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\begin{center}
{\myfont WAVELETS IN OPTIMIZATION AND APPROXIMATIONS}
\vspace{.8\baselineskip}
{\bf A. N. Fedorova and M. G. Zeitlin}\\
Computational Mechanics Group,
Institute of Problems of
Mechanical Engineering,\\
Russian Academy of Sciences,
Russia, 199178, St.~Petersburg, V.O., Bolshoj pr. 61.\\
E-mail: zeitlin@math.ipme.ru, anton@math.ipme.ru
\end{center}
\vspace{.5\baselineskip}
\noindent{\bf Abstract.}
We give the explicit time description of the following problems:
dynamics of storage rings, optimal dynamics for some important electromechanical
system, Galerkin approximation for beam oscillations in liquid,
computations of Melnikov functions for perturbed Hamiltonian systems.
All these problems are reduced to the problem of
the solving of the systems of differential equations with polynomial
nonlinearities with or without some constraints.
The first main part of our construction is some variational approach
to this problem, which reduces initial problem to the problem of
the solution of functional equations at the first stage and some
algebraical problems at the second stage.
We consider also two private cases of our general construction.
In the first case (particular) we have the solution as a series on shifted
Legendre polynomials, which is parameterized by the solution
of reduced algebraical system of equations.
In the second case (general) we have the solution in a compactly
supported wavelet basis. Multiresolution expansion is the second main
part of our construction.
The solution is parameterized by solutions of two reduced algebraical
problems, the first one is the same as in the first case and
the second one is some linear
problem, which is obtained from one of the next wavelet
constructions: Fast Wavelet Transform, Stationary
Subdivision Schemes, the method of Connection
Coefficients.
\vspace{2\baselineskip}
\noindent We give the explicit time description of the following
problems: dynamics and optimal dynamics for important nonlinear dynamical
systems, Ga\-lerkin approximation for some class of partial
dif\-ferential equations, computations of Melnikov function for
perturbed Hamiltonian systems. All these problems are reduced to the
problem of the solving of the sys\-tems of differential equations
with polynomial non\-li\-nearities with or without some constraints.
The first main part of our construction is some variational
approach to this problem, which reduces initial problem to the
problem of the solution of functional equations at the first stage
and some algebraical problems at the second stage.
We consider also two private cases of our general construction.
In the first case (particular) we have for Riccati type equations
the solution as a series on shifted
Legendre polynomials, which is parametrized by the solution
of reduced algebraical (also Riccati) system of
equations [5].
In the second case (general polynomial sys\-tems) we have
the solution in a compactly supported wavelet basis [8], [11]-[13].
Multiresolution
expansion is the second main part of our construction. In this case the
solution is parametrized by solutions of two reduced algebraic
problems, one as in the first case and the second is some linear
problem, which is obtained from one of the next wavelet
constructions: Fast Wavelet Transform (FWT) [1], Stationary
Subdivision Schemes (SSS) [3], the method of Connection
Coefficients (CC) [15].
We use our general construction for solution of important technical
problems: minimization of energy and detecting signals from
oscillations of a submarine and also for calculations of orbital
motions in storage rings [6], [7].
Our initial problem comes from very impor\-tant technical problem
-- minimization of energy in electromechanical system with
enormous expense of energy. That is synchronous drive of the
mill--the electrical machine with the mill as load. It
is described by Park system of equations [5]:
\begin{eqnarray*}
\frac{di_k}{dt}=\sum_\ell A_\ell i_\ell+\sum_{r,s} A_{r,s}i_r i_s+A_k(t),
\end{eqnarray*}
where $A_\ell,(k,\ell,r,s=\overline {1,6})$ are constants,
$A_k(t), (k=\overline{1,5})$ are
explicit functions of time,
$A_6(i_6,t)=a+di_6+bi^2_6$ is analytical approximation for the
mechanical moment of the mill. In our case we consider
${i_1,i_2}$ as the controlling variables.
Because we consider the energy optimization, we use the
next general form of energy functional in our electromechanical
system
$$Q=\int^t_{t_0}[K_1(i_1,i_2)+K_2(\dot i_1,\dot i_2)]dt,$$
where $K_1,K_2$ are quadratic forms.
