rt t sst - - PowerPoint PPT Presentation

SLIDE 1

✶

▼❚▼❉✵✷ ❖♣❡r❛t✐♦♥❛❧ ❉❛t❛ ❆ss✐♠✐❧❛t✐♦♥ ❚❡❝❤♥✐q✉❡s

P❛rt ■✿ ❱❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ✭❘♦ss ❇❛♥♥✐st❡r✮ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ✭❛✮ ■♥✈❡rs❡ ♣r♦❜❧❡♠s ✭❜✮ ◆♦t❛t✐♦♥ ✭❝✮ ❍✐st♦r② ♦❢ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✐♥ ♠❡t❡♦r♦❧♦❣✐❝❛❧ ♦♣❡r❛✲ t✐♦♥s ❛♥❞ t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❝②❝❧❡ ✭❞✮ ❚❤❡ s❝❛❧❡✴❝❤❛❧❧❡♥❣❡s ♦❢ t❤❡ ♦♣❡r❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✷✳ ❱❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s ✭❱❆❘✮ ✭❛✮ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ✭❜✮ ❊rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s ✭❝✮ ❈♦st ❢✉♥❝t✐♦♥s ❛♥❞ s✐♠♣❧✐✜❝❛t✐♦♥s ❢♦r ♦♣❡r❛t✐♦♥❛❧ ❛s✲ s✐♠✐❧❛t✐♦♥ ✭❞✮ ❖♣t✐♠❛❧ ✐♥t❡r♣♦❧❛t✐♦♥ ❛♥❞ ♣❤②s✐❝❛❧ s♣❛❝❡ ❛♥❛❧②s✐s s②st❡♠s ✸✳ ❆✲♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ ❇✲♠❛tr✐① ✭❛✮ ❚❤❡ ♥✉❧❧ s♣❛❝❡ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦♣❡r❛t♦r ❛♥❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ❛✲♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥ ✭❜✮ ❚❤❡ r♦❧❡ ♦❢ t❤❡ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✭❝✮ ❙♣❛t✐❛❧ ❛s♣❡❝ts ✭✐♥✈❡rs❡ ▲❛♣❧❛❝✐❛♥s✱ ❞✐✛✉s✐♦♥ ♦♣❡r❛✲ t♦rs✮ ✭❞✮ ▼✉❧t✐✈❛r✐❛t❡ ❛s♣❡❝ts ❛♥❞ ❜❛❧❛♥❝❡ ✭❡✮ ❈♦♥tr♦❧ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s ❛♥❞ t❤❡ ✐♠♣❧✐❡❞ ❇✲♠❛tr✐① ✭❢✮ ❈♦♥❞✐t✐♦♥✐♥❣ ✹✳ ❖♣❡r❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s ✺✳ ▼❡❛s✉r✐♥❣ t❤❡ ❇✲♠❛tr✐① ✭❛✮ ❆♥❛❧②s✐s ♦❢ ✐♥♥♦✈❛t✐♦♥s ✭❜✮ ◆▼❈ ♠❡t❤♦❞ ✭❝✮ ▼♦♥t❡✲❈❛r❧♦ ✭❡♥s❡♠❜❧❡✮ ♠❡t❤♦❞ ✻✳ ❍②❜r✐❞ ✭✈❛r✴❡♥s❡♠❜❧❡✮ ❢♦r♠✉❧❛t✐♦♥s ✭❛✮ ❇❛s✐❝ ✐❞❡❛s ✭❜✮ ■♥❝♦r♣♦r❛t✐♥❣ ❛ s✐♠♣❧❡ ❤②❜r✐❞ s❝❤❡♠❡ ✐♥ ❱❆❘ ✭❝✮ ■♥❝♦r♣♦r❛t✐♥❣ ❛ ❧♦❝❛❧✐③❡❞ ❤②❜r✐❞ s❝❤❡♠❡ ✐♥ ❱❆❘ ✼✳ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❞✐❛❣♥♦st✐❝s ✭❛✮ ❚❤❡ ❇❡♥♥❡t✲❚❛❧❛❣r❛♥❞ t❤❡♦r❡♠ ✭❜✮ ❉❡sr♦③✐❡r ❞✐❛❣♥♦st✐❝s

SLIDE 2

✷

❋✉rt❤❡r r❡❛❞✐♥❣

✬ ✫ ✩ ✪

• ❇❡♥♥❡tt ❆✳❋✳✱ ✷✵✵✷✱ ■♥✈❡rs❡ ▼♦❞❡❧✐♥❣ ♦❢ t❤❡ ❖❝❡❛♥ ❛♥❞ ❆t♠♦s♣❤❡r❡

✭❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❛♥❞ r❡♣r❡s❡♥t❡rs ✲ s❡❝t✐♦♥s ✶✳✷✱ ✶✳✸✮✳

• ❉❛❧❡② ❘✳✱ ✶✾✾✶✱ ❆t♠♦s♣❤❡r✐❝ ❉❛t❛ ❆♥❛❧②s✐s ✭❤✐st♦r✐❝❛❧ ❛s♣❡❝ts ❛♥❞ ❜❛s✐❝

✐❞❡❛s ✲ ❝❤❛♣t❡rs ✶✱ ✶✸✮✳

• ❑❛❧♥❛② ❊✳✱ ✷✵✵✸✱ ❆t♠♦s♣❤❡r✐❝ ▼♦❞❡❧✐♥❣✱ ❉❛t❛ ❆ss✐♠✐❧❛t✐♦♥ ❛♥❞ Pr❡✲

❞✐❝t❛❜✐❧✐t② ✭❜❛s✐❝ ❛s♣❡❝ts ♦❢ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✲ ❝❤❛♣t❡r ✺✮✳

• ▲❡✇✐s ❏✳▼✳✱ ▲❛❦s❤♠✐✈❛r❛❤❛♥ ❙✳✱ ❉❤❛❧❧ ❙✳❑✳✱ ✷✵✵✻✱ ❉②♥❛♠✐❝ ❞❛t❛ ❛s✲

s✐♠✐❧❛t✐♦♥✿ ❛ ▲❡❛st ❙q✉❛r❡s ❆♣♣r♦❛❝❤ ✭❛♣♣❧✐❝❛t✐♦♥s ✲ ❝❤❛♣t❡rs ✸✱✹✱ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❛❧❣♦r✐t❤♠s ✲ ❝❤❛♣t❡r ✶✾✮✳

• ❙❝❤❧❛tt❡r ❚✳❲✳✱ ✷✵✵✵✱ ❱❛r✐❛t✐♦♥❛❧ ❆ss✐♠✐❧❛t✐♦♥ ♦❢ ▼❡t❡♦r♦❧♦❣✐❝❛❧ ❖❜s❡r✲

✈❛t✐♦♥s ✐♥ t❤❡ ▲♦✇❡r ❆t♠♦s♣❤❡r❡✿ ❛ ❚✉t♦r✐❛❧ ♦♥ ❍♦✇ ✐t ❲♦r❦s✱ ❏♦✉r♥❛❧ ♦❢ ❆t♠♦s♣❤❡r✐❝ ❛♥❞ ❙♦❧❛r✲❚❡rr❡str✐❛❧ P❤②s✐❝s ✻✷✱ ♣♣✳ ✶✵✺✼✲✶✵✼✵✳

• ▼❛t❤❡♠❛t✐❝s ❆✐❞❡ ▼❡♠♦✐r ❤❛♥❞♦✉t✳

◆♦t❡ t❤❛t ♣❛❣❡ ♥✉♠❜❡rs ♦♥ t❤❡ s❧✐❞❡s ❛♥❞ ♦♥ t❤❡ ❤❛♥❞♦✉ts ❞♦ ♥♦t ❛❧✇❛②s ♠❛t❝❤✳

SLIDE 3

✸

✶✳ ■♥tr♦❞✉❝t✐♦♥

✶✭❛✮ ■♥✈❡rs❡ ♣r♦❜❧❡♠s

❋✐❡❧❞ ❊①❛♠♣❧❡ ✐♥✈❡rs❡ ♣r♦❜❧❡♠ t♦ ❜❡ s♦❧✈❡❞ ▼❡❞✐❝❛❧ ❞✐❛❣♥♦s✐s ❲❤❛t ✐s t❤❡ ✸✲❉ str✉❝t✉r❡ ♦❢ ❜✐♦❧♦❣✐❝❛❧ t✐ss✉❡s ❢r♦♠ ❳✲r❛② ✐♠❛❣❡s ✭❈❆❚ s❝❛♥✮❄ ❙❡✐s♠♦❧♦❣② ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ s✉❜t❡rr❛♥❡❛♥ ♣r♦♣❡rt✐❡s ❢r♦♠ s❡✐s♠✐❝ ❞❛t❛ ✭❡✳❣✳ ♣♦r♦s✐t②✱ ❤②❞r♦❝❛r❜♦♥ ❝♦♥t❡♥t✮ ❆str♦♣❤②s✐❝s ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ t❤❡ ✐♥t❡r♥❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❙✉♥ ❢r♦♠ s✉r❢❛❝❡ ♦❜s❡r✈❛t✐♦♥s ❆str♦♥♦♠② ❖r❜✐t ❞❡t❡r♠✐♥❛t✐♦♥ ❢r♦♠ ♦❜s❡r✈❛t✐♦♥s ❆str♦♥❛✉t✐❝s ▲❛♥❞✐♥❣ ❛ s♣❛❝❡❝r❛❢t s❛❢❡❧② ♦♥ ❛♥♦t❤❡r ♣❧❛♥❡t P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ❉❡t❡r♠✐♥❛t✐♦♥ ♦❢ ✉♥❦♥♦✇♥ ♠♦❞❡❧ ♣❛r❛♠❡t❡rs ❆t♠♦s♣❤❡r✐❝ ♣♦❧❧✉t✐♦♥ ❲❤❛t ✐s t❤❡ s♦✉r❝❡✴s✐♥❦ ✜❡❧❞ ♦❢ ❛♥ ❛t♠♦s♣❤❡r✐❝ ♣♦❧❧✉t❛♥t❄ ❆t♠♦s♣❤❡r✐❝ r❡tr✐❡✈❛❧s ❲❤❛t ✐s t❤❡ ✈❡rt✐❝❛❧ ♣r♦✜❧❡ ♦❢ ❛t♠♦s♣❤❡r✐❝ q✉❛♥t✐t✐❡s ❢r♦♠ r❡♠♦t❡❧② s❡♥s❡❞ ♦❜s❡r✈❛t✐♦♥s❄ ❲❡❛t❤❡r ❢♦r❡❝❛st✐♥❣ ❲❤❛t ❛r❡ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭❡✳❣✳ u✱ v✱ T✱ p✱ q✱ ❝❧♦✉❞✱ ❙❙❚✱ s❛❧✐♥✐t②✮ ♦❢ ❛♥ ❛t♠♦s♣❤❡r❡ ♦r ♦❝❡❛♥ ❢♦r❡❝❛st ♠♦❞❡❧ t❤❛t ❛❣r❡❡s ✇✐t❤ t❤❡ ❧❛t❡st ♦❜s❡r✈❛t✐♦♥s❄

