Picard Iteration์œผ๋กœ ํ•ด์˜ ์กด์žฌ์„ฑ ์ฆ๋ช…ํ•˜๊ธฐ

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๋ณต์ˆ˜์ „๊ณตํ•˜๊ณ  ์žˆ๋Š” ์ˆ˜ํ•™๊ณผ์˜ ํ•™๋ถ€ ์กธ์—…์‹œํ—˜์— ๋ฏธ๋ถ„๋ฐฉ์ •์‹์ด ์žˆ๋Š” ์ค„ ์•Œ๊ณ , ์‹œํ—˜ ์ค€๋น„๋„ ํ•  ๊ฒธ ๋ณตํ•™ํ•  ๋•Œ โ€œ์ƒ๋ฏธ๋ถ„๋ฐฉ์ •์‹โ€ ๊ณผ๋ชฉ์„ ์‹ ์ฒญํ–ˆ์Šต๋‹ˆ๋‹ค. ๋‚˜์ค‘์— ์•Œ๊ณ ๋ณด๋‹ˆ ๋ฏธ๋ถ„๋ฐฉ์ •์‹์€ ์กธ์—…์‹œํ—˜ ๊ณผ๋ชฉ์ด ์•„๋‹ˆ์—ˆ์Šต๋‹ˆ๋‹คโ€ฆ OTLโ€ฆ ๊ทธ๋ž˜๋„ ์ด์™• ์‹œ์ž‘ํ•œ ๊ฒƒ ํฌ๊ธฐ๋ž€ ์—†์Šต๋‹ˆ๋‹ค!! ๐Ÿ’ช ์œผ๋ž์ฐจ!! ์ƒ๋ฏธ๋ถ„๋ฐฉ์ •์‹ ํฌ์ŠคํŠธ ์ „์ฒด ๋ณด๊ธฐ

๋“ค์–ด๊ฐ€๋ฉฐ

๊ฒฝ๊ณ ํ•˜๋Š”๋ฐ ์—ฌ๊ธฐ์„œ๋ถ€ํ„ฐ ์ง„์งœ ์™„์ „ํžˆ ์ƒˆ๋กœ์šด ๋‚ด์šฉ์ž…๋‹ˆ๋‹คโ€ฆ;; ์ง€๊ธˆ๊นŒ์ง€๋Š” ๋ฏธ๋ถ„๋ฐฉ์ •์‹์˜ ์‹ฌํ™” ๋ฒ„์ „์„ ํ•˜๋Š” ๋Š๋‚Œ์ด์—ˆ๋‹ค๋ฉด, ์—ฌ๊ธฐ์„œ๋ถ€ํ„ฐ ์ง„์งœ MATH4xx ๊ณผ๋ชฉ์˜ ์œ„์—„์ด ๋ญ”์ง€ ์ž‘์‚ด๋‚˜๊ฒŒ ๋Š๋‚„ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค ใ…‹ใ…‹

์ด ์ฑ•ํ„ฐ์˜ ๋ชฉํ‘œ๋Š” ODE์˜ solution์ด ์กด์žฌ(Existence)ํ•˜๊ณ  ๊ทธ๋ฆฌ๊ณ  ์œ ์ผ(Uniqueness)ํ•˜๋‹ค๋Š” ๊ฒƒ์„ ๋ณด์ด๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ๊ทธ๋Ÿฐ๋ฐ ์ €๋Š” ๊ฐ์ž(๐Ÿฅ”)๋‹ˆ๊นŒ ๊ทธ ์ฃผ๋ณ€ ๊ณ๋‹ค๋ฆฌ๋ถ€ํ„ฐ ๋‹ค๊ฐ€๊ฐ€๋ณด๋„๋ก ํ•˜๊ฒ ์Šต๋‹ˆ๋‹ค.

[Existence and Uniqueness์˜ ๊ณ๋‹ค๋ฆฌ๋“ค]

์ˆœ์„œ๋Š” ์ƒ๊ด€์—†์Šต๋‹ˆ๋‹ค.