Moreover, we consider the optimization problem with some
constraints, which are motivated by technical reasons. After the
manipulations from the theory of
optimal control, we reduce the problem of energy minimization to
the some nonlinear system of equations. Thus
for the Lagrangian optimization
we have the system of 13 equations (12--
differential equations, 1--functional one). For
the Hamiltonian optimization we have the
system of 12 equations (10--differential equations, 2--algebraical
ones). In both cases obtained systems of equations are
the systems of Riccati type. As result of solution of
equations of optimal dynamics we have:
1. the explicit time dependence of the controlling
variables
$u(t)=\{ i_1(t),i_2(t)\}$, which give
2. the optimum of corresponding functional of energy and
3. explicit time dynamics of the controllable variables
$\{ i_3,i_4,i_5,i_6\} (t)$.
Next we consider the construction of explicit time
solution. The obtained solutions are given in the next form:
$$ i_k(t)=i_k(0)+\sum^N_{i=1}\lambda_k^i X_i(t),$$
where in our first case we have
$X_i(t)=Q_i(t)$, where $Q_i(t)$ are shifted Legendre
polynomials [14] and $\lambda_k^i$ are roots of reduced
algebraic system of equations. In wavelet case $X_i(t)$
correspond to multiresolution expansions in the base of
compactly supported wavelets and $\lambda_k^i$ are the roots of
corresponding algebraic Riccati systems with coefficients, which
are given by FWT, SSS or CC constructions. According to the
variational method of [14] to give the reduction from
differential to algebraical system of equations we need compute
the objects $\gamma ^j_a(i_b)$ and $\mu_{ji}$, where in
Lagrangian case $a=\overline{1,13},b=\overline{1,13}$. We compute it
by the formulae:
\begin{eqnarray*}
\gamma ^j_a(i_b)=t_f\int
^1_0\phi_a(i_b,\tau)X_j(\tau)d\tau, \quad
\mu_{ji}=\int\limits_0^1X_i'(\tau)X_j(\tau)d\tau,
\end{eqnarray*}
where
$\phi_a$ is RHS of initial equations.
Then the reduced algebraical system has the form:
$$
\sum\limits_{i=1}^N \mu_{ji}\lambda_a^i-\gamma_a^j(\lambda_b)=0,
$$
where coefficients of algebraical systems are constructed from
objects:
\begin{eqnarray}
\sigma_i&\equiv&\int^1_0X_i(\tau)d\tau=(-1)^{i+1},\\
\nu_{ij}&\equiv&\int^1_0X_i(\tau)X_j(\tau)d\tau=
\sigma_i\sigma_j+\frac{\delta_{ij}}{ (2j+1)},\nonumber\\
\beta_{klj}&\equiv&\int^1_0X_k(\tau)X_l(\tau)X_j(\tau)
d\tau=\sigma_k\sigma_l\sigma_j+ \alpha_{klj}+
\frac{\sigma_k\delta_{jl}}{ 2j+1}+
\frac{\sigma_l\delta_{kj}}{ 2k+1}+
\frac {\sigma_j\delta_{kl}}{ 2l+1},\nonumber\\
\alpha_{klj}&\equiv&\int^1_0X^\ast_kX^\ast_lX^\ast_jd\tau=
\frac{R\bigl(1/2(j+k-l)\bigr)
R\bigl(1/2(j-k+l)\bigr)
R\bigl(1/2(-j+k+l)\bigr)}{(j+k+l+1)R\bigl(1/2(i+j+k)\bigr)},\nonumber
\end{eqnarray}
if $j+k+l=2m, m\in{\it Z}$, and
$\alpha_{klj}=0$ if $ j+k+l=2m+1 $;
$ R(i)=(2i)!/(2^ii!)^2$, $X_i=\sigma_i+X_i^\ast$, where the
second equality in the formulae for $\sigma,\nu,\beta,\alpha$
hold for the first case.
Now we give construction for the
computations of objects (1) in the wavelet case.
We use some constructions from multiresolution analysis: a sequence of
successive approximation closed subspaces $V_j$
$$
...V_2\subset V_1\subset V_0\subset V_{-1}\subset V_{-2}\subset ...
$$
satisfying the following properties:
\begin{eqnarray*}
\displaystyle\bigcap_{j\in{\bf Z}}V_j=0,\quad
\overline{\displaystyle\bigcup_{j\in{\bf Z}}}V_j=L^2({\bf R}),
\end{eqnarray*}
where from $f(x)\in V_j$ we have $f(2x)\in V_{j+1}$.
There is a function $\varphi\in V_0$ such that\\ \{${\varphi_{0,k}(x) =
\varphi(x-k)}_{k\in{\bf Z}}$\} forms a Riesz basis for $V_0$.