SLIDE 4

✹

✶✭❜✮ ◆♦t❛t✐♦♥

xA ❛♥❛❧②s✐s st❛t❡ xB ❜❛❝❦❣r♦✉♥❞ st❛t❡ δx ✐♥❝r❡♠❡♥t❛❧ st❛t❡ ❙♦♠❡t✐♠❡s x ❛♥❞ y ❛r❡ ❢♦r ♦♥❧② ♦♥❡ t✐♠❡ x✲✈❡❝t♦rs ❤❛✈❡ n ❡❧❡♠❡♥ts ✐♥ t♦t❛❧ y✲✈❡❝t♦rs ❤❛✈❡ p ❡❧❡♠❡♥ts ✐♥ t♦t❛❧

SLIDE 5

✺

✶✭❝✮ ❍✐st♦r② ♦❢ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✐♥ ♠❡t❡♦r♦❧♦❣✐❝❛❧ ♦♣❡r❛t✐♦♥s ❛♥❞ t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❝②❝❧❡

❙✉❜❥❡❝t✐✈❡ ✬❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✬ ✶✾✶✵s✱ ✶✾✷✵s

• ▲❋ ❘✐❝❤❛r❞s♦♥ ✭✶✾✷✷✮ ❛tt❡♠♣t❡❞ ❛ ❤✐♥❞✲❝❛st ✭❜② ❤❛♥❞✦✮

❢♦r ✷✵t❤ ▼❛② ✶✾✶✵✳

• Pr✐♠✐t✐✈❡ ❡q✉❛t✐♦♥✲❜❛s❡❞ ❢♦r❡❝❛st ♠♦❞❡❧✿ r❡s♦❧✉t✐♦♥ ∆λ =

3◦✱ ∆φ = 1.8◦✱ ✺ ✈❡rt✐❝❛❧ ❧❡✈❡❧s✳

• ✬❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥✬ ✇❛s ❞♦♥❡ ❢♦r ♠❛ss ✈❛r✐❛❜❧❡s (T, p) s❡♣✲

❛r❛t❡❧② ❢r♦♠ ✇✐♥❞ ✈❛r✐❛❜❧❡s (u, v) ✭✐✳❡✳ ✉♥✐✈❛r✐❛t❡✮ ❜② ✐♥✲ t❡r♣♦❧❛t✐♥❣ ♦❜s❡r✈❛t✐♦♥s s✉❜❥❡❝t✐✈❡❧②✳

• ❆ ❞✐s❛str♦✉s ❢♦r❡❝❛st✿ ∆P/∆t ≈ 145 ❤P❛ /6 ❤♦✉rs✳
• ❈❛t❛str♦♣❤✐❝ ❣r♦✇t❤ r❛t❡ ♥♦t ❞✉❡ t♦ t❤❡ ♠♦❞❡❧✱ ❜✉t ❞✉❡

t♦ ✐♥❛❞❡q✉❛t❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✕ t❤❡ ♠❛ss ❛♥❞ ✇✐♥❞ ✇❡r❡ ♦✉t ♦❢ ❜❛❧❛♥❝❡✳

• ❇❥❡r❦♥❡s✱ ✶✾✶✶✱ ❞❡s❝r✐❜❡❞ t❤❡ ❛♥❛❧②s✐s ♣r♦❜❧❡♠ ❛s✱ ✏❚❤❡

✉❧t✐♠❛t❡ ♣r♦❜❧❡♠ ✐♥ ▼❡t❡♦r♦❧♦❣②✑✳

SLIDE 6

✻ ❙✉❝❝❡ss❡s ✐♥ ◆❲P✱ ✶✾✹✵s

• ❙✉❝❝❡ss ✇✐t❤ ✜❧t❡r❡❞ ❞②♥❛♠✐❝❛❧ ♠♦❞❡❧s ❝♦♥t❛✐♥✐♥❣ ❜❛❧✲

❛♥❝❡❞ ♠♦t✐♦♥ ♦♥❧② ✭❡✳❣✳ ❜❛r♦tr♦♣✐❝ ✈♦rt✐❝✐t② ❡q✉❛t✐♦♥✮✱ ❡✈❡♥ ✇✐t❤ s✉❜❥❡❝t✐✈❡ ❛♥❛❧②s✐s✳

• ❇❱❊ ✐s ❧❡ss ❛❝❝✉r❛t❡ t❤❛♥ t❤❡ ♣r✐♠✐t✐✈❡ ❡q✉❛t✐♦♥s✱ ❜✉t ✐s

✐♥s❡♥s✐t✐✈❡ t♦ ✐♠❜❛❧❛♥❝❡s ✐♥ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭t❤❡r❡ ❛r❡ ♥♦ ❣r❛✈✐t② ✇❛✈❡s ✐♥ t❤❡ ❇❱❊✮✳

• ❊◆■❆❈ ✭❊❧❡❝tr♦♥✐❝ ◆✉♠❡r✐❝❛❧ ■♥t❡❣r❛t♦r ❛♥❞ ❈♦♠♣✉t❡r✮✳
• ❬❲❡ ♥♦✇ ✉s❡ ♣r✐♠✐t✐✈❡ ❡q✉❛t✐♦♥s ❢♦r ◆❲P✱ ❜✉t ✇✐t❤ ❉❆

t❤❛t ✐♥❤✐❜✐ts ✐♠❜❛❧❛♥❝❡✳❪

SLIDE 7

✼ ❇❡❣✐♥♥✐♥❣s ♦❢ ♦❜❥❡❝t✐✈❡ ❛♥❛❧②s✐s✿ ♣♦❧②♥♦♠✐❛❧ ✜tt✐♥❣✱ ❧❛t❡ ✶✾✹✵s

• ❋✐t ❛ ♣♦❧②♥♦♠✐❛❧ ❡①♣❛♥s✐♦♥ t♦ ♦❜s❡r✈❛t✐♦♥s✳
• ▼❛❞❡ ♥♦ ❛❝❝♦✉♥t ♦❢ ♦❜s❡r✈❛t✐♦♥ ❛❝❝✉r❛❝②✳
• ❉✐✛❡r❡♥t ✈❛r✐❛❜❧❡s tr❡❛t❡❞ ✐♥❞❡♣❡♥❞❡♥t❧② ✭✉♥✐✈❛r✐❛t❡✮✳
• ❉✐r❡❝t ♦❜s❡r✈❛t✐♦♥s ♦♥❧②✳
• ❯♥r❡❛❧✐st✐❝ ✈❛❧✉❡s ✐♥ ❞❛t❛ ✈♦✐❞s✳

SLIDE 8

✽ ❈r❡ss♠❛♥ ❛♥❛❧②s✐s ✴ ♠❡t❤♦❞ ♦❢ s✉❝❝❡ss✐✈❡ ❝♦rr❡❝t✐♦♥s✱ ✶✾✺✵s✱ ✶✾✻✵s

• ❯s❡ ♣r✐♦r ❦♥♦✇❧❡❞❣❡ ✭❛ ❜❛❝❦❣r♦✉♥❞ st❛t❡✮✳
• Pr♦✈✐❞❡s ✐♥❢♦r♠❛t✐♦♥ ✐♥ ❞❛t❛ ✈♦✐❞s✳
• Pr✐♦r ❦♥♦✇❧❡❞❣❡ ❝❛♥ ❝♦♠❡ ❢r♦♠ ❝❧✐♠❛t♦❧♦❣② ♦r ❛ ♣r❡✈✐♦✉s

❢♦r❡❝❛st✳

• ▲❛tt❡r ❧❡❛❞s ♦♥ t♦ t❤❡ ✬❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❝②❝❧❡✬✳

x0

i

= ✜rst ❣✉❡ss ✭❜❛❝❦❣r♦✉♥❞✮ xn+1

i

= xn

i +

Kn

i

k=1 W n ik(yk − ˜

xn

k)

Kn

i

k=1 W n ik + ǫ2 k

• xn

i ❡st✐♠❛t❡ ♦❢ ✜❡❧❞ ❛t ❣r✐❞ ♣♦✐♥t i ❛❢t❡r t❤❡ nt❤ ✐t❡r❛t✐♦♥✳

• ˜

xn

k ✜❡❧❞ ✈❛❧✉❡ ❛t ❣r✐❞ ❧♦❝❛t✐♦♥ ❝❧♦s❡st t♦ ♦❜s❡r✈❛t✐♦♥ k✳

• W n

ik ✇❡✐❣❤t ♦❢ ✐♥✢✉❡♥❝❡ ♦❢ ♦❜s❡r✈❛t✐♦♥ k ♦♥ ❣r✐❞ ♣♦✐♥t i

✭r❡❞✉❝❡s ✇✐t❤ ❞✐st❛♥❝❡✮✳

• Kn

i ♥✉♠❜❡r ♦❢ ♦❜s❡r✈❛t✐♦♥s ✇✐t❤✐♥ ❞✐st❛♥❝❡ Rn ♦❢ ❣r✐❞

♣♦✐♥t i✳

• yk kt❤ ♦❜s❡r✈❛t✐♦♥ ✈❛❧✉❡✳
• ǫk ❝♦♥tr♦❧s t❤❡ ❞❡❣r❡❡ ♦❢ ✐♥✢✉❡♥❝❡ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥s ♦♥

t❤❡ ❛♥❛❧②s✐s ✭❞✐♠✐♥✐s❤✐♥❣ ✐♥✢✉❡♥❝❡ ❛s ǫ → ∞✮✳

SLIDE 9

✾ ◆✉❞❣✐♥❣ ✭◆❡✇t♦♥✐❛♥ r❡❧❛①❛t✐♦♥✮✱ ✶✾✼✵s ✲ ♣r❡s❡♥t

• ❆❧❧♦✇s t❤❡ ❛♥❛❧②s✐s t♦ ❜❡ ❝♦♠❜✐♥❡❞ ✇✐t❤ t❤❡ ❜❛❝❦❣r♦✉♥❞

st❛t❡ s♠♦♦t❤❧②✳

• ❘❡❧✐❡s ♦♥ ❛♥ ✐♥t❡r♠❡❞✐❛t❡ ❛♥❛❧②s✐s✱ xint ✭❡✳❣✳ ❢r♦♠ ❙❈▼✮✳
• xint t♦ ❜❡ ✐♥tr♦❞✉❝❡❞ ♦✈❡r ❛ t✐♠❡s❝❛❧❡ τ✳
• ▼♦❞❡❧ ❡q✉❛t✐♦♥s✿

∂x ∂t = f(x),

• ✳✳✳ ❛r❡ ♠♦❞✐✜❡❞ t♦✿

∂x ∂t = f(x) − x − xint τ .

✬ ✫ ✩ ✪

❊①❛♠♣❧❡ ✇✐t❤ ❛ s❝❛❧❛r ✭x✮ ❢♦r ❛ ♣❡rs✐st❡♥❝❡ ♠♦❞❡❧ ✭f(x) = 0✮✿ dx dt = −x − xint τ , ⇒ x(t) = xint + (x(0) − xint) exp − t τ .