The Existence and Uniqueness Theorem

Consider the initial value problem

\[X' = F(X), \quad X(0) = X_0\]

where $X_0 \in \mathbb{R}^n$. Supp. that $F: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is $C^1$.

Then there exists a โ€œuniqueโ€ solution of this initial value problem. More precisely, there exists $a > 0$ and a unique solution

\[X: (-a, a) \rightarrow \mathbb{R}^n\]

of this differential equation satisfying the initial condition $X(0) = X_0$.

์ด๋•Œ, $C^1$์€ โ€œContinuously Differentiable Functionโ€์ž…๋‹ˆ๋‹ค. ๊ทธ๋ฆฌ๊ณ  $F(X)$๋Š” ๋ฒกํ„ฐ ํ•„๋“œ๋กœ

\[F(X) = (f_1, (x_1, ..., x_n), ..., f_n(x_1, ..., x_n))\]

์„ ๋งŒ์กฑํ•ฉ๋‹ˆ๋‹ค.

Road to the theorem

์šฐ๋ฆฌ์˜ ๋ชฉํ‘œ๋Š” ์œ„์˜ ์ •๋ฆฌ๋ฅผ ์ดํ•ดํ•˜๊ณ , ์ฆ๋ช…ํ•ด๋ณด๋Š” ๊ฒƒ์ž…๋‹ˆ๋‹ค. ๋‚ด์šฉ์ด ์–ด๋ ค์šธ ์ˆ˜๋„ ์žˆ๊ฒ ์ง€๋งŒ, ํฌ๊ธฐํ•˜์ง€ ์•Š๊ณ  ์ „์ง„ํ•ด๋ด…์‹œ๋‹ค! ๐Ÿƒโ€โ™‚๏ธโ€โžก๏ธ ๋‚ด์šฉ์„ ๋‹ค ์ดํ•ดํ•˜์ง€ ๋ชป ํ•ด๋„ ๊ดœ์ฐฎ๋‹ค!! (๋‚˜์—๊ฒŒ ํ•˜๋Š” ๋ง ใ…‹ใ…‹)

Continuous Differential Functions are Locally Lipschitz

Supp. that the function $F: \Omega \rightarrow \mathbb{R}^n$ is $C^1$.

Then $F$ is locally Lipschitz.

* ์ด๋•Œ, ํ•จ์ˆ˜์˜ ์ •์˜์—ญ $\Omega$๋Š” ์ฝคํŒฉํŠธ ์ง‘ํ•ฉ์ด๋‹ค.

Let $x_0 \in \Omega$ and let $B_{\epsilon} := \left\{ x: | x - x_0 | \le \epsilon \right\}$ with small $\epsilon > 0$ s.t. $B_{\epsilon} \subset \Omega$.

$B_{\epsilon}$์ด convex ํ•˜๋ฏ€๋กœ, $B_{\epsilon}$ ์•ˆ์— ์กด์žฌํ•˜๋Š” ๋‘ ์  $Y$, $Z$๋ฅผ ์—ฐ๊ฒฐํ•˜๋Š” straight line ์—ญ์‹œ $B_{\epsilon}$ ์•ˆ์— ์กด์žฌํ•˜๊ฒŒ ๋ฉ๋‹ˆ๋‹ค. ๊ทธ๋ฆฌ๊ณ  ๊ทธ line๋Š” ์•„๋ž˜์™€ ๊ฐ™์ด ์ •์˜๋ฉ๋‹ˆ๋‹ค.

\[Y + s U \in B_{\epsilon}\]

(์ด๋•Œ, $U$๋Š” ๋ฐฉํ–ฅ ๋ฒกํ„ฐ๋กœ $U = Z - Y$, ๊ทธ๋ฆฌ๊ณ  $s$๋Š” $0 \le s \le 1$๋ผ๋Š” ๋งค๊ฐœ๋ณ€์ˆ˜.)

