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\begin{document}
\title{ WAVELET APPROACH TO ACCELERATOR PROBLEMS, I. \\
POLYNOMIAL DYNAMICS}
\author{A.~Fedorova and M.~Zeitlin,
Institute of Problems of Mechanical Engineering,\\
Russian Academy of Sciences, 199178, Russia, St.~Petersburg,\\
V.O., Bolshoj pr., 61, e-mail: zeitlin@math.ipme.ru\\
Z.~Parsa, Dept. of Physics, Bldg.~901A, Brookhaven National Laboratory,\\
Upton, NY 11973-5000, e-mail: parsa@bnl.gov}
\maketitle
\begin{abstract}
This is the first part of a series of talks in which we
present applications of methods from
wavelet analysis to polynomial approximations for
a number of accelerator physics problems.
In the general case we have the solution as
a multiresolution expansion in the base of compactly
supported wavelet basis. The solution is parametrized by
solutions of two reduced algebraical problems, one is
nonlinear and the second 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.
\end{abstract}
In this paper we consider the problem of
calculation of orbital motion in storage rings.
The key point in
the solution of this problem is the use of the methods of
wavelet analysis,
relatively novel set of mathematical methods, which gives
us a possibility to work with well-localized bases in functional
spaces and with the general type of operators (including pseudodifferential)
in such bases.
Our problem as many related problems in the framework of
our type of approximations of complicated physical nonlinearities
is reduced to the problem of the solving of the systems of
differential equations with polynomial nonlinearities with or
without some constraints.
In this paper we consider as the main example the particle motion in
storage rings in standard approach.
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:
\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}
Then we use series expansion of function $f(p_\sigma)$
and the corresponding expansion of RHS of equations (1).
In the following we take into account only an arbitrary
polynomial (in terms of dynamical variables) expressions and
neglecting all nonpolynomial types of expressions, i.e. 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).
The first main part of our construction is some variational approach
to this problem, which reduces initial problem to the problem of
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 equations
(particular quadratic approximations) the solution
as a series on shifted
Legendre polynomials, which is parameterized by the solution
of reduced algebraical (also Riccati) system of equations.
This is only an example of general construction.
In the second case (general polynomial system) 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, one as in the first case and the second is some linear
problem, which is obtained from one of the next wavelet
construction: Fast Wavelet Transform (FWT), Stationary
Subdivision Schemes (SSS), the method of Connection
Coefficients (CC).
Our problems may be formulated as the systems of ordinary differential
equations
%\begin{eqnarray}
$
{dx_i}/{dt}=f_i(x_j,t), \quad (i,j=1,...,n)
$
%\end{eqnarray}
with fixed initial conditions $x_i(0)$, where $f_i$ are not more
than polynomial functions of dynamical variables $x_j$
and have arbitrary dependence of time. Because of time dilation
we can consider only next time interval: $0\leq t\leq 1$.
Let us consider a set of
functions
%\begin{eqnarray}
$
\Phi_i(t)=x_i{dy_i}/{dt}+f_iy_i
$
%\end{eqnarray}
and a set of functionals
%\begin{eqnarray}
$
F_i(x)=\int_0^1\Phi_i (t)dt-x_iy_i\mid^1_0,
$
%\end{eqnarray}
where $y_i(t) (y_i(0)=0)$ are dual variables.
It is obvious that the initial system and the system $F_i(x)=0$
are equivalent.
In part 3 we consider symplectization of this approach.
Now we consider formal expansions for $x_i, y_i$:
\begin{eqnarray}
x_i(t)=x_i(0)+\sum_k\lambda_i^k\varphi_k(t)\quad
y_j(t)=\sum_r \eta_j^r\varphi_r(t),
\end{eqnarray}
where because of initial conditions we need only $\varphi_k(0)=0$.
Then we have the following reduced algebraical system
of equations on the set of unknown coefficients $\lambda_i^k$ of
expansions (2):
\begin{eqnarray}
\sum_k\mu_{kr}\lambda^k_i-\gamma_i^r(\lambda_j)=0
\end{eqnarray}
Its coefficients are
%\begin{eqnarray}
$
\mu_{kr}=\int_0^1\varphi_k'(t)\varphi_r(t)dt,\quad
\gamma_i^r=\int_0^1f_i(x_j,t)\varphi_r(t)dt.
$
%\end{eqnarray}
Now, when we solve system (3) and determine
unknown coefficients from formal expansion (2) we therefore
obtain the solution of our initial problem.
It should be noted if we consider only truncated expansion (2) with N terms
then we have from (3) the system of $N\times n$ algebraical equations and
the degree of this algebraical system coincides
with degree of initial differential system.
So, we have the solution of the initial nonlinear
(polynomial) problem in the form
\begin{eqnarray}
x_i(t)=x_i(0)+\sum_{k=1}^N\lambda_i^k X_k(t),
\end{eqnarray}
where coefficients $\lambda_i^k$ are roots of the corresponding
reduced algebraical problem (3).
Consequently, we have an parametrization of solution of initial problem
by solution of reduced algebraical problem (3). But in general case,
when the problem of computations of coefficients of reduced algebraical
system (3) is not solved explicitly as in the quadratic case, which we
shall consider below, we have also parametrization of solution (4) by
solution of corresponding problems, which appear when we need to calculate
coefficients of (3).