SLIDE 10

✶✵ ❖♣t✐♠❛❧ ✐♥t❡r♣♦❧❛t✐♦♥✱ ✶✾✼✵s ✴ ✶✾✽✵s

★ ✧ ✥ ✦

xA = xB + BHT(R + HBHT)−1(y − h(xB))

• ❆ ❢♦r♠❛❧ ✇❛② ♦❢ ❝♦♠❜✐♥✐♥❣ ♦❜s❡r✈❛t✐♦♥s ❛♥❞ ♠♦❞❡❧s✳
• ■♥t✐♠❛t❡❧② r❡❧❛t❡❞ t♦ ♠❡t❤♦❞ ♦❢ ❧❡❛st sq✉❛r❡s✳
• ❘❡♣r❡s❡♥ts ✉♥❝❡rt❛✐♥t✐❡s ♦❢ ❛❧❧ ✐♥❢♦r♠❛t✐♦♥✳
• ❚♦♦ ❡①♣❡♥s✐✈❡ t♦ s♦❧✈❡ ❢♦r t❤❡ ❣❧♦❜❛❧ s②st❡♠ ✭s♦❧✈❡ ❢♦r

♣❛t❝❤❡s ❛♥❞ ❣❧✉❡ t♦❣❡t❤❡r ❢♦r ✬❣❧♦❜❛❧✬ ❛♥❛❧②s✐s✮✳

• ◆❡❡❞ ❛❝❝✉r❛t❡ ❡st✐♠❛t❡s ♦❢ B ❛♥❞ R ♠❛tr✐❝❡s✳
• xA ❛♥❛❧②s✐s st❛t❡ ✭♣♦st❡r✐♦r✮ ∈ Rn✳
• xB ❜❛❝❦❣r♦✉♥❞ st❛t❡ ✭♣r✐♦r✮ ∈ Rn✳
• B ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✭❛❝❝♦✉♥ts ❢♦r ✉♥✲

❝❡rt❛✐♥t② ✐♥ xB✮ ∈ Rn×n✳

• y ♦❜s❡r✈❛t✐♦♥ ✈❡❝t♦r ∈ Rp✳
• h ♦❜s❡r✈❛t✐♦♥ ♦♣❡r❛t♦r Rn → Rp✳
• H ❏❛❝♦❜✐❛♥ ♦❢ h ∈ Rp×n✳
• R ♦❜s❡r✈❛t✐♦♥ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✭❛❝❝♦✉♥ts ❢♦r ✉♥✲

❝❡rt❛✐♥t② ✐♥ y✮ ∈ Rp×p✳

SLIDE 11

✶✶ ❱❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s ✭❱❆❘✮✱ ✶✾✾✵s ✴ ✷✵✵✵s

• ❇r♦❛❞❧② s♣❡❛❦✐♥❣ ✭✐♥ t❤❡ ❝❛s❡ ♦❢ ✸❉✲❱❆❘✮ ❛ ✇❛② ♦❢ s♦❧✈✐♥❣

t❤❡ ❖■ ❡q✉❛t✐♦♥s ❡✣❝✐❡♥t❧②✳

• ❈♦♥str✉❝t ❛ ❝♦st ❢✉♥❝t✐♦♥❛❧✱ J[x] ❛s t❤❡ s✉♠ ♦❢ sq✉❛r❡s ♦❢

❞❡✈✐❛t✐♦♥s ❢r♦♠ ❞❛t❛✳

• ❆♥❛❧②s✐s ✐s ❞❡✜♥❡❞ ❛s t❤❡ x t❤❛t ♠✐♥✐♠✐③❡s J[x]✳
• B ✐s ♥♦t ❛♣♣❧✐❡❞ ❛s ❛♥ ❡①♣❧✐❝✐t ♠❛tr✐①✱ ❜✉t ✐s ✐♥st❡❛❞

♠♦❞❡❧❧❡❞ ✭s❡❡ ❧❛t❡r✮✳

• ❊✣❝✐❡♥t ❡♥♦✉❣❤ ❢♦r ❛ tr✉❧② ❣❧♦❜❛❧ ❛♥❛❧②s✐s✳
• ❙t✐❧❧ ♥❡❡❞ ❛❝❝✉r❛t❡ ❡st✐♠❛t❡s ♦❢ B ❛♥❞ R ♠❛tr✐❝❡s✳ B ✐s

✉s✉❛❧❧② st❛t✐❝✳

• ❱❛r✐❛♥ts✿ ✶❉✲❱❆❘ ✴ ✸❉✲❱❆❘ ✴ ✹❉✲❱❆❘ ✴ ❡t❝✳ ✭s❡❡ ❧❛t❡r✮✳
• ❊①❛♠♣❧❡ ❢♦r str♦♥❣ ❝♦♥str❛✐♥t ✹❉✲❱❆❘✿

J(x) = 1 2 (x − xB) TB−1 (x − xB) + 1 2

T

• t=0

(y(t) − ht[Mt←0(x)])T R−1

t

× (y(t) − ht[Mt←0(x)]) , ✇❤❡r❡ x ✐s t❤❡ st❛t❡ ✈❡❝t♦r ❛t t = 0✳ P❛rt ■ ♦❢ t❤✐s ❝♦✉rs❡ ✐s ♠❛✐♥❧② ❛❜♦✉t ✈❛r✐❛t✐♦♥❛❧ ♠❡t❤♦❞s✳

SLIDE 12

✶✷ ❊♥s❡♠❜❧❡ ♠❡t❤♦❞s ✷✵✵✵s ✴ ✷✵✶✵s

• ❚❤❡ s♣r❡❛❞ ✐♥ ❛♥ ❡♥s❡♠❜❧❡ ♦❢ N ❜❛❝❦❣r♦✉♥❞ ❢♦r❡❝❛sts ❤❛s

✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ❜❛❝❦❣r♦✉♥❞ ✉♥❝❡rt❛✐♥t②✱ ♠❡♠❜❡r i x(i)✳

• ❋❧♦✇✲❞❡♣❡♥❞❡♥t ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s✱ Pf✳
• ❋♦r♠✉❧❛t✐♦♥ st❛rts ✇✐t❤ t❤❡ ❖■ ❡q✉❛t✐♦♥ ✭B → Pf✮✱ ❜✉t

❢♦r ❛♥ ❡♥s❡♠❜❧❡ ♦❢ st❛t❡s✳

• ❉♦❡s ♥♦t ♥❡❡❞ t❤❡ Pf✲♠❛tr✐① ❡①♣❧✐❝✐t❧②✳
• ❙❡✈❡r❡ r❛♥❦ ❞❡✜❝✐❡♥❝② ♣r♦❜❧❡♠s ✇✐t❤ Pf ❞✉❡ t♦ ✉♥❞❡rs❛♠✲

♣❧✐♥❣ ✭✉s❡✱ ❡✳❣✳✱ ❧♦❝❛❧✐③❛t✐♦♥ t❡❝❤♥✐q✉❡s t♦ ♦✈❡r❝♦♠❡✮✳

• ❉❡t❡r♠✐♥✐st✐❝ ✭sq✉❛r❡✲r♦♦t✮ ❛♥❞ ♥♦♥✲❞❡t❡r♠✐♥✐st✐❝ ✭♥♦♥✲

sq✉❛r❡✲r♦♦t✮ ❢♦r♠✉❧❛t✐♦♥s ❡①✐st ✲ s❡❡ ♣❛rt ■■ ♦❢ t❤❡ ❝♦✉rs❡✳ Pf ≈ Pf

(N) =

1 N − 1

N

• i=1
• x(i) − x

x(i) − x T , x ≈ 1 N

N

• i=1

x(i).

SLIDE 13

✶✸ ❍②❜r✐❞ ♠❡t❤♦❞s ✷✵✶✵s

• ❈♦♠❜✐♥❡ t❤❡ r♦❜✉st♥❡ss ♦❢ t❤❡ B✲♠❛tr✐① ✇✐t❤ t❤❡ ✢♦✇✲❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ Pf✲♠❛tr✐①✳
• ▼♦st s✐♠♣❧❡ ✐s t❤❡ ❛r✐t❤♠❡t✐❝ ❛✈❡r❛❣❡✿

PH = αB + (1 − α)Pf

(N)

• ❙♦❧✈❡ ❛ ❱❆❘✲❧✐❦❡ ♣r♦❜❧❡♠ ❜✉t B → PH✳
• ❙t✐❧❧ ♥❡❡❞ ❧♦❝❛❧✐③❛t✐♦♥ ♠❡t❤♦❞s✳
• ❖t❤❡r ❛♣♣r♦❛❝❤❡s ❡①✐st t♦♦✳
• ❯s❡s ♠❡t❤♦❞s t❤❛t ❛✈♦✐❞ t❤❡ ♥❡❡❞ t♦ ❤♦❧❞ ❧❛r❣❡ ♠❛tr✐❝❡s ❡①♣❧✐❝✐t❧②✳

SLIDE 14

✶✹ ❚❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❝②❝❧❡

SLIDE 15

✶✺

✷✳ ❱❛r✐❛t✐♦♥❛❧ t❡❝❤♥✐q✉❡s

✷✭❛✮ ❚❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s

❚❤✐s s❡❝t✐♦♥ t❡❛❝❤❡s ✉s ❢♦r♠❛❧❧② ❛❜♦✉t t❤❡ ✈❛r✐❛t✐♦♥❛❧ s♦❧✉t✐♦♥ ♦❢ ❛♥ ✐♥✈❡rs❡ ♣r♦❜❧❡♠✱ ❜❛❝❦✇❛r❞ ✭♦r ❛❞❥♦✐♥t✮ ✈❛r✐❛❜❧❡s ❛♥❞ t❤❡ str♦♥❣ ❛♥❞ ✇❡❛❦ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥s✳ ❚❤❡ ♠❡t❤♦❞ ♦❢ r❡♣r❡s❡♥t❡rs✱ ✉s❡❞ t♦ s♦❧✈❡ t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s✱ ✐s ✐♥tr♦❞✉❝❡❞✳ ❙t❛t❡♠❡♥t ♦❢ ♣r♦❜❧❡♠ ❲❤❛t ✐s t❤❡ ♦♣t✐♠❛❧ st❛t❡✱ φ(x, t) ♦❢ t❤❡ ✶✲❉ s②st❡♠ ✇❤♦s❡ ❞②♥❛♠✐❝s ❛r❡ ❣♦✈❡r♥❡❞ ❜② ∂φ ∂t + u∂φ ∂x − F = e, ✭✶✮ ✇❤✐❝❤ ❧✐❡s ❝❧♦s❡ t♦ s♦♠❡ ❣✐✈❡♥ ♦❜s❡r✈❛t✐♦♥s✱ s♦♠❡ ✐♥✐t✐❛❧ ❝♦♥❞✐✲ t✐♦♥s ❛♥❞ s♦♠❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s❄ ❚❤❡ ✭✐♠♣❡r❢❡❝t❧② ❦♥♦✇♥✮ ✐♥❢♦r♠❛t✐♦♥ ✇❡ ❤❛✈❡ ❛❜♦✉t t❤❡ s②s✲ t❡♠ ✐s ✭s❡❡ ❋✐❣✳✮✿

• φ(x, 0) ≈ I(x) ✐♠♣❡r❢❡❝t❧② ❦♥♦✇♥ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ✭✐✳❝✳s✮✱

0 ≤ x ≤ L✱

• φ(0, t) ≈ B(t) ✐♠♣❡r❢❡❝t❧② ❦♥♦✇♥ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s