์ด straight line ์œ„์—์„œ์˜ ํ•จ์ˆซ๊ฐ’์„ $\psi(s) = F(Y + sU)$๋ผ๊ณ  ์ •์˜ํ•ด๋ด…์‹œ๋‹ค. (๋ฏธ์ 2์˜ ์„ ์ ๋ถ„(Line integral)์ด ๋– ์˜ค๋ฅด๋„ค์š”) ์ด๊ฑธ ๋ฏธ๋ถ„ํ•˜๋ฉด Differential on Vector Field์—์„œ ํ–ˆ๋˜๋Œ€๋กœ

\[\psi'(s) = DF_{Y+sU}(U)\]

๊ฐ€ ๋ฉ๋‹ˆ๋‹ค. ํ‘œ๊ธฐ๊ฐ€ ํ•ญ์ƒ ํ—ท๊ฐˆ๋ฆฌ๋Š”๋ฐ, $DF_{Y+sU}$๋Š” ์  $Y+sU$ ์œ„์—์„œ์˜ Jacobian์„ ๋งํ•˜๊ณ , $U$๋Š” ์†๋ฏธ๋ถ„์— ์˜ํ•ด ๋ฐ–์œผ๋กœ ๋‚˜์˜จ ๋ฒกํ„ฐ์ž…๋‹ˆ๋‹ค. ์ด ๋‘˜์„ ๋‚ด์ ํ•œ ๊ฒƒ์ด $\psiโ€™(s)$ ์ž…๋‹ˆ๋‹ค.

์ด์ œ ์ฒ˜์Œ์— ์žก์•˜๋˜ ๋‘ ์  $Y, Z \in B_{\epsilon}$์— ๋Œ€ํ•œ ๋‘ ํ•จ์ˆ˜๊ฐ’์˜ ์ฐจ์ด์ธ $F(Z) - F(Y)$๋ฅผ ํ™•์ธํ•ด๋ด…์‹œ๋‹ค.

\[\begin{aligned} F(Z) - F(Y) &= \psi(1) - \psi(0) \\ &= \int_0^1 \psi'(s) \, ds \\ &= \int_0^1 DF_{Y+sU}(U) \, ds \end{aligned}\]

์ด๋•Œ, ์ง‘ํ•ฉ $B_{\epsilon}$๊ฐ€ compact ํ•˜๋ฏ€๋กœ, ๊ทธ ์ •์˜์—ญ ์•ˆ์—์„œ ํ•จ์ˆ˜๊ฐ’ $F(X)$์€ Minimum๊ณผ Maximum์ด ์กด์žฌํ•ฉ๋‹ˆ๋‹ค. ๊ทธ๋ฆฌ๊ณ  $B_{\epsilon}$์— ๋Œ€ํ•œ Jacobian์˜ ๋…ธ๋ฆ„ $K$๋ฅผ ์ •์˜ํ•ฉ๋‹ˆ๋‹ค: $K = \sup_{x \in B_{\epsilon}} | DF_x | < + \infty$.

์ด์ œ ์•„๋ž˜์™€ ๊ฐ™์€ ๋ถ€๋“ฑ์‹์„ ๋งŒ์กฑํ•ฉ๋‹ˆ๋‹ค.

\[\|F(Z) - F(Y)\| = \left\| \int_0^1 DF_{Y+sU}(U) \, ds \right\| \le \int_0^1 K \| U \| \, ds = K \| Z - Y \|\]

๋ถ€๋“ฑ์‹์„ ์ž˜ ์ •๋ฆฌํ•˜๋ฉด, ์•„๋ž˜์™€ ๊ฐ™์ด Lipschitz ๋ถ€๋“ฑ์‹์— ๋Œ€ํ•œ ๊ฒฐ๊ณผ๋ฅผ ์–ป๊ฒŒ ๋ฉ๋‹ˆ๋‹ค.

\[\frac{\| F(Z) - F(Y) \|}{\| Z - Y \|} \le K\]