We use compactly supported wavelet basis: orthonormal basis
for functions in $L^2({\bf R})$ [2]. As usually $\varphi(x)$ is
a scaling function, $\psi(x)$ is a wavelet function, where
$\varphi_i(x)=\varphi(x-i)$. Scaling or refinement equations that defines
$\varphi,\psi$ are
\begin{eqnarray*}
\varphi(x)&=&\sum\limits^{N-1}_{k=0}a_k\varphi(2x-k)=
\sum\limits^{N-1}_{k=0}a_k\varphi_k(2x),\\
\psi(x)&=&\sum\limits^{N-2}_{k=-1}(-1)^k a_{k+1}\varphi(2x+k).
\end{eqnarray*}
Let be $ f : {\bf R}\longrightarrow {\bf C}$ and the wavelet
expansion is
\begin{eqnarray}
f(x)=\sum\limits_{\ell\in{\bf Z}}c_\ell\varphi_\ell(x)+
\sum\limits_{j=0}^\infty\sum\limits_{k\in{\bf
Z}}c_{jk}\psi_{jk}(x).
\end{eqnarray}
The indices $k, \ell$ and $j$ represent translation and scaling, respectively
\begin{eqnarray*}
\varphi_{jl}(x)=2^{j/2}\varphi(2^j x-\ell),\quad
\psi_{jk}(x)=2^{j/2}\psi(2^j x-k).
\end{eqnarray*}
The set $\{\varphi_{j,k}\}_{k\in {\bf Z}}$ forms a Riesz basis for $V_j$.
Let $W_j$ be the orthonormal complement of $V_j$ with respect to $V_{j+1}$.
Just as $V_j$ is spanned by dilation and translations of the scaling function,
so $W_j$ are spanned by translations and dilation of the mother wavelet
$\psi_{jk}(x)$.
All expansions which we used are based on the following properties:
\begin{eqnarray*}
& &\{\varphi_{jk}\}_{j\geq 0, k\in {\bf Z}} \quad \mbox{ is an orthonormal basis
for} L^2({\bf
R}),\\
& &V_{j+1}=V_j\bigoplus W_j,\qquad
L^2({\bf R})=\overline{V_0\displaystyle\bigoplus^\infty_{j=0} W_j},\\
& &\{\varphi_{0,k},\psi_{j,k}\}_{j\geq 0,k\in {\bf Z}} \quad \mbox{ is
an orthonormal basis for} L^2({\bf R}),\\
& & \{\varphi_{j,k},\psi_{\ell,k}; 0\leq j\leq J\leq \ell, k\in{\bf Z}\}
\mbox{is an orthonormal basis for} L^2({\bf R}),\\
& & \displaystyle\int^\infty_{-\infty}\varphi(x)dx=1.
\end{eqnarray*}
If in formulae (2) $c_{jk}=0$ for $j\geq J$, then $f(x)$ has an alternative
expansion in terms of dilated scaling functions only
\begin{eqnarray}
f(x)=\sum\limits_{\ell\in {\bf Z}}c_{J\ell}\varphi_{J\ell}(x)
\end{eqnarray}
This is a finite wavelet expansion, it can be written solely in
terms of translated scaling functions. We use wavelet $\psi(x)
$, which has $k$ vanishing moments
\begin{eqnarray*}
\int x^k \psi(x)dx=0,
\end{eqnarray*}
or, equivalently
\begin{eqnarray*}
x^k=\sum c_\ell\varphi_\ell(x) \quad \mbox{ for each $k$,}\quad
0\leq k\leq K
\end{eqnarray*}
Also we have the shortest possible support: scaling function
$DN$ (where $N$ is even integer) will have support $[0,N-1]$ and
$N/2$ vanishing moments.
There exists $\lambda>0$ such that $DN$ has $\lambda N$
continuous derivatives; for small $N,\lambda\geq 0.55$.
To solve our second associated linear problem we need to
evaluate derivatives of $f(x)$ in terms of $\varphi(x)$.
Let be $
\varphi^n_\ell=d^n\varphi_\ell(x)/dx^n
$.
