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collocation_method.lyx
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2302 lines (1651 loc) · 38.1 KB
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
\lyxformat 544
\begin_document
\begin_header
\textclass optbook
\end_header
\begin_body
\begin_layout Section
Collocation Method
\end_layout
\begin_layout Standard
From the definition of (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:def_T_d_operator"
plural "false"
caps "false"
noprefix "false"
\end_inset
), we note that
\begin_inset Formula $\mathcal{T}_{\varphi}(x)$
\end_inset
depends only on
\begin_inset Formula $x(t)$
\end_inset
at
\begin_inset Formula $t=c_{j}$
\end_inset
.
Therefore, we can only need to calculate
\begin_inset Formula $\mathcal{T}_{\varphi}(x)(t)$
\end_inset
at
\begin_inset Formula $t=c_{j}$
\end_inset
.
To simplify the notation, for any
\begin_inset Formula $x\in\mathcal{C}([0,T],\R^{d})$
\end_inset
, we define a corresponding matrix
\begin_inset Formula $[x]\in\R^{d\times D}$
\end_inset
by
\begin_inset Formula $[x]_{i,j}=x_{i}(c_{j})$
\end_inset
.
For any
\begin_inset Formula $d\times D$
\end_inset
matrix
\begin_inset Formula $X$
\end_inset
, we define
\begin_inset Formula $F(X,c)$
\end_inset
as an
\begin_inset Formula $d\times D$
\end_inset
matrix
\begin_inset Formula
\begin{equation}
F(X,c)_{i,j}=F(X_{*,j},c_{j})_{i}.\label{eq:bF}
\end{equation}
\end_inset
where
\begin_inset Formula $X_{*,j}$
\end_inset
is the
\begin_inset Formula $j$
\end_inset
-th column of
\begin_inset Formula $X$
\end_inset
.
We also define
\begin_inset Formula $A_{\varphi}$
\end_inset
as a
\begin_inset Formula $D\times D$
\end_inset
matrix
\begin_inset Formula
\begin{equation}
(A_{\varphi})_{i,j}=\int_{0}^{c_{j}}\varphi_{i}(s)\d s.\label{eq:Aphi}
\end{equation}
\end_inset
Note that
\begin_inset Formula $A_{\phi}$
\end_inset
can be pre-computed.
By inspecting the definition of (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:def_T_d_operator"
plural "false"
caps "false"
noprefix "false"
\end_inset
), (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:bF"
plural "false"
caps "false"
noprefix "false"
\end_inset
) and (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:Aphi"
plural "false"
caps "false"
noprefix "false"
\end_inset
), we have that
\begin_inset Formula
\[
[\mathcal{T}_{\varphi}(x)]=v\cdot1_{D}^{\top}+F([x],c)A_{\varphi}
\]
\end_inset
where
\begin_inset Formula $1_{D}$
\end_inset
is a column of all 1 vector of length
\begin_inset Formula $D$
\end_inset
.
Hence, we can apply the map
\begin_inset Formula $\mathcal{T}_{\varphi}$
\end_inset
by simply multiply
\begin_inset Formula $F([x],c)$
\end_inset
by a pre-compute
\begin_inset Formula $D\times D$
\end_inset
matrix
\begin_inset Formula $A_{\varphi}$
\end_inset
.
For the basis we consider here, each iteration takes only
\begin_inset Formula $\tilde{O}(dD)$
\end_inset
which is nearly linear in the size of our representation of the solution.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{algorithm2e}[H]
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
caption{
\end_layout
\end_inset
\begin_inset Formula $\mathtt{CollocationMethod\}}$
\end_inset
\begin_inset ERT
status open
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
SetAlgoLined
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Input:
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $F,v,T,\text{\ensuremath{\varphi}},c$
\end_inset
\family default
\series default
\shape default
\size default
\emph default
\bar default
\strikeout default
\xout default
\uuline default
\uwave default
\noun default
\color inherit
.
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $N=\left\lceil \log\left(\frac{T}{\epsilon}\max_{s\in[0,T]}\left\Vert F(v,s)\right\Vert \right)\right\rceil $
\end_inset
.