$\blacksquare$

Integral form of the differential equation

์šฐ๋ฆฌ๊ฐ€ ์ง€๊ธˆ๊นŒ์ง€ ์‚ดํŽด๋ณธ ๋ฏธ๋ถ„๋ฐฉ์ •์‹์€ $Xโ€™ = F(X)$์˜ ๊ผด์ด์—ˆ์Šต๋‹ˆ๋‹ค. ๊ทธ๋Ÿฐ๋ฐ, ์ด๊ฒƒ์„ ๋งค๊ฐœ๋ณ€์ˆ˜ํ™” ํ•˜๊ณ , ์ ๋ถ„ ๊ผด๋กœ ๋ฐ”๊พธ๋ฉด ๋ฏธ๋ถ„๋ฐฉ์ •์‹๊ณผ ๋™์น˜์ธ ์ ๋ถ„๋ฐฉ์ •์‹์„ ์–ป์„ ์ˆ˜ ์žˆ์Šต๋‹ˆ๋‹ค.

์ž…๋ ฅ ๋ณ€์ˆ˜ $X$๋ฅผ $t$๋กœ ๋งค๊ฐœ๋ณ€์ˆ˜ํ™” ํ•˜๋ฉด, $Xโ€™(t) = F(X(t))$๋ฅผ ๋งŒ์กฑํ•˜๊ฒŒ ๋ฉ๋‹ˆ๋‹ค. $X(0) = X_0$. ๊ทธ๋ฆฌ๊ณ  ์ด๊ฒƒ์„ ์ ๋ถ„๊ผด๋กœ ๋ฐ”๊พธ๋ฉด

\[X(t) = X_0 + \int_0^t F(X(s)) \, ds\]

ํ•ด์˜ ์กด์žฌ์„ฑ ์ฆ๋ช…์„ ์œ„ํ•ด ์šฐ๋ฆฌ๋Š” ์š” ์ ๋ถ„๊ผด์„ ์‚ฌ์šฉํ•  ์˜ˆ์ •์ž…๋‹ˆ๋‹ค.

Assumptions to prove

์กด์žฌ์„ฑ ์ฆ๋ช…์„ ์œ„ํ•ด ์•„๋ž˜์™€ ๊ฐ™์ด ์„ธํŒ…ํ•ฉ๋‹ˆ๋‹ค.

  • ์ดˆ๊ธฐ๊ฐ’ $X_0$๋ฅผ ์ค‘์‹ฌ์œผ๋กœํ•˜๊ณ , ๋ฐ˜์ง€๋ฆ„ $\rho > 0$์ธ closed ball $O_\rho$๋ฅผ ์ •์˜ํ•จ.
  • ๋ฒกํ„ฐ ํ•„๋“œ $F(X)$๊ฐ€ $O_\rho$ ์•ˆ์— ๋Œ€ํ•ด Lipschitz Constant $K$๋ฅผ ๊ฐ€์ง.
  • ๋ฒกํ„ฐ ํ•„๋“œ $F(X)$์˜ ์ƒํ•œ์ด $O_\rho$ ์•ˆ์— ์กด์žฌํ•˜๋Š”๋ฐ, ์ด๋ฅผ $M$์ด๋ผ๊ณ  ํ•จ.
  • ๊ตฌ๊ฐ„ $J = [-a, a]$๋ฅผ ์ •์˜ํ•˜๋Š”๋ฐ, $a$๋Š” $a < \min \{ \rho/M, 1/K \}$์—ฌ์•ผ ํ•จ.

Function Sequence

$J = [-a, a]$ ๋ฒ”์œ„ ์•ˆ์—์„œ ํ•จ์ˆ˜์—ด $\left\{U_0, U_1, โ€ฆ\right\}$๋ฅผ ์ •์˜ํ•ฉ๋‹ˆ๋‹ค. ์ด๊ฒƒ์€ Picard Iteration์— ์˜ํ•ด ์ •์˜๋˜๋Š” ํ•จ์ˆ˜์—ด์ž…๋‹ˆ๋‹ค.