We derive the wavelet - Galerkin approximation of a
differentiated $f(x)$ as
$
f^d(x)=\sum_\ell c_l\varphi_\ell^d(x)
$
and values $\varphi_\ell^d(x)$ can be expanded in terms of
$\varphi(x)$
$$
\phi_\ell^d(x)=\sum\limits_m\lambda_m\varphi_m(x),
$$
where
$$
\lambda_m=\int\limits_{-\infty}^{\infty}\varphi_\ell^d(x)\varphi_m(x)dx.
$$
The coefficients $\lambda_m$ are 2-term connection
coefficients [15]. In general we need to find
\begin{eqnarray*}
& &\Lambda(\ell_1,\ell_2,...,\ell_n,d_1,d_2,...,d_n)=
\Lambda^{d_1 d_2 ...d_n}_{\ell_1 \ell_2 ...\ell_n}=
\int\limits_{-\infty}^{\infty}\prod\varphi^{d_i}_{\ell_i}(x)dx
\end{eqnarray*}
For our Riccati case we need to evaluate two and three
connection coefficients $(d_i\geq 0)$
\begin{eqnarray*}
\Lambda_\ell^{d_1
d_2}=\int^\infty_{-\infty}\varphi^{d_1}(x)\varphi_\ell^{d_2}(x)dx,\quad
\Lambda^{d_1 d_2
d_3}=\int\limits_{-\infty}^\infty\varphi^{d_1}(x)\varphi_
\ell^{d_2}(x)\varphi^{d_3}_m(x)dx.
\end{eqnarray*}
According to [15] we use the next construction. When $N$ in
scaling equation is a finite even positive integer the function
$\varphi(x)$ has compact support contained in $[0,N-1]$.
For a fixed triple $(d_1,d_2,d_3)$ only some $\Lambda_{\ell
m}^{d_1 d_2 d_3}$ are nonzero : $2-N\leq \ell\leq N-2,\quad
2-N\leq m\leq N-2,\quad |\ell-m|\leq N-2$ . There are
$M=3N^2-9N+7$ such pairs $(\ell,m)$. Let $\Lambda^{d_1 d_2 d_3}$
be an M-vector, whose components are numbers $\Lambda^{d_1 d_2
d_3}_{\ell m}$. Then we have the first key result: $\Lambda$
satisfy the system of equations
\begin{eqnarray}
A\Lambda^{d_1 d_2 d_3}&=&2^{1-d}\Lambda^{d_1 d_2 d_3},\
d=d_1+d_2+d_3,\\
A_{\ell,m;q,r}&=&\sum\limits_p a_p a_{q-2\ell+p}a_{r-2m+p}\nonumber
\end{eqnarray}
By moment equations we have created a system of $M+d+1$
equations in $M$ unknowns. It has rank $M$ and we can obtain
unique solution by combination of LU decomposition and QR
algorithm.
The second key result gives us the 2-term connection
coefficients $(d=d_1+d_2)$:
\begin{eqnarray}
A\Lambda^{d_1 d_2}=2^{1-d}\Lambda^{d_1 d_2},\quad
A_{\ell,q}=\sum\limits_p a_p a_{q-2\ell+p}
\end{eqnarray}
We use for modelling D6, D8, D10 functions and programs RADAU and DOPRI for
testing.
Also, we use FWT and SSS for computing coefficients of reduced
algebraic systems.
As a result we obtained the explicit time solution of optimal
control problem in form (3). The generalization to arbitrary polynomial systems is
evidently.
Instead of two reduced linear problems (4), (5) for computational objects
(1), we have analogous additional linear problems.
Analogously we consider in wavelet approach related problems:
computations in Galerkin approximations and routes to chaos in Melnikov
approach [9], [10]. These problems are related to
a problem of detecting
signals from an oscillating submarine.
In 2-mode Galerkin approximation for beam contacting with ideal
compressible liquid in a channel we have
the next system of equations:
\begin{eqnarray*}
\dot x_1&=&x_2, \quad \dot x_3=x_4, \quad \dot x_5=r, \quad \dot x_6=s, \\
\dot x_2&=&-ax_1-b[\cos(x_5)+\cos(x_6)]x_1-dx^3_1-
mdx_1x^2_3- px_2-\varphi(x_5) \\
\dot x_4&=&ex_3-f[\cos(x_5)+\cos(x_6)]x_3- gx_3^3-
kx_1^2x_3-gx_4-
\psi(x_5) \\
\end{eqnarray*}
or in Hamiltonian form
\begin{eqnarray*}
\dot{x}=J\cdot\nabla H(x)+\varepsilon g(x,\Theta), \quad
\dot{\Theta}=\omega, (x,\Omega)\in R^4\times T^2,
\end{eqnarray*}
for $\varepsilon=0 $ we have:
\begin{eqnarray}
\dot{x}=J\cdot\nabla H(x),\quad
\dot\Theta=\omega
\end{eqnarray}
We solve two problems related with these systems of equations.