Let
\begin_inset Formula $(A_{\varphi})_{i,j}=\int_{0}^{c_{j}}\varphi_{i}(s)\d s.$
\end_inset
\end_layout
\begin_layout Standard
\family roman
\series medium
\shape up
\size normal
\emph off
\bar no
\strikeout off
\xout off
\uuline off
\uwave off
\noun off
\color none
\begin_inset Formula $X^{(0)}\leftarrow v\cdot1_{D}^{\top}.$
\end_inset
\end_layout
\begin_layout Standard
For
\begin_inset Formula $j=1,2,\cdots,N-1$
\end_inset
:
\begin_inset Formula
\[
X^{(j)}\leftarrow v\cdot1_{D}^{\top}+F(X^{(j-1)},c)A_{\varphi}.
\]
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Formula $x^{(N)}(t)\leftarrow v+\int_{0}^{t}\sum_{i=1}^{D}F(X_{*,i}^{(N)},c_{i})\varphi_{i}(s)\mathrm{d}s$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
Return
\end_layout
\end_inset
\begin_inset Formula $x^{(N)}$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
end{algorithm2e}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
We state our guarantee for a first-order ODE.
It can be extended to a
\begin_inset Formula $k$
\end_inset
'th order ODE.
\end_layout
\begin_layout Theorem
\begin_inset CommandInset label
LatexCommand label
name "thm:first_order_ode"
\end_inset
Let
\begin_inset Formula $x^{*}(t)$
\end_inset
be the solution of an
\begin_inset Formula $d$
\end_inset
dimensional ODE
\begin_inset Formula
\begin{align*}
x(0)=v,\frac{\d x(t)}{\d t}=F(x(t),t)\quad\text{for all \ensuremath{0\leq t\leq T}}.
\end{align*}
\end_inset
We are given a
\begin_inset Formula $D$
\end_inset
dimensional subspace
\begin_inset Formula $\mathcal{V}\subset\mathcal{C}([0,T],\R)$
\end_inset
, node points
\begin_inset Formula $\{c_{j}\}_{j=1}^{D}\subset[0,T]$
\end_inset
and a
\begin_inset Formula $\gamma_{\varphi}$
\end_inset
bounded basis
\begin_inset Formula $\{\varphi_{j}\}_{j=1}^{D}\subset\mathcal{V}$
\end_inset
(Definition
\begin_inset CommandInset ref
LatexCommand ref
reference "def:basis"
plural "false"
caps "false"
noprefix "false"
\end_inset
).
Let
\begin_inset Formula $L$
\end_inset
and
\begin_inset Formula $\epsilon>0$
\end_inset
be such that there exists a function
\begin_inset Formula $q\in\mathcal{V}$
\end_inset
such that
\begin_inset Formula
\begin{align*}
\left\Vert q(t)-\frac{\d}{\d t}x^{*}(t)\right\Vert \leq\frac{\epsilon}{T},\forall t\in[0,T]
\end{align*}
\end_inset
and for any
\begin_inset Formula $y,z\in\R^{d}$
\end_inset
,
\begin_inset Formula
\begin{align*}
\|F(y,t)-F(z,t)\|\leq L\|y-z\|,\forall t\in[0,T].
\end{align*}
\end_inset
Assume
\begin_inset Formula $\gamma_{\varphi}LT\leq1/2$
\end_inset
.
Then Algorithm
\begin_inset Formula $\textsc{CollocationMethod}$
\end_inset
outputs a function
\begin_inset Formula $x^{(N)}\in\mathcal{V}$
\end_inset
such that
\begin_inset Formula
\begin{align*}
\max_{t\in[0,T]}\|x^{(N)}(t)-x^{*}(t)\|\leq20\gamma_{\varphi}\epsilon
\end{align*}
\end_inset
using
\begin_inset Formula $O\left(D\log\left(\frac{T}{\epsilon}\max_{s\in[0,T]}\left\Vert F(v,s)\right\Vert \right)\right)$
\end_inset
evaluations of
\begin_inset Formula $F$
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status collapsed
\begin_layout Plain Layout
Next we state the general result for a
\begin_inset Formula $k$
\end_inset
-th order ODE.
We prove this via a reduction from higher order ODE to first-order ODE.
See the proof in Appendix
\begin_inset space ~
\end_inset
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:app_ode"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\end_layout
\begin_layout Plain Layout
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
begin{theorem}[$k$-th order ODE]
\end_layout
\end_inset
\begin_inset CommandInset label
LatexCommand label
name "thm:kth_order_ode"
\end_inset
Let
\begin_inset Formula $x^{*}(t)\in\R^{d}$
\end_inset
be the solution of the ODE
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
%
\backslash
eqref{eq:kth_order_ode}.