✭❜✳❝✳s✮✱ 0 ≤ t ≤ T✱ ❛♥❞

• ym ✐♠♣❡r❢❡❝t ♦❜s❡r✈❛t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✭❛ ❞✐r❡❝t ♦❜s❡r✈❛✲

t✐♦♥ ♦❢ φ(xm, tm)✮✱ 1 ≤ m ≤ p✳ ❚❤❡ ❛✲♣r✐♦r✐ st❛t❡ φB(x, t) s❛t✐s✜❡s t❤❡ ❦♥♦✇♥ ❜✐ts ♦❢ t❤❡ ♣r♦❜❧❡♠ ✭t❤❡ s♣❡❝✐✜❡❞ ✐✳❝✳s ❛♥❞ ❜✳❝✳s✱ ❛♥❞ ✭✶✮ ✇✐t❤ e = 0✮✿ ∂φB ∂t +u∂φB ∂x −F = 0, φB(x, 0) = I(x), φB(0, t) = B(t). ✭✷✮

SLIDE 16

✶✻ ❚❤❡ str♦♥❣ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥ ❚❤❡ str♦♥❣ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥ ✐♠♣♦s❡s t❤❡ ❦♥♦✇♥ ♣❛rts ♦❢ t❤❡ s②st❡♠ ❡q✉❛t✐♦♥s ❡①❛❝t❧② ✭t❤❛t ✐s ✇❡ ❛ss✉♠❡ t❤❛t t❤❡r❡ ✐s ♥♦ ♠♦❞❡❧ ❡rr♦r✱ ❡✈❡♥ t❤♦✉❣❤ ✐♥ r❡❛❧✐t② t❤❡r❡ ♥❡❛r❧② ❛❧✇❛②s ✐s✮✳ ❲❡ st✐❧❧ ❛❧❧♦✇ ❢♦r ✐♠♣❡r❢❡❝t✐♦♥s ✐♥ t❤❡ ♦t❤❡r ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ t❤♦✉❣❤ ✭✐✳❝✳s✱ ❜✳❝✳s ❛♥❞ ♦❜s❡r✈❛t✐♦♥s✮✳ ■♥ ♦r❞❡r t♦ ✜♥❞ t❤❡ ♦♣t✐♠❛❧ s♦❧✉t✐♦♥ t♦ t❤❡ ♣r♦❜❧❡♠ ✐♥ t❤✐s ❢♦r♠✉❧❛t✐♦♥✱ ❝♦♥str✉❝t ❛ ❢✉♥❝t✐♦♥❛❧ f[φ]✿ f[φ] = Wic

L

ˆ

x=0

dx{φ(x, 0) − I(x)}2 + Wbc

T

ˆ

t=0

dt{φ(0, t) − B(t)}2 + Wob

p

• i=1

{φ(xi, ti) − yi}2. ✭✸✮ ❲❡ ❛s❦ t❤❡ q✉❡st✐♦♥✿ ✇❤❛t φ(x, t) ♠❛❦❡s f[φ] st❛t✐♦♥❛r②✱ s✉❜❥❡❝t t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦❞❡❧ ❝♦♥str❛✐♥t✿ g(x, t) = ∂φ ∂t + u∂φ ∂x − F = 0 ? ✭✹✮ ❚❤❡ ❝♦♥str❛✐♥❡❞ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✐♥tr♦❞✉❝❡s ❛ ♥❡✇ ❢✉♥❝t✐♦♥❛❧✱ J✱ ✇❤✐❝❤ ✐s t❤❡ s✉♠ ♦❢ f ❛♥❞ ❛❧❧ ♦❢ t❤❡ ❝♦♥str❛✐♥t t❡r♠s✳ ❊❛❝❤ ❝♦♥str❛✐♥t t❡r♠ r❡♣r❡s❡♥ts t❤❡ ❝♦♥str❛✐♥t ❛t ❛ ♣♦s✐t✐♦♥ ❛♥❞ t✐♠❡ ❛♥❞ ✐s r❡♣r❡s❡♥t❡❞ ❛s t❤❡ ♣r♦❞✉❝t ♦❢ g(x, t) ❛♥❞ t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r✱ 2λ(x, t)✿ J[φ, λ] = f[φ] + 2

L

ˆ

x=0

dx

T

ˆ

t=0

dt λ(x, t)g(x, t), = Wic

L

ˆ

x=0

dx{φ(x, 0) − I(x)}2 + Wbc

T

ˆ

t=0

dt{φ(0, t) − B(t)}2 + Wob

p

• i=1

{φ(xi, ti) − yi}2 + 2

L

ˆ

x=0

dx

T

ˆ

t=0

dt λ(x, t) ∂φ ∂t + u∂φ ∂x − F

• .

✭✺✮ ❚❤✐s ✐s st❛♥❞❛r❞ ✈❛r✐❛t✐♦♥❛❧ ❝❛❧❝✉❧✉s ✭s❡❡✱ ❡✳❣✳✱ t❤❡ ❆✐❞❡ ▼❡♠♦✐r ❤❛♥❞♦✉t✮✳ ❚❤❡ ♣r♦❜❧❡♠ ♥♦✇ ✐s t♦ ♠✐♥✐♠✐③❡ ✭✺✮ ✇✳r✳t✳ t❤❡ ✜❡❧❞s φ(x, t) ❛♥❞ λ(x, t)✳

SLIDE 17

✶✼ ❱❛r✐❛t✐♦♥s ♦❢ J ✭str♦♥❣ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥✮ ❈♦♥str✉❝t ✈❛r✐❛t✐♦♥s ♦❢ J ❛❜♦✉t s♦♠❡ r❡❢❡r❡♥❝❡ ✜❡❧❞s ˆ φ, ˆ λ✱ ✐✳❡✳ J[ˆ φ + δφ, ˆ λ + δλ] = J[ˆ φ, ˆ λ] + δJ|ˆ

φ,ˆ λ✱ ✇❤❡r❡✿

δJ|ˆ

φ,ˆ λ = L

ˆ

x=0

dx

T

ˆ

t=0

dt ∂J ∂φ|ˆ

φ,ˆ λδφ + L

ˆ

x=0

dx

T

ˆ

t=0

dt ∂J ∂λ|ˆ

φ,ˆ λδλ + O(δφ2, δλ2, δφδλ),

= 2Wic

L

ˆ

x=0

dx{ˆ φ(x, 0) − I(x)}δφ(x, 0) + 2Wbc

T

ˆ

t=0

dt{ˆ φ(0, t) − B(t)}δφ(0, t) + 2Wob

p

• i=1

{ˆ φ(xi, ti) − yi}δφ(xi, ti) + (∗) 2

L

ˆ

x=0

dx

T

ˆ

t=0

dt ˆ λ(x, t) ∂δφ ∂t + u∂δφ ∂x

• +

2

L

ˆ

x=0

dx

T

ˆ

t=0

dt δλ(x, t)

• ∂ ˆ

φ ∂t + u∂ ˆ φ ∂x − F

• + O(δφ2, δλ2, δφδλ).

✭✻✮ ❲❡ ✐♠♣♦s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ♦♥ ˆ λ✿ ˆ λ(x, T) = 0, ˆ λ(L, t) = 0✳

SLIDE 18

✶✽ ❈❤❛♥❣✐♥❣ ❢♦r♠ ✭str♦♥❣ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥✮

b

ˆ

a

vdu dxdx = [uv]b

a − b

ˆ

a

udv dxdx. ✭✼✮ ❯s✐♥❣ t❤✐s t♦ r❡✇r✐t❡ t❤❡ ✜rst t❡r♠ ✐♥ ✭✯✮✿

T

ˆ

t=0

dt ˆ λ(x, t)∂δφ ∂t = [δφˆ λ]T

0 − T

ˆ

t=0

dt δφ∂ˆ λ ∂t , = δφ(x, T)ˆ λ(x, T) − δφ(x, 0)ˆ λ(x, 0) −

T

ˆ

t=0

dt δφ∂ˆ λ ∂t , ✭✽✮ ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ✐♥ ✭✯✮✿

L

ˆ

x=0

dx ˆ λ(x, t)u∂δφ ∂x = u[δφˆ λ]L

0 − L

ˆ

x=0

dx uδφ∂ˆ λ ∂x, = uδφ(L, t)ˆ λ(L, t) − uδφ(0, t)ˆ λ(0, t) −

L

ˆ

x=0

dx uδφ∂ˆ λ ∂x. ✭✾✮ ◆♦t❡ ❛❧s♦ ❢♦r t❤❡ ♦❜s❡r✈❛t✐♦♥ t❡r♠✿ {ˆ φ(xi, ti) − yi}δφ(xi, ti) =

L

ˆ

x=0

dx

T

ˆ

t=0

dt {ˆ φ(xi, ti) − yi}δφ(x, t)δ(x − xi)δ(t − ti). ✭✶✵✮ ❯s✐♥❣ ✭✽✮✱ ✭✾✮ ❛♥❞ ✭✶✵✮ ✐♥ ✭✶✶✮ ❛♥❞ ♥♦t✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ ˆ λ ❣✐✈❡♥ ❛❢t❡r ✭✶✶✮✿

SLIDE 19

✶✾ δJ|ˆ

φ,ˆ λ = 2Wic L

ˆ

x=0

dx{ˆ φ(x, 0) − I(x)}δφ(x, 0) + 2Wbc

T

ˆ

t=0

dt{ˆ φ(0, t) − B(t)}δφ(0, t) + 2Wob

L

ˆ

x=0

dx

T

ˆ

t=0

dt

p

• i=1

{ˆ φ(xi, ti) − yi}δφ(x, t)δ(x − xi)δ(t − ti) − 2

L

ˆ

x=0

dx ˆ λ(x, 0)δφ(x, 0) − 2

L

ˆ

x=0

dx

T

ˆ

t=0

dt ∂ˆ λ ∂t δφ(x, t) − 2

T

ˆ

t=0

dt uˆ λ(0, t)δφ(0, t) − 2

T

ˆ

t=0

dt

L

ˆ

x=0

dx u∂ˆ λ ∂xδφ(x, t) + 2

L

ˆ

x=0

dx

T

ˆ

t=0

dt

• ∂ ˆ

φ ∂t + u∂ ˆ φ ∂x − F

• δλ(x, t) + O(δφ2, δλ2, δφδλ).