We need to compute explicit time solution for perturbed and unperturbed
systems. The solution for unperturbed system (6) we use next for computing
Melnikov functions
\begin{eqnarray}
M(\Theta)&=&\int\limits_{-\infty}^{\infty}\nabla
H(\bar{x}_{0}(t))\wedge g(\bar{x}_{0}(t),\omega t+\Theta)dt\\
M^{m/n}(t_0)&=&\int\limits^{mT}_0 DH(x_{\alpha}(t))\wedge(x_\alpha(t),t+t_0)dt
\end{eqnarray}
which we use for detecting chaotic and quasiperiodic regimes of
oscillations [9], [10]. In comparison with
wavelet expansion
on the real line which we use in optimal control problem and in
calculation of Galerkin approximation, in Melnikov function approach
for the computation of (quasi)periodic regimes (8)
we need to use periodized wavelet expansion, i.e. wavelet expansion
on finite interval [2]. Also in the solution of perturbed system we have
some problem with variable coefficients.
For solving last problem we need to consider one more refinement equation
for scaling function $\phi_2(x)$:
$$
\phi_2 (x)=\sum\limits^{N-1}_{k=0} a^2_k \phi_2 (2x-k)
$$
and corresponding wavelet expansion for variable coefficients $b(t)$:
$$
\sum\limits_k B_k^j (b)\phi_2(2^jx-k),
$$
where $B_k^j (b)$ are functionals supported in a small neighborhood of
$2^{-j}k$ [4].
The solution of the first problem consists in periodizing. In this case
we expand (quasi)pe\-rio\-dic orbit into periodized wavelets
defined by [2]:
$$
\phi^{per}_{-j,k}(x)=2^{j/2}\sum\limits_{Z} \phi(2^jx+2^j\ell-k).
$$
All these modifications lead only to transformations of coefficients of
reduced algebraic system, but general scheme remains the same.
All this approaches we also use in very important now nonlinear dynamical
problems from accelerator physics.
As an example we consider a problem of computations of orbital motions
in storage rings.
Starting from Hamiltonian, which described classical dynamics in
storage rings
and using Serret--Frenet parametrization, we have
after standard manipulations with truncation of
power series expansion of square root
the corresponding equations of motion [6]:
\begin{eqnarray}
&&\frac{d}{ds}x=
\frac{p_x+H\cdot z}{[1+f(p_\sigma)]}; \\
&&\frac{d}{ds}p_x=
\frac{[p_z-H\cdot x]}{[1+f(p_\sigma)]}\cdot H -[K^2_x+g]\cdot x+N\cdot z\nonumber\\
&&
+K_x\cdot f(p_\sigma)-\frac{\lambda}{2}\cdot(x^2-z^2)-
\frac{\mu}{6}(x^3-3xz^2);\nonumber\\
&&\frac{d}{ds}z=
\frac{p_z-H\cdot x}{[1+f(p_\sigma)]}; \nonumber\\
&&\frac{d}{ds}p_z=
-\frac{[p_x+H\cdot z]}{[1+f(p_\sigma)]}\cdot H-
[K^2_z-g]\cdot z \nonumber\\
&&+N\cdot x+K_z\cdot f(p_\sigma)-\lambda\cdot xz-
\frac{\mu}{6}(z^3-3x^2z);\nonumber\\
&&\frac{d}{ds}\sigma=
1-[1+K_x\cdot x+K_z\cdot z]\cdot f^\prime(p_\sigma)- \nonumber\\
&&\frac{1}{2}\cdot\frac{[p_x+H\cdot z]^2+[p_z-H\cdot x]^2}{[1+f(p_\sigma)]^2}
\cdot f^\prime(p_\sigma);\nonumber\\
&&\frac{d}{ds}p_\sigma=-
\frac{1}{\beta_0^2}\cdot\frac{eV(s)}{E_0}\cdot\sin\left[h\cdot\frac{2\pi}{L}
\cdot\sigma+\varphi\right]. \nonumber
\end{eqnarray}
We use series expansion of function $f(p_\sigma)$
and the corresponding expansion of RHS of equations (9).
Then taking into account only an arbitrary
polynomial (in terms of dynamical variables) expressions and
neglecting all nonpolynomial types of expressions, we
consider such approximations of RHS, which are not more than polynomial
functions in dynamical variables and arbitrary functions of
independent variable $s$ ("time" in our case, if we consider
our system of equations as dynamical problem). After applying our
approaches we also have description of complicated accelerator
dynamics according to formulae (3), (7), (8) [6], [7].
This research was supported in part by the National Science Foundation under Grant
No. PHY94-07194.
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