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\begin_inset Formula
\begin{align*}
%\label{eq:kth_{o}rder_{o}de}
\frac{\d^{k}}{\d t^{k}}x(t)= & ~F\left(\frac{\d^{k-1}}{\d t^{k-1}}x(t),\cdots,x(t),t\right)\\
\frac{\d^{i}}{\d t^{i}}x(0)= & ~v_{i},\forall i\in\{k-1,\cdots,1,0\}.
\end{align*}
\end_inset
where
\begin_inset Formula $F:\R^{kd+1}\rightarrow\R^{d}$
\end_inset
,
\begin_inset Formula $x(t)\in\R^{d}$
\end_inset
, and
\begin_inset Formula $v_{0},v_{1},\cdots,v_{k-1}\in\R^{d}$
\end_inset
.
\end_layout
\begin_layout Plain Layout
We are given a
\begin_inset Formula $D$
\end_inset
dimensional subspace
\begin_inset Formula $\mathcal{V}\subset\mathcal{C}([0,T],\R)$
\end_inset
, node points
\begin_inset Formula $\{c_{j}\}_{j=1}^{D}\subset[0,T]$
\end_inset
and a
\begin_inset Formula $\gamma_{\varphi}$
\end_inset
bounded basis
\begin_inset Formula $\{\varphi_{j}\}_{j=1}^{D}\subset\mathcal{V}$
\end_inset
(Definition
\begin_inset CommandInset ref
LatexCommand ref
reference "def:basis"
plural "false"
caps "false"
noprefix "false"
\end_inset
).
Given some
\begin_inset Formula $L$
\end_inset
and
\begin_inset Formula $\epsilon>0$
\end_inset
such that
\end_layout
\begin_layout Plain Layout
1.
For
\begin_inset Formula $i\in[k]$
\end_inset
, there exists a function
\begin_inset Formula $q^{(i)}\in\mathcal{V}$
\end_inset
such that
\begin_inset Formula
\begin{align*}
\left\Vert q^{(i)}(t)-\frac{\d^{i}}{\d t^{i}}x^{*}(t)\right\Vert \leq\frac{\epsilon}{T^{i}},\forall t\in[0,T].
\end{align*}
\end_inset
\end_layout
\begin_layout Plain Layout
2.
For any
\begin_inset Formula $y,z\in\R^{kd}$
\end_inset
,
\begin_inset Formula
\begin{align*}
\|F(y,t)-F(z,t)\|\leq\sum_{i=1}^{k}L_{i}\|y_{i}(t)-z_{i}(t)\|,\forall t\in[0,T].
\end{align*}
\end_inset
\end_layout
\begin_layout Plain Layout
Assume
\begin_inset Formula $\gamma_{\varphi}LT\leq1/8$
\end_inset
with
\begin_inset Formula $L=\sum_{i=1}^{k}L_{i}^{1/i}$
\end_inset
.
Then, we can find functions
\begin_inset Formula $\{q^{(i)}\}_{i\in\{0,1,\cdots,k-1\}}\subset\mathcal{V}$
\end_inset
such that
\begin_inset Formula
\begin{align*}
\max_{t\in[0,T]}\left\Vert q^{(i)}(t)-\frac{\d^{i}}{\d t^{i}}x^{*}(t)\right\Vert _{p}=20(1+2k)\gamma_{\varphi}\frac{\epsilon}{T^{i}},\forall i\in\{0,1,\cdots,k-1\}.
\end{align*}
\end_inset
The algorithm takes
\begin_inset Formula $O(D\log(C/\epsilon))$
\end_inset
evaluations of
\begin_inset Formula $F$
\end_inset
where
\begin_inset Formula
\[
C=(4\gamma_{\varphi}T)^{k}\cdot\max_{s\in[0,T]}\left\Vert F(v_{k-1},v_{k-2},\cdots,v_{0},s)\right\Vert +\sum_{i=1}^{k-1}(4\gamma_{\varphi}T)^{i}\left\Vert v_{i}\right\Vert .
\]
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
end{theorem}
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
Note that the statement is a bit awkward.