✭✶✶✮

SLIDE 20

✷✵ ❚❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ str♦♥❣ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥ ❙❡tt✐♥❣ t❤❡ ❧✐♥❡❛r ♣❛rt ♦❢ ✭✶✶✮ t♦ ③❡r♦✱ ❛♥❞ ✉s✐♥❣ t❤❡ ❝♦♥❞✐t✐♦♥s ♦♥ ˆ λ✱ ❣✐✈❡s ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ str♦♥❣ ❝♦♥str❛✐♥t✿

✬ ✫ ✩ ✪

∂ ˆ φ ∂t + u∂ ˆ φ ∂x − F = 0, ✭✶✷✮ Wic{ˆ φ(x, 0) − I(x)} − ˆ λ(x, 0) = 0, ✭✶✸✮ Wbc{ˆ φ(0, t) − B(t)} − uˆ λ(0, t) = 0, ✭✶✹✮ Wob

p

• i=1

{ˆ φ(xi, ti) − yi}δ(x − xi)δ(t − ti) −

• ∂ˆ

λ ∂t + u∂ˆ λ ∂x

• = 0,

✭✶✺✮ ˆ λ(x, T) = 0, ✭✶✻✮ ˆ λ(L, t) = 0. ✭✶✼✮ ✭✶✷✮ ✐s ❦♥♦✇♥ ❛s t❤❡ ❢♦r✇❛r❞ ❡q✉❛t✐♦♥✱ ❛♥❞ ✭✶✸✮✴✭✶✹✮ ❛r❡ ✐ts ✐♥✐t✐❛❧✴❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s✳ ✭✶✺✮ ✐s ❦♥♦✇♥ ❛s t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥✱ ❛♥❞ ✭✶✻✮✴✭✶✼✮ ❛r❡ ✐ts ❝♦♥❞✐t✐♦♥s✳ ❚❤❡ s♦❧✉t✐♦♥ t♦ t❤❡s❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❢♦r ˆ φ s♦❧✈❡s t❤❡ ♦r✐❣✐♥❛❧ ♣r♦❜❧❡♠ t❤❛t ✇❡ ♣♦s❡❞ ✐♥ ❙❡❝t✐♦♥ ✶ ✭✇✐t❤ t❤❡ ❛ss✉♠♣t✐♦♥ ♦❢ ❛ ♣❡r❢❡❝t ♠♦❞❡❧✮✳

SLIDE 21

✷✶ ❚❤❡ ✇❡❛❦ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥ ❚❤❡ ✇❡❛❦ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥ ✐♠♣♦s❡s t❤❡ ❦♥♦✇♥ ♣❛rts ♦❢ t❤❡ s②st❡♠ ❡q✉❛t✐♦♥s ❛♣♣r♦①✐♠❛t❡❧②✳ J[φ] = Wic

L

ˆ

x=0

dx{φ(x, 0) − I(x)}2 + Wbc

T

ˆ

t=0

dt{φ(0, t) − B(t)}2 + Wob

p

• i=1

{φ(xi, ti) − yi}2 + We

L

ˆ

x=0

dx

T

ˆ

t=0

dt ∂φ ∂t + u∂φ ∂x − F 2 . ✭✶✽✮ ❱❛r✐❛t✐♦♥s ♦❢ J ✭✇❡❛❦ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥✮ ❈♦♥str✉❝t ✈❛r✐❛t✐♦♥s ♦❢ J ❛❜♦✉t s♦♠❡ r❡❢❡r❡♥❝❡ ✜❡❧❞ ˆ φ✱ ✐✳❡✳ J[ˆ φ + δφ] = J[ˆ φ] + δJ|ˆ

φ✱ ✇❤❡r❡✿

δJ|ˆ

φ = L

ˆ

x=0

dx

T

ˆ

t=0

dt ∂J ∂φ|ˆ

φδφ + O(δφ2),

= 2Wic

L

ˆ

x=0

dx{ˆ φ(x, 0) − I(x)}δφ(x, 0) + 2Wbc

T

ˆ

t=0

dt{ˆ φ(0, t) − B(t)}δφ(0, t) + 2Wob

L

ˆ

x=0

dx

T

ˆ

t=0

dt

p

• i=1

{ˆ φ(xi, ti) − yi}δφ(x, t)δ(x − xi)δ(t − ti) + (∗) 2We

L

ˆ

x=0

dx

T

ˆ

t=0

dt

• ∂ ˆ

φ ∂t + u∂ ˆ φ ∂x − F ∂δφ ∂t + u∂δφ ∂x

• + O(δφ2).

✭✶✾✮ ❉❡✜♥❡✿ ˆ µ(x, t) = We

• ∂ ˆ

φ ∂t + u∂ ˆ φ ∂x − F

• ,

✭✷✵✮ ✭❧✐❦❡ t❤❡ λ(x, t) ✐♥ ✭✺✮✮ ✇❤❡r❡ ˆ µ(x, T) = 0, ˆ µ(L, t) = 0✳

SLIDE 22

✷✷ ❈❤❛♥❣✐♥❣ ❢♦r♠ ✭✇❡❛❦ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥✮ ◆♦t❡ t❤❛t ✭✯✮ ♦❢ ✭✷✶✮ ✐s ❧✐❦❡ ✭✯✮ ♦❢ ✭✻✮ s♦ ❝❤❛♥❣❡ t❤❡ ❢♦r♠ ✉s✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥ ❜② ♣❛rts ❢♦r♠✉❧❛ ✭✼✮✱ ♠❛❦✐♥❣ ✭✷✶✮ ✐♥t♦✿ δJ|ˆ

φ = 2Wic L

ˆ

x=0

dx{ˆ φ(x, 0) − I(x)}δφ(x, 0) + 2Wbc

T

ˆ

t=0

dt{ˆ φ(0, t) − B(t)}δφ(0, t) + 2Wob

L

ˆ

x=0

dx

T

ˆ

t=0

dt

p

• i=1

{ˆ φ(xi, ti) − yi}δφ(x, t)δ(x − xi)δ(t − ti) − 2

L

ˆ

x=0

dx ˆ µ(x, 0)δφ(x, 0) − 2

L

ˆ

x=0

dx

T

ˆ

t=0

dt ∂ˆ µ ∂t δφ(x, t) − 2

T

ˆ

t=0

dt uˆ µ(0, t)δφ(0, t) − 2

T

ˆ

t=0

dt

L

ˆ

x=0

dx u∂ˆ µ ∂xδφ(x, t) + O(δφ2). ✭✷✶✮ ❚❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ✇❡❛❦ ❝♦♥str❛✐♥t ❢♦r♠✉❧❛t✐♦♥ ❙❡tt✐♥❣ t❤❡ ❧✐♥❡❛r ♣❛rt ♦❢ ✭✷✶✮ t♦ ③❡r♦✱ ✉s✐♥❣ t❤❡ ♠♦❞❡❧ ❡q✉❛t✐♦♥ ✭✶✮✱ ❛♥❞ ❞❡✜♥✐t✐♦♥ ✭✷✶✮ ❣✐✈❡s ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ❢♦r t❤❡ ✇❡❛❦ ❝♦♥str❛✐♥t✿

✬ ✫ ✩ ✪

∂ ˆ φ ∂t + u∂ ˆ φ ∂x − F = W −1

e

ˆ µ, ✭✷✷✮ Wic{ˆ φ(x, 0) − I(x)} − ˆ µ(x, 0) = 0, ✭✷✸✮ Wbc{ˆ φ(0, t) − B(t)} − uˆ µ(0, t) = 0, ✭✷✹✮ Wob

p

• i=1

{ˆ φ(xi, ti) − yi}δ(x − xi)δ(t − ti) − ∂ˆ µ ∂t + u∂ˆ µ ∂x

• = 0,

✭✷✺✮ ˆ µ(x, T) = 0, ✭✷✻✮ ˆ µ(L, t) = 0. ✭✷✼✮

SLIDE 23

✷✸ ❙♦❧✈✐♥❣ t❤❡ ✇❡❛❦✲❝♦♥str❛✐♥t ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ r❡♣r❡s❡♥t❡rs ❚❤❡ ❢♦r✇❛r❞ ❡q✉❛t✐♦♥ ✭✷✷✮ ✐s s♦❧✈❡❞ ❢♦r ˆ φ(x, t) ✬✉♣✇❛r❞s ❛♥❞ t♦ t❤❡ r✐❣❤t✬ ✭s✐♥❝❡ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ˆ φ ❛r❡ ❣✐✈❡♥ ❢♦r x = 0 ❛♥❞ t = 0✱ s❡❡ ❋✐❣✳✮✱ ❛♥❞ t❤❡ ❜❛❝❦✇❛r❞ ❡q✉❛t✐♦♥ ✭✷✺✮ ✐s s♦❧✈❡❞ ❢♦r ˆ µ(x, t) ✬❞♦✇♥✇❛r❞s ❛♥❞ t♦ t❤❡ ❧❡❢t✬ ✭s✐♥❝❡ t❤❡ ❝♦♥❞✐t✐♦♥s ❢♦r ˆ φ ❛r❡ ❣✐✈❡♥ ❢♦r x = L ❛♥❞ t = T✱ s❡❡ ❋✐❣✳✮✳ Pr♦❜❧❡♠✿ ■♥ ♦r❞❡r t♦ s♦❧✈❡ ✭✷✷✮ ❢♦r ˆ φ(x, t)✱ ˆ µ(x, t) ✐s ♥❡❡❞❡❞✱ ❜✉t ✐♥ ♦r❞❡r t♦ s♦❧✈❡ ✭✷✺✮ ❢♦r ˆ µ(x, t)✱ ˆ φ(x, t) ✐s ♥❡❡❞❡❞✦ ❚❤❡ s❡t ♦❢ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s ♠✉st ❜❡ ❛❧❧ s♦❧✈❡❞ t♦❣❡t❤❡r✳ ❘❡❝✐♣❡ ❢♦r t❤❡ s♦❧✉t✐♦♥ ✉s✐♥❣ t❤❡ ♠❡t❤♦❞ ♦❢ r❡♣r❡s❡♥t❡rs ✶✳ ❙♦❧✈❡ t❤❡ ❜❛❝❦❣r♦✉♥❞ ♣r♦❜❧❡♠ ✭✷✮ ❢♦r φB(x, t)✳ ❚❤✐s ✐s ❛♥ ❡①❡r❝✐s❡ ✐♥ s♦❧✈✐♥❣ ♣❛rt✐❛❧ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s ✭P❉❊s✮ ❛♥❛❧②t✐❝❛❧❧② ♦r ♥✉♠❡r✐❝❛❧❧②✳ ✷✳ ❉❡✜♥❡ t❤❡ p ❢♦r✇❛r❞ r❡♣r❡s❡♥t❡r ❢✉♥❝t✐♦♥s ❛♥❞ t❤❡ p ❜❛❝❦✇❛r❞ r❡♣r❡s❡♥t❡r ❢✉♥❝t✐♦♥s ✭♦♥❡ ❡❛❝❤ ♣❡r ♦❜s❡r✈❛t✐♦♥✮ ❛s✿ ❋♦r✇❛r❞ r❡♣r❡s❡♥t❡r ❢✉♥❝t✐♦♥ ri(x, t) ❇❛❝❦✇❛r❞ r❡♣r❡s❡♥t❡r ❢✉♥❝t✐♦♥ αi(x, t)

• 1 ≤ i ≤ p.

