prevu, _Complex double ** tmp1, _Complex double ** tmp2, _Complex double ** tmp3, _Complex double ** tmp4, _Complex double ** tmp5, _Complex double ** tmp6, _Complex double ** tmp7, _Complex double ** tmp8, _Complex double ** tmp9, _Complex double** tmp10, fft_vect* fft1, fft_vect* fft2, fft_vect* ifft, int K1, int K2, int N1, int N2, unsigned int nthreads, unsigned int algorithm);
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int lyapunov_init_tmps(double** lyapunov, double** lyapunov_avg, double ** Du_prod, double ** Du, _Complex double ** u, _Complex double ** prevu, _Complex double ** tmp1, _Complex double ** tmp2, _Complex double ** tmp3, _Complex double ** tmp4, _Complex double ** tmp5, _Complex double ** tmp6, _Complex double ** tmp7, _Complex double ** tmp8, _Complex double ** tmp9, double** tmp10, fft_vect* fft1, fft_vect* fft2, fft_vect* ifft, int K1, int K2, int N1, int N2, unsigned int nthreads, unsigned int algorithm);
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// release vectors
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int lyapunov_free_tmps(double* lyapunov, double* lyapunov_avg, _Complex double* Du_prod, _Complex double* Du, _Complex double* u, _Complex double* prevu, _Complex double* tmp1, _Complex double* tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, _Complex double* tmp8, _Complex double* tmp9, _Complex double* tmp10, fft_vect fft1, fft_vect fft2, fft_vect ifft, unsigned int algorithm);
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int lyapunov_free_tmps(double* lyapunov, double* lyapunov_avg, double* Du_prod, double* Du, _Complex double* u, _Complex double* prevu, _Complex double* tmp1, _Complex double* tmp2, _Complex double* tmp3, _Complex double* tmp4, _Complex double* tmp5, _Complex double* tmp6, _Complex double* tmp7, _Complex double* tmp8, _Complex double* tmp9, double* tmp10, fft_vect fft1, fft_vect fft2, fft_vect ifft, unsigned int algorithm);
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#endif
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