Instead of finding a function whose derivatives are same as the derivatives
of
\begin_inset Formula $x^{*}$
\end_inset
, the algorithm approximates the derivatives of
\begin_inset Formula $x^{*}$
\end_inset
individually.
This is because we do not know if derivatives/integrals of functions in
\begin_inset Formula $\mathcal{V}$
\end_inset
remain in
\begin_inset Formula $\mathcal{V}$
\end_inset
.
For piece-wise polynomials, we can approximate the
\begin_inset Formula $j$
\end_inset
-th derivative of the solution by taking
\begin_inset Formula $(k-j)$
\end_inset
-th iterated integral of
\begin_inset Formula $q^{(k)}$
\end_inset
, which is still a piece-wise polynomial.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
%
\backslash
tat{Santosh: See if this explanation is clear.}
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
In section
\begin_inset CommandInset ref
LatexCommand ref
reference "sec:piecewise_poly"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we give a basis for piece-wise polynomials (Lemma
\begin_inset CommandInset ref
LatexCommand ref
reference "lem:basis_piecewise_poly"
plural "false"
caps "false"
noprefix "false"
\end_inset
).
Using this basis, we have the following Theorem.
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\begin_layout Plain Layout
\begin_inset ERT
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\begin_layout Plain Layout
\backslash
define
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{
\end_layout
\end_inset
thm:kthorderodepiecewise
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{
\end_layout
\end_inset
Theorem
\begin_inset ERT
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\begin_layout Plain Layout
}
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{
\end_layout
\end_inset
\family roman
(
\begin_inset Formula $k$
\end_inset
-th order ODE)
\family default
Let
\begin_inset Formula $x^{*}(t)\in\R^{d}$
\end_inset
be the solution of the ODE
\begin_inset Formula
\begin{align*}
\frac{\d^{k}}{\d t^{k}}x(t)= & ~F\left(\frac{\d^{k-1}}{\d t^{k-1}}x(t),\cdots,x(t),t\right)\\
\frac{\d^{i}}{\d t^{i}}x(0)= & ~v_{i},\forall i\in\{k-1,\cdots,1,0\}.
\end{align*}
\end_inset
where
\begin_inset Formula $F:\R^{kd+1}\rightarrow\R^{d}$
\end_inset
,
\begin_inset Formula $x(t)\in\R^{d}$
\end_inset
, and
\begin_inset Formula $v_{0},v_{1},\cdots,v_{k-1}\in\R^{d}$
\end_inset
.
Given some
\begin_inset Formula $L$
\end_inset
and
\begin_inset Formula $\epsilon>0$
\end_inset
such that
\end_layout
\begin_layout Plain Layout
1.
There exists a piece-wise polynomial
\begin_inset Formula $q(t)$
\end_inset
such that
\begin_inset Formula $q(t)$
\end_inset
on
\begin_inset Formula $[T_{j-1},T_{j}]$
\end_inset
is a degree
\begin_inset Formula $D_{j}$
\end_inset
polynomial with
\begin_inset Formula
\[
0=T_{0}<T_{1}<\cdots<T_{n}=T
\]
\end_inset
and that
\begin_inset Formula
\begin{align*}
\left\Vert q(t)-\frac{\d^{k}}{\d t^{k}}x^{*}(t)\right\Vert \leq\frac{\epsilon}{T^{k}},\forall t\in[0,T]
\end{align*}
\end_inset
\end_layout
\begin_layout Plain Layout
2.
The algorithm knows about the intervals
\begin_inset Formula $[T_{j-1},T_{j}]$
\end_inset
and the degree
\begin_inset Formula $D_{j}$
\end_inset
for all
\begin_inset Formula $j\in[n]$
\end_inset
.
\end_layout
\begin_layout Plain Layout
3.
For any
\begin_inset Formula $y,z\in\R^{kd}$
\end_inset
,
\begin_inset Formula
\begin{align*}
\|F(y,t)-F(z,t)\|\leq\sum_{i=1}^{k}L_{i}\|y_{i}-z_{i}\|,\forall t\in[0,T].
\end{align*}
\end_inset
Assume
\begin_inset Formula $LT\leq1/16000$
\end_inset
with
\begin_inset Formula $L=\sum_{i=1}^{k}L_{i}^{1/i}$
\end_inset
.
Then, we can find a piece-wise polynomial