❚❤❡ ♠♦❞✐✜❡❞ ❡q✉❛t✐♦♥s t❤❛t t❤❡s❡ r❡♣r❡s❡♥t❡rs s❛t✐s❢② ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s✱ ❜✉t ❤❛✈❡ ˆ φ → ri✱ ˆ µ → αi✱ F = 0✱ I(x) = 0✱ B(t) = 0 ❛♥❞ r❡♣❧❛❝❡ t❤❡ ♦❜s❡r✈❛t✐♦♥s ✇✐t❤ ❛ s✐♥❣❧❡ ✐♠♣✉❧s❡ ❛t t❤❡ ♣♦s✐t✐♦♥ ❛♥❞ t✐♠❡ ♦❢ t❤❡ it❤ ♦❜s❡r✈❛t✐♦♥ ✭Wob p

i=1{ˆ

φ(xi, ti) − yi}δ(x − xi)δ(t − ti) → δ(x − xi)δ(t − ti)✮✳ ∂ri ∂t + u∂ri ∂x = W −1

e αi,

✭✷✽✮ Wicri(x, 0) − αi(x, 0) = 0, ✭✷✾✮ Wbcri(0, t) − uαi(0, t) = 0, ✭✸✵✮ δ(x − xi)δ(t − ti) − ∂αi ∂t + u∂αi ∂x

• = 0,

✭✸✶✮ αi(x, T) = 0, ✭✸✷✮ αi(L, t) = 0. ✭✸✸✮

SLIDE 24

✷✹ ✸✳ ❙t❛rt ✇✐t❤ t❤❡ ❜❛❝❦✇❛r❞ r❡♣r❡s❡♥t❡rs✳ ❙♦❧✈❡ ✭✸✶✮✱ ✭✸✷✮ ❛♥❞ ✭✸✸✮ ❢♦r ❡❛❝❤ i ✬❞♦✇♥✇❛r❞s ❛♥❞ t♦ t❤❡ ❧❡❢t✬ ✭❛❣❛✐♥ ❛♥ ❡①❡r❝✐s❡ ✐♥ s♦❧✈✐♥❣ P❉❊s✮✳ ❚❤✐s ❣✐✈❡s t❤❡ p ❜❛❝❦✇❛r❞ r❡♣r❡s❡♥t❡rs✱ αi(x, t)✳ ■♥ t❤❡ ♠♦❞✐✜❡❞ ❡q✉❛t✐♦♥s✱ t❤❡ ❜❛❝❦✇❛r❞ r❡♣r❡s❡♥t❡rs ❞♦ ♥♦t ❞❡♣❡♥❞ ✉♣♦♥ t❤❡ ❢♦r✇❛r❞ r❡♣r❡s❡♥t❡rs✱ ri(x, t)✳ ✹✳ ◆♦✇ ✜♥❞ t❤❡ ❢♦r✇❛r❞ r❡♣r❡s❡♥t❡rs✳ ❙♦❧✈❡ ✭✷✽✮✱ ✭✷✾✮ ❛♥❞ ✭✸✵✮ ❢♦r ❡❛❝❤ i ✬✉♣✇❛r❞s ❛♥❞ t♦ t❤❡ r✐❣❤t✬ ✭❛❣❛✐♥ ❛♥ ❡①❡r❝✐s❡ ✐♥ s♦❧✈✐♥❣ P❉❊s✮✳ ❚❤✐s ❣✐✈❡s t❤❡ p ❢♦r✇❛r❞ r❡♣r❡s❡♥t❡rs✱ ri(x, t)✱ ✇❤✐❝❤ ❝❛♥ ❜❡ ❢♦✉♥❞ ❜❡❝❛✉s❡ t❤❡ αi ❛r❡ ♥♦✇ ❦♥♦✇♥✳ ✺✳ ▲♦♦❦ ❢♦r ❛ s♦❧✉t✐♦♥ ♦❢ ˆ φ(x, t) ✭t❤❡ ✜❡❧❞ t❤❛t ✇❡ ❛r❡ r❡❛❧❧② ✐♥t❡r❡st❡❞ ✐♥✮ t❤❛t ✐s ❛ ❧✐♥❡❛r ❝♦♠❜✐♥❛t✐♦♥ ♦❢ t❤❡ ❢♦r✇❛r❞ r❡♣r❡s❡♥t❡r ❢✉♥❝t✐♦♥s✿ ˆ φ(x, t) = φB(x, t) +

p

• i=1

βiri(x, t), ✭✸✹✮ ✇❤❡r❡ t❤❡ βi ❛r❡ t❤❡ ❝♦❡✣❝✐❡♥ts ✇❤✐❝❤ ❛r❡ ❞❡t❡r♠✐♥❡❞ ❜② ✐♥s✐st✐♥❣ t❤❛t ˆ φ(x, t) s❛t✐s✜❡s t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s✳ ✻✳ ❚♦ ♠❛❦❡ ✭✸✹✮ s❛t✐s❢② t❤❡ ❊✉❧❡r✲▲❛❣r❛♥❣❡ ❡q✉❛t✐♦♥s✱ ❛❝t ✇✐t❤ ∂/∂t + u∂/∂x ♦♥ ✭✸✹✮✱ t❤❡♥ ✉s❡ ✭✷✷✮✱ ✭✷✮ ❛♥❞ ✭✸✶✮✿ ∂ ˆ φ ∂t + u∂ ˆ φ ∂x = ∂φB ∂t + u∂φB ∂x +

p

• i=1

βi ∂ri ∂t + u∂ri ∂x

• ,

⇒ F + W −1

e

ˆ µ = F +

p

• i=1

βiW −1

e αi,

⇒ ˆ µ(x, t) =

p

• i=1

βiαi(x, t). ✭✸✺✮ ✼✳ ❙✉❜st✐t✉t❡ ✭✸✺✮ ✐♥t♦ ✭✷✺✮✱ t❤❡♥ ✉s❡ ✭✸✹✮ ❛♥❞ ✭✸✶✮✿ Wob

p

• i=1

{ˆ φ(xi, ti) − yi}δ(x − xi)δ(t − ti) =

p

• i=1

βi ∂αi ∂t + u∂αi ∂x

• ,

⇒ Wob

p

• i=1

{φB(xi, ti) +

p

• j=1

βjrj(xi, ti) − yi}δ(x − xi)δ(t − ti) =

p

• i=1

βiδ(x − xi)δ(t − ti). ✭✸✻✮

SLIDE 25

✷✺ ✽✳ ❊q✉❛t❡ ❝♦❡✣❝✐❡♥ts ♦❢ ✐♠♣✉❧s❡s ✐♥ ✭✸✻✮✿ Wob

• φB(xi, ti) +

p

• j=1

βjrj(xi, ti) − yi

• = βi,

⇒ Wob {φB(xi, ti) − yi} +

p

• j=1

{Wobrj(xi, ti) − δij} βj = 0, ✭✸✼✮ ✇❤❡r❡ δij ✐s t❤❡ ❑r♦♥❡❝❦❡r ❞❡❧t❛✲❢✉♥❝t✐♦♥✳ ❚❤✐s ✐s t❤❡ ❡q✉❛t✐♦♥ t❤❛t ✇❡ ❤❛✈❡ t♦ s♦❧✈❡ ❢♦r t❤❡ βi ❝♦❡✣❝✐❡♥ts✳ ❖♥❝❡ t❤❡s❡ ❛r❡ ❦♥♦✇♥✱ t❤❡ s♦❧✉t✐♦♥ ❝❛♥ ❜❡ ❜✉✐❧t ✉s✐♥❣ ✭✸✹✮✳ ❋✐♥❞✐♥❣ t❤❡ ❝♦❡✣❝✐❡♥ts ❊q✉❛t✐♦♥ ✭✸✼✮ ✐s t❤❡ r❡♠❛✐♥✐♥❣ ❡q✉❛t✐♦♥ t♦ s♦❧✈❡✳ ❲❡ ✇✐❧❧ ✉s❡ s♦♠❡ ❧✐♥❡❛r ❛❧❣❡❜r❛ ✭✈❡❝t♦rs ❛♥❞ ♠❛tr✐❝❡s✮ t♦ ❞♦ t❤✐s✳ ❚❤✐s ✐s ❛ st❛♥❞❛r❞ ♣r♦❝❡❞✉r❡ ✐♥ ❛ ✇✐❞❡ r❛♥❣❡ ♦❢ ♥✉♠❡r✐❝❛❧ ❛♥❛❧②s✐s ♣r♦❜❧❡♠s✳ ▲❡t t❤❡ ✈❡❝t♦rs y ∈ Rp✱ β ∈ Rp ❛♥❞ φob

B ∈ Rp ✭❜♦❧❞ s②♠❜♦❧s✮ r❡♣r❡s❡♥t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦❧❧❡❝t✐♦♥s ♦❢ ✐♥❢♦r♠❛t✐♦♥✿

y =     y1 y2 ✳ ✳ ✳ yp     , β =     β1 β2 ✳ ✳ ✳ βp     , φob

B =

    φB(x1, t1) φB(x2, t2) ✳ ✳ ✳ φB(xp, tp)     . ❚❤❡s❡ r❡♣r❡s❡♥t ✭r❡s♣❡❝t✐✈❡❧②✮ t❤❡ ♦❜s❡r✈❛t✐♦♥s✱ t❤❡ ✭❛s ②❡t✮ ✉♥❦♥♦✇♥ ❝♦❡✣❝✐❡♥ts t❤❛t ✇❡ ❛r❡ tr②✐♥❣ t♦ ✜♥❞ ❛♥❞ t❤❡ ❜❛❝❦❣r♦✉♥❞ ✈❛❧✉❡s ❛t t❤❡ ♦❜s❡r✈❛t✐♦♥ ♣♦s✐t✐♦♥s ❛♥❞ t✐♠❡s✳ ❚❤❡ ❡q✉❛t✐♦♥s r❡♣r❡s❡♥t❡❞ ❜② ✭✸✼✮ ✭1 ≤ i ≤ p✮ ♠❛② ❜❡ ✇r✐tt❡♥ ✐♥ ❧✐♥❡❛r ❛❧❣❡❜r❛✐❝ ❢♦r♠✿ Wob

• φob

B − y

• + (WobP − I) β = 0,

✇❤❡r❡ P ∈ Rp×p ✐s P =     r1(x1, t1) r2(x1, t1) · · · rp(x1, t1) r1(x2, t2) r2(x2, t2) · · · rp(x2, t2) ✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳ r1(xp, tp) r2(xp, tp) · · · rp(xp, tp)     , ❛♥❞ I ∈ Rp×p ✐s t❤❡ ✐❞❡♥t✐t② ♠❛tr✐①✳ ❆❧❧ ♦❢ t❤❡s❡ ✈❡❝t♦rs ❛♥❞ ♠❛tr✐❝❡s ❛r❡ ❦♥♦✇♥ ❡①❝❡♣t ❢♦r β✳ Pr♦✈✐❞✐♥❣ t❤❛t t❤❡ ♠❛tr✐① WobP − I ✐s ❢✉❧❧ r❛♥❦✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥ ✐s ❢♦✉♥❞ t♦ ❜❡ β = Wob (WobP − I)−1 y − φob

B

• .

SLIDE 26

✷✻

SLIDE 27

✷✼

✷✭❝✮ ❈♦st ❢✉♥❝t✐♦♥s ❛♥❞ s✐♠♣❧✐✜❝❛t✐♦♥s ❢♦r ♦♣❡r❛t✐♦♥❛❧ ❛ss✐♠✐❧❛t✐♦♥

❲❡❛❦ ❝♦♥str❛✐♥t ✹❉✲❱❆❘ Jwc[x] = 1 2 [x(0) − xB(0)]T Pf−1 [x(0) − xB(0)] + 1 2

T

• t=0

[y(t) − ht (x(t))]T R−1

t

[y(t) − ht (x(t))] + 1 2

T

• t=1

T

• t′=1

[x(t) − Mt←t−1 (x(t − 1))] (Q−1)tt′ [x(t′) − Mt′←t′−1 (x(t′ − 1))] . ❍❡r❡ x ✐s ❝❛❧❧❡❞ t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ ❛♥❞ ✐s t❤❡ ✹❉ st❛t❡ ✈❡❝t♦r✿

SLIDE 28

✷✽ ❙✐♠♣❧✐✜❝❛t✐♦♥ ✸✿ ❆ss✉♠❡ t❤❛t t❤❡ ♥✉♠❡r✐❝❛❧ ♠♦❞❡❧ ✐s ♣❡r❢❡❝t ✭str♦♥❣ ❝♦♥str❛✐♥t ✹❉✲❱❆❘✮ Jsc[x] = 1 2(x − xB)TB−1(x − xB) + 1 2

T

• t=0

[y(t) − ht (Mt←0(x))]T R−1

t

× [y(t) − ht (M t←0(x))] , Mt←0(x) =

• Mt←t−1 (· · · M2←1 (M1←0(x)))

t > 0 I t = 0 . ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ♠❛❦✐♥❣ Qtt → 0 ✐♥ t❤❡ ✇❡❛❦ ❝♦♥str❛✐♥t ❝♦st ❢✉♥❝t✐♦♥✳

SLIDE 29

✷✾ ❙✐♠♣❧✐✜❝❛t✐♦♥ ✹✿ ■♥❝r❡♠❡♥t❛❧ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥

✛ ✚ ✘ ✙

x(t) = xref

k (t) + δx(t).