As we shall see, these problems may be explicitly solved in wavelet approach.
Next we consider the construction of explicit time
solution for our problem. The obtained solutions are given
in the form (4),
where in our first case we have
$X_k(t)=Q_k(t)$, where $Q_k(t)$ are shifted Legendre
polynomials and $\lambda_k^i$ are roots of reduced
quadratic system of equations. In wavelet case $X_k(t)$
correspond to multiresolution expansions in the base of
compactly supported wavelets and $\lambda_k^i$ are the roots of
corresponding general polynomial system (3) with coefficients, which
are given by FWT, SSS or CC constructions. According to the
variational method to give the reduction from
differential to algebraical system of equations we need compute
the objects $\gamma ^j_a$ and $\mu_{ji}$,
which 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\\
\mu_{ji}&\equiv&\int
X'_i(\tau)X_j(\tau)d\tau=\sigma_jF_1(i,0)+F_1(i,j),\nonumber\\
& &F_1(r,s)=[1-(-1)^{r+s}]\hat{s}(r-s-1),\nonumber\\
& &\hat{s}(p)=\left\{ \begin{array}{ll}
1, & \quad p\geq 0\\
0, & \quad p < 0
\end{array}
\right. \nonumber \\
\beta_{klj}&\equiv&\int^1_0X_k(\tau)X_l(\tau)X_j(\tau)
d\tau=\sigma_k\sigma_l\sigma_j+ \nonumber\\
& & \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=\nonumber\\
& & \frac{1}{ (j+k+l+1)R\bigl(1/2(i+j+k)\bigr)}\times\nonumber\\
& & R\bigl(1/2(j+k-l)\bigr)
R\bigl(1/2(j-k+l)\bigr)\times\nonumber\\
& &R\bigl(1/2(-j+k+l)\bigr),\nonumber
\end{eqnarray}
if $j+k+l=2m , m\in{\it Z}$,and
$\alpha_{klj}=0$ if $ j+k+l=2m+1 $;
where
$ R(i)=(2i)!/(2^ii!)^2$, $Q_i=\sigma_i+P_i^\ast$, where the
second equality in the formulae for $\sigma, \nu, \mu, \beta,\alpha$
hold for the first case.
Now we give construction for
computations of objects(5) 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$,
$\overline{\displaystyle\bigcup_{j\in{\bf Z}}}V_j=L^2({\bf R})$,
$ f(x)\in V_j <=> f(2x)\in V_{j+1}
$
%\end{eqnarray*}
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 $L^2({\bf R})$. As usually $\varphi(x)$ is
a scaling function, $\psi(x)$ is a wavelet function, where
$\varphi_i(x)=\varphi(x-i)$. Scaling relation 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
$$
\varphi_{jl}(x)=2^{j/2}\varphi(2^j x-\ell),
\psi_{jk}(x)=2^{j/2}\psi(2^j x-k)
$$
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 are $W_j$ spanned by translations and dilation of the mother wavelet
$\psi_{jk}(x)$.
If in formulae (6) $c_{jk}=0$ for $j\geq J$, then $f(x)$ has an alternative
expansion in terms of dilated scaling functions only
$
f(x)=\sum\limits_{\ell\in {\bf Z}}c_{J\ell}\varphi_{J\ell}(x)
$.
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
$
\int x^k \psi(x)d(x)=0$, or equivalently
$x^k=\sum c_\ell\varphi_\ell(x)$ for each $k$,
$0\leq k\leq K$.
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
$\varphi(x)$
\begin{eqnarray*}
\phi_\ell^d(x)=\sum\limits_m\lambda_m\varphi_m(x),\quad
\lambda_m=\int\limits_{-\infty}^{\infty}\varphi_\ell^d(x)\varphi_m(x)dx
\end{eqnarray*}
The coefficients $\lambda_m$ are 2-term connection
coefficients. In general we need to find
\begin{eqnarray}
\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 Riccati case we need to evaluate two and three
connection coefficients
\begin{eqnarray*}
\Lambda_\ell^{d_1
d_2}&=&\int^\infty_{-\infty}\varphi^{d_1}(x)\varphi_\ell^{d_2}(x)dx,
\quad d_i\geq 0, \\
\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 CC method [7] 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}
\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:
\begin{eqnarray*}
A\Lambda^{d_1 d_2}&=&2^{1-d}\Lambda^{d_1 d_2},\quad d=d_1+d_2,\\
A_{\ell,q}&=&\sum\limits_p a_p a_{q-2\ell+p}
\end{eqnarray*}
For nonquadratic case we have analogously additional linear problems for
objects (7).
Also, we use FWT and SSS for computing coefficients of reduced
algebraic systems.
We use for modelling D6,D8,D10 functions and programs RADAU and
DOPRI for testing.
As a result we obtained the explicit time solution (4) of our
problem. In comparison with
wavelet expansion
on the real line which we use now and in
calculation of Galerkin approximation, Melnikov function approach, etc also
we need to use periodized wavelet expansion, i.e. wavelet expansion
on finite interval. 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
$2^{-j}k$.
The solution of the first problem consists in periodizing. In this case
we use expansion into periodized wavelets
defined by
$
\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.
Extendeed version and related results may be found in [1]-[6].
This research was supported in part by the National Science Foundation under
Grant No.~PHY94-07194.
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\end{document}