์ดˆ๊ธฐ์—” $U_0(t) = X_0$์ž…๋‹ˆ๋‹ค. Iteration์„ ํ•œ๋ฒˆ ๋Œ๋ฉด,

\[U_1(t) = X_0 + \int_0^t F(U_0(s)) \, ds = X_0 + t F(X_0)\]

๋กœ ์ •์˜๋ฉ๋‹ˆ๋‹ค. ์ด๋•Œ, $U_k(t)$๊ฐ€ ๋‹ค์‹œ $O_{\rho}$์— ์†ํ•˜๋Š”์ง€ ํ™•์ธํ•ด๋ด…์‹œ๋‹ค. ๋ชจ๋“  $U_k(t) \in O_\rho$๋ฅผ ๋งŒ์กฑํ•ด์•ผ ๊ฐ™์€ ์กฐ๊ฑด ์œ„์—์„œ Iteration์„ ๊ณ„์†ํ•  ์ˆ˜ ์žˆ๊ธฐ ๋•Œ๋ฌธ์ž…๋‹ˆ๋‹ค.

\[\begin{aligned} U_1(t) &= X_0 + t F(X_0) \\ \| U_1(t) - X_0 \| &= \| t \| \cdot \| F(X_0) \| \le a \cdot M < \rho \end{aligned}\]

์ด๊ฒƒ์€ $U_1(t)$๊ฐ $X_0$๋ฅผ ์ค‘์‹ฌ์œผ๋กœ ํ•˜๋Š” ๋‹ซํžŒ ์› $O_{\rho}$์— ์†ํ•œ๋‹ค๋Š” ๊ฒƒ์„ ๋งํ•ฉ๋‹ˆ๋‹ค: $U_1(t) \in O_{\rho}$.

Picard Iteration ๋ฐฉ์‹์— ๋”ฐ๋ผ ์ด ๊ณผ์ •์„ ๋ฐ˜๋ณตํ•˜๋ฉด, ๊ท€๋‚ฉ๋ฒ•์— ์˜ํ•ด $U_k(t)$๋Š” ์•„๋ž˜์™€ ๊ฐ™์ด ์ •์˜๋˜๊ณ 

\[U_{k+1}(t) = X_0 + \int_0^t F(U_{k}(s)) \, ds\]

๊ฐ $U_k(t)$๋Š” $O_{\rho}$์— ํฌํ•จ๋˜๊ฒŒ ๋ฉ๋‹ˆ๋‹ค. ($U_1(t)$์—์„œ ํ–ˆ๋˜ ๋ฐฉ์‹๋Œ€๋กœ ์ •๋ฆฌํ•˜๋ฉด ๋ฉ๋‹ˆ๋‹ค.)

๋”ฐ๋ผ์„œ, ํ•จ์ˆ˜์—ด $\left\{ U_k (t) \right\}$๋Š” $J = [-a, a]$ ์œ„์—์„œ well-defined ์ž…๋‹ˆ๋‹ค.

Convergence of Function Sequence

์œ„์˜ ๊ณผ์ •์—์„œ ํ•จ์ˆ˜์—ด $\left\{ U_k (t) \right\}$์ด well-defined์ธ ๊ฒƒ์„ ํ™•์ธ ํ–ˆ์Šต๋‹ˆ๋‹ค. ์ด์ œ $U_{k}(t)$๊ฐ€ solution์ธ $X(t)$์— ์ˆ˜๋ ดํ•œ๋‹ค๋Š” ๊ฒƒ์„ ๋ณด์—ฌ์•ผ ํ•ฉ๋‹ˆ๋‹ค. ํ•จ์ˆ˜์—ด์ด ์ˆ˜๋ ดํ•˜๋Š”์ง€ ๋ณด์ด๊ธฐ ์œ„ํ•ด ์•„๋ž˜์˜ ๊ฒƒ์„ ๋ณด์—ฌ์•ผ ํ•˜๋Š”๋ฐ

\[\lim_{k\rightarrow\infty}\| U_{k+1}(t) - U_{k}(t) \| < + \infty\]

References

https://youtu.be/Zxr6Wekwxh0?si=k3uo7A_srkM8Us7R

https://people.math.wisc.edu/~aseeger/319/notes2.pdf ^์ฝ์–ด๋ด์•ผ ํ•จ