▲✐♥❡❛r✐③✐♥❣ t❤❡ ❢♦r❡❝❛st ♠♦❞❡❧✿ x(t) = Mt←t−1(x(t − 1)), xref

k (t) + δx(t) = Mt←t−1

• xref

k (t − 1) + δx(t − 1)

• ,

≃ Mt←t−1

• xref

k (t − 1)

• + Mt←t−1δx(t − 1),

δx(t) = Mt←t−1δx(t − 1), ✇❤❡r❡ t❤❡ r❡❢❡r❡♥❝❡ st❛t❡ xref

k (t) ≡ Mt←t−1

• xref

k (t − 1)

• ,

❛♥❞ Mt←t−1 ≡ ∂Mt←t−1 (x(t − 1)) ∂x(t − 1)

• xref

k

∈ Rn×n, ✇✐t❤ ♠❛tr✐① ❡❧❡♠❡♥ts {Mt←t−1}ij = ∂ {Mt←t−1 (x(t − 1))}i ∂ {x(t − 1)}j

• xref

k

. ▲✐♥❡❛r✐③✐♥❣ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦♣❡r❛t♦r✿ ymo(t) = ht(x(t)), = ht

• xref

k (t) + δx(t)

• ,

≃ ht

• xref

k (t)

• + Htδx(t),

≃ ymo

ref,k(t) + Htδx(t),

δymo(t) = Htδx(t), ✇❤❡r❡ δymo(t) ≡ ymo(t) − ymo

ref,k(t),

❛♥❞ Ht ≡ ∂ht(x(t)) ∂x(t)

• xref

k

∈ Rp×n, ✇✐t❤ ♠❛tr✐① ❡❧❡♠❡♥ts {Ht}ij = ∂ {ht(x(t))}i ∂ {x(t)}j

• xref

k

.

SLIDE 30

✸✵ ❇② ✇r✐t✐♥❣ t❤❡ ❜❛❝❦❣r♦✉♥❞ ❛s ❛ ♣❡rt✉r❜❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ r❡❢❡r❡♥❝❡ st❛t❡✱ xB(t) ≡ xref

k (t) + δxB(t)✱ ❛♥❞ ❞❡✜♥✐♥❣

δy(t) ≡ y(t) − ht

• Mt←0(xref

k )

• ✱ t❤❡ str♦♥❣ ❝♦♥str❛✐♥t ❝♦st ❢✉♥❝t✐♦♥ ❜❡❝♦♠❡s✿

✬ ✫ ✩ ✪

J4Dinc[δx] = 1 2(δx − δxB)TB−1(δx − δxB) + 1 2

T

• t=0

[δy(t) − HtMt←0δx]T R−1

t

[δy(t) − HtMt←0δx] .

• ❚❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ ✐s δx = δx(0) ✐♥ t❤✐s ✐♥❝r❡♠❡♥t❛❧ ❢♦r♠✉❧❛t✐♦♥✳
• ▲❛t❡r ✇❡ ✇✐❧❧ ❝❛❧❧ δy(t) − HtMt←0δx t❤❡ r❡s✐❞✉❛❧ ✈❡❝t♦r✱ r(t)✳
• J4Dinc[δx] ✐s ❡①❛❝t❧② q✉❛❞r❛t✐❝ ❛♥❞ s♦ ✐s ❡❛s✐❡r t♦ ♠✐♥✐♠✐③❡ t❤❛♥ J4D[δx]✳
• ■❢ t❤❡ ✈❛❧✉❡ ♦❢ δx t❤❛t ♠✐♥✐♠✐③❡s t❤✐s ✐s δxA ✭✬✐♥♥❡r ❧♦♦♣✬✮✱ t❤❡♥ t❤❡ ❛♥❛❧②s✐s ✐s

xA = xref

k + δxA.

• ❙❡t xref

k+1(t) = xA ❛♥❞ r❡♣❡❛t ✭✬♦✉t❡r ❧♦♦♣✬✮✳

SLIDE 31

✸✶ ❙✐♠♣❧✐✜❝❛t✐♦♥ ✻✿ ✸❉✲❱❆❘

SLIDE 32

✸✷

✸✳ ❆✲♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ ❇✲♠❛tr✐①

✸✭❛✮ ❚❤❡ ♥✉❧❧ s♣❛❝❡ ♦❢ t❤❡ ♦❜s❡r✈❛t✐♦♥ ♦♣❡r❛t♦r ❛♥❞ t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ❛✲♣r✐♦r✐ ✐♥❢♦r♠❛t✐♦♥

SLIDE 33

✸✸ P❤②s✐❝❛❧ ❡①❛♠♣❧❡ ♦❢ ❛♥ ♦❜s❡r✈❛t✐♦♥ ♦♣❡r❛t♦r ❛♥❞ ♥✉❧❧ s♣❛❝❡ ▲❡t x =

• u

v

• =
• ✉♥✐❢♦r♠ ③♦♥❛❧ ✇✐♥❞ − ↔+

✉♥✐❢♦r♠ ♠❡r✐❞✐♦♥❛❧ ✇✐♥❞ +

−

• ,

y = ♠❡❛s✉r❡♠❡♥t ♦❢ ✇✐♥❞ ❝♦♠♣♦♥❡♥t ✐♥ ❛ ❞✐r❡❝t✐♦♥ θ ❢r♦♠ ❊, σ2

y = ❊rr♦r ✈❛r✐❛♥❝❡ ♦❢ ♠❡❛s✉r❡♠❡♥t.

❚❤✐s ♠❡❛s✉r❡♠❡♥t ✐s ❣✐✈❡♥ ❡✳❣✳ ❜② ❛ ❉♦♣♣❧❡r r❛❞❛r ✐♥str✉♠❡♥t✳ ˆ u ✐s t❤❡ ✉♥✐t ✈❡❝t♦r ✐♥ t❤❡ ❧✐♥❡ ♦❢ s✐❣❤t ♦❢ t❤❡ r❛❞❛r ❜❡❛♠✱ ˆ u =

• cos θ

sin θ

• .

SLIDE 34

✸✹

✸✭❜✮ ❚❤❡ r♦❧❡ ♦❢ t❤❡ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①

SLIDE 35

✸✺

✸✭❝✮ ❙♣❛t✐❛❧ ❛s♣❡❝ts ✭✐♥✈❡rs❡ ▲❛♣❧❛❝✐❛♥s✱ ❞✐✛✉s✐♦♥ ♦♣❡r❛t♦rs✮

■♥✈❡rs❡ ▲❛♣❧❛❝✐❛♥s ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦r♠ ♦❢ ❛ B✲♠❛tr✐① ❢♦r ❛ s✐♥❣❧❡ ✜❡❧❞ ✭✉♥✐✈❛r✐❛t❡✮✿ COR = γ

• I + l4

2 (∇2)2 −1 , B = COV = Σγ

• I + l4

2 (∇2)2 −1 Σ, B−1 = γ−1Σ−1

• I + l4

2 (∇2)2

• Σ−1,

✭✇❤❡r❡ l ✐s t❤❡ ✭❝❤♦s❡♥✮ ❝♦rr❡❧❛t✐♦♥ ❧❡♥❣t❤✲s❝❛❧❡✮✳ ❲❤❛t ✐s t❤❡ r❡s✉❧t ♦❢ ❛❝t✐♥❣ ✇✐t❤ COR ♦♥ t❤❡ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ f(x) ✐♥ ✶✲❉❄ ▲❡t g(x) = COR{f(x)} = γ

• 1 + l4

2 d4 dx4 −1 f(x). ❚❤✐s ❝❛♥ ❜❡ ❡❛s✐❧② s♦❧✈❡❞ ✐♥ ❋♦✉r✐❡r s♣❛❝❡✿ f(x) = 1 √ 2π ˆ dk ¯ f(k)eikx g(x) = 1 √ 2π ˆ dk ¯ g(k)eikx, f(x) = γ−1

• 1 + l4

2 d4 dx4

• g(x),

ˆ dk ¯ f(k)eikx = γ−1

• 1 + l4

2 d4 dx4 ˆ dk ¯ g(k)eikx, = ˆ dk ¯ g(k)γ−1

• 1 + l4k4

2

• eikx.

▼✉❧t✐♣❧② ❡❛❝❤ s✐❞❡ ❜② e−ik′x✱ ✐♥t❡❣r❛t❡ ♦✈❡r x✱ ❛♥❞ ✉s❡ ♦rt❤♦❣♦✲ ♥❛❧✐t② ♦❢ ❝♦♠♣❧❡① ❡①♣♦♥❡♥t✐❛❧s✿ ¯ g(k) = γ

• 1 + l4k4

2 −1 ¯ f(k). ■♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ t❤✐s t♦ ❣❡t t❤❡ r❡s✉❧t ✐♥ x✲s♣❛❝❡✿ g(x) = I.F.T.

• γ
• 1 + l4k4

2 −1 ¯ f(k)

• ,

= I.F.T.

• ¯

c(k) ¯ f(k)

• ,

= 1 2π ˆ dx′ c(x − x′)f(x′), ❜② t❤❡ ❝♦♥✈♦❧✉t✐♦♥ t❤❡♦r❡♠ ♦❢ ❋♦✉r✐❡r tr❛♥s❢♦r♠s✳ c(x) ✐s t❤❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ γ/(1 + l4k4/2)✳

SLIDE 36

✸✻

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 60
• 40
• 20

20 40 60 wavenumber 1 / (1 + l^4 k^4) [small l] 0.2 0.4 0.6 0.8 1

• 60
• 40
• 20

20 40 60 wavenumber 1 / (1 + l^4 k^4) [large l] 0.2 0.4 0.6 0.8 1

• 3
• 2
• 1

1 2 3 Relative position IFT{1 / (1 + l^4 k^4)} [small l] 0.7 0.75 0.8 0.85 0.9 0.95 1

• 3
• 2
• 1

1 2 3 Relative position IFT{1 / (1 + l^4 k^4)} [large l]

(keep all but smallest scales) Filter away very large wavenumbers only Filter away all except very small wavenumbers (keep large−scales) SMALL LENGTHSCALE LARGE LENGTHSCALE SPECTRAL SPACE REAL SPACE

SLIDE 37

✸✼ ❉✐✛✉s✐♦♥ ♦♣❡r❛t♦rs ❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❢♦r ✐♥t❡❣r❛t✐♦♥ ❢r♦♠ t = 0 t♦ T✿ ∂g(x, t) ∂t − κ∂2g(x, t) ∂x2 = 0, κ ✿ ❞✐✛✉s✐♦♥ ❝♦✲❡✣❝✐❡♥t✱ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ g(x, 0) = f(x). ❚❤❡ ❞✐✛✉s✐♦♥ ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ✐♥t❡❣r❛t❡❞ ❛♥❛❧②t✐❝❛❧❧② ✐♥ ❋♦✉r✐❡r s♣❛❝❡✳ ❋♦r ✇❛✈❡♥✉♠❜❡r k✿ ∂¯ g(k, t) ∂t + κk2¯ g(k, t), ¯ g(k, 0) = ¯ f(k). ■♥t❡❣r❛t❡ ❢r♦♠ t = 0 t♦ T✿ ˆ T

t=0

d ln ¯ g(k, t) + κk2 ˆ T

t=0

dt = 0, ln ¯ g(k, T) − ln ¯ g(k, 0) + κk2T = 0, ¯ g(k, T) = ¯ f(k) exp(−κk2T). ❚♦ ✜♥❞ t❤❡ s♦❧✉t✐♦♥ ✐♥ r❡❛❧ s♣❛❝❡✱ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ t❤❡ ❛❜♦✈❡✳ ❚❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡ ✐s ❛ ♣r♦❞✉❝t ♦❢ ❢✉♥❝t✐♦♥s ✐♥ ❋♦✉r✐❡r s♣❛❝❡✱ s♦ ✉s❡ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ t❤❡♦r❡♠ ❛❣❛✐♥✿ g(x, T) = 1 2π ˆ dx′ f(x′) c(x − x′). c(x) ✐s ❤❡r❡ t❤❡ ✐♥✈❡rs❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ exp(−κk2T)✱ ✇❤✐❝❤ ✐s

• π/κT exp(−x2/4κT) ✭❛ ●❛✉ss✐❛♥ ❢✉♥❝t✐♦♥ ✇✐t❤ ❧❡♥❣t❤✲

s❝❛❧❡ √ 2κT✮✳ ❚❤❡ s♦❧✉t✐♦♥ ✐s t❤✉s✿ g(x, T) = 1 √ 4πκT ˆ dx′ f(x′) exp(−(x − x′)2/4κT). ◆♦t❡ t❤❡ ❝♦rr❡s♣♦♥❞❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ❝♦♥✈♦❧✉t✐♦♥ ❛♥❞ ❛❝t✐♦♥ ✇✐t❤ ❛ ❤♦♠♦❣❡♥❡♦✉s ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✭❛s ✐♥ t❤❡ ♣r❡✈✐♦✉s s❡❝t✐♦♥ ♦♥ ✐♥✈❡rs❡ ▲❛♣❧❛❝✐❛♥s✮✱ ✇❤✐❝❤ ♠❡❛♥s t❤❛t t❤❡ str✉❝t✉r❡ ❢✉♥❝t✐♦♥s ❤❛✈❡ t❤❡ ❢♦r♠✿ 1 √ 4πκT exp(−(x − x′)2/4κT).

SLIDE 38

✸✽

✸✭❞✮ ▼✉❧t✐✈❛r✐❛t❡ ❛s♣❡❝ts ❛♥❞ ❜❛❧❛♥❝❡

❊①❛♠♣❧❡ ✇✐t❤ ♣❡r❢❡❝t ❣❡♦str♦♣❤✐❝ ❜❛❧❛♥❝❡ ❋♦r ✢♦✇s ✇✐t❤ s♠❛❧❧ ❘♦ss❜② ♥✉♠❜❡r✱ Ro = U/fL ≪ 1✱ t❤❡ ♠♦♠❡♥t✉♠ ❡q✉❛t✐♦♥s ❛♣♣r♦①✐♠❛t❡ t♦ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣♥♦st✐❝ ❡q✉❛t✐♦♥s✿ v = 1 fρ ∂p ∂x, u = − 1 fρ ∂p ∂y, ✭t❤✐s ✐s ❣❡♦str♦♣❤✐❝ ❜❛❧❛♥❝❡✮✳

SLIDE 39

✸✾ ◆♦✇ ❞❡r✐✈❡ t❤❡ ♠✉❧t✐✈❛r✐❛t❡ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s ❜❡t✇❡❡♥ ♣♦s✐t✐♦♥s i ❛♥❞ j✿ p − p ❝♦✈s✿ δpiδpj = σ2

pµij ✭❜② ❞❡✜♥✐t✐♦♥✮,

p − u ❝♦✈s✿ δpiδuj = − 1 fρ

• δpi

∂δpj ∂yj

• = − 1

fρ ∂ ∂yj δpiδpj = −σ2

p

fρ ∂µij ∂yj , p − v ❝♦✈s✿ δpiδvj = 1 fρ

• δpi

∂δpj ∂xj

• = 1

fρ ∂ ∂xj δpiδpj = σ2

p

fρ ∂µij ∂xj , u − p ❝♦✈s✿ δuiδpj = − 1 fρ ∂δpi ∂yi δpj

• = − 1

fρ ∂ ∂yi δpiδpj = −σ2

p

fρ ∂µij ∂yi , u − u ❝♦✈s✿ δuiδuj = 1 f 2ρ2 ∂δpi ∂yi ∂δpj ∂yj

• =

1 f 2ρ2 ∂2 ∂yi∂yj δpiδpj = σ2

p

f 2ρ2 ∂2µij ∂yi∂yj , u − v ❝♦✈s✿ δuiδvj = − 1 f 2ρ2 ∂δpi ∂yi ∂δpj ∂xj

• = − 1

f 2ρ2 ∂2 ∂yi∂xj δpiδpj = − σ2

p

f 2ρ2 ∂2µij ∂yi∂xj , v − p ❝♦✈s✿ δviδpj = 1 fρ ∂δpi ∂xi δpj

• = 1

fρ ∂ ∂xi δpiδpj = σ2

p

fρ ∂µij ∂xi , v − u ❝♦✈s✿ δviδuj = − 1 f 2ρ2 ∂δpi ∂xi ∂δpj ∂yj

• = − 1

f 2ρ2 ∂2 ∂xi∂yj δpiδpj = − σ2

p

f 2ρ2 ∂2µij ∂xi∂yj , v − v ❝♦✈s✿ δviδvj = 1 f 2ρ2 ∂δpi ∂xi ∂δpj ∂xj

• =

1 f 2ρ2 ∂2 ∂xi∂xj δpiδpj = σ2

p

f 2ρ2 ∂2µij ∂xi∂xj .

SLIDE 40

✹✵ ◆♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ✜rst ❛♥❞ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ µ✿ ∂µij ∂xi = −µij (xi − xj) L2 ∂µij ∂xj = µij (xi − xj) L2 , ∂µij ∂yi = −µij (yi − yj) L2 , ∂µij ∂yj = µij (yi − yj) L2 , ∂2µij ∂xi∂xj = µij L2

• 1 − (xi − xj)2

L2

• ,

∂2µij ∂yi∂yj = µij L2

• 1 − (yi − yj)2

L2

• ,

∂2µij ∂yi∂xj = −µij (xi − xj)(yi − yj) L4 , ∂2µij ∂xi∂yj = −µij (xi − xj)(yi − yj) L4 . ❊①❛♠♣❧❡ str✉❝t✉r❡ ❢✉♥❝t✐♦♥s ❣✐✈✐♥❣ t❤❡ ♦✉t♣✉t ✜❡❧❞ ✭p✱ u ♦r v ❞♦✇♥ t❤❡ s✐❞❡✮ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❛ ♣♦✐♥t ✐♥ t❤❡ ❝❡♥tr❡ ♦❢ t❤❡ ❞♦✲ ♠❛✐♥ ✭❡✐t❤❡r ♦❢ p✱ u ♦r v ❛❧♦♥❣ t❤❡ t♦♣✮✳ ❘❡❞ ✐s ♣♦s✐t✐✈❡✱ ❜❧✉❡ ✐s ♥❡❣❛t✐✈❡✳

SLIDE 41

✹✶

✸✭❡✮ ❈♦♥tr♦❧ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠s ❛♥❞ t❤❡ ✐♠♣❧✐❡❞ ❇✲♠❛tr✐①

✬ ✫ ✩ ✪

❙♦❧✈✐♥❣ ❛ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✉s✐♥❣ ❈❱❚s ✐♥✈♦❧✈❡s t❤❡ ❢♦❧❧♦✇✐♥❣ st❡♣s✿

• ❆ss✉♠❡ t❤❛t ✇❡ ❦♥♦✇ t❤❡ ❈❱❚✱ U✱ ❛♥❞ ✐ts ❛❞❥♦✐♥t ❛♥❞ t❤❛t t❤❡② ❛r❡ ♣r❛❝t✐❝❛❧ t♦ ❛♣♣❧②✳
• ▼✐♥✐♠✐③❡ J[δχ] ✇✐t❤ r❡s♣❡❝t t♦ ✈❛r②✐♥❣ δχ✳ ❚❤❡ ❝♦st ❢✉♥❝t✐♦♥ ✐s✿

J[δχ] = 1 2δχTδχ + 1 2

T

• t=0

[δy(t) − HtMt←0Uδχ]T R−1

t

[δy(t) − HtMt←0Uδχ] .

• ❚❤❡ ❛♥❛❧②s✐s ✐♥❝r❡♠❡♥t ✐♥ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ s♣❛❝❡ t❤❛t ♠✐♥✐♠✐③❡s t❤❡ ❛❜♦✈❡ ✐s δχA✳
• ❚❤❡ ❛♥❛❧②s✐s ✐♥ ♠♦❞❡❧ s♣❛❝❡ ✐s xA = xB + UδχA✳
• ❚❤✐s ✐s ❡q✉✐✈❛❧❡♥t t♦ ♠✐♥✐♠✐③✐♥❣ t❤❡ ♦r✐❣✐♥❛❧ ❝♦st ❢✉♥❝t✐♦♥ J[δx] ✇✐t❤ t❤❡ ✐♠♣❧✐❡❞ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①

Bimp = UUT✳

SLIDE 42

✹✷ ❊①❛♠♣❧❡ ♦❢ t❤❡ ❈❱❚ ♠❡t❤♦❞ t♦ ♠♦❞❡❧ ❤♦r✐③♦♥t❛❧ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s ✭❡✳❣✳ ❢♦r ♣r❡ss✉r❡✱ ♣✮ ❙❡❡ ❋✐❣✳ ❢♦r ❞❡✜♥✐t✐♦♥s ♦❢ ❛♥❣❧❡s ❛♥❞ ❧❡♥❣t❤s ✐♥ r❡❛❧ ❛♥❞ ❋♦✉r✐❡r s♣❛❝❡s✳ ◆♦t❡ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ∆r =

• ∆x

∆y

• = ∆r
• cos θx

sin θx

• ,

k =

• kx

ky

• = K
• cos θk

sin θk

• ,

dk = KdKdθk. ❲❤❛t ✐s t❤❡ s❛✈✐♥❣ ♦❢ t❤✐s ❈❱❚ ♠❡t❤♦❞ ♦❢ ♠♦❞❡❧❧✐♥❣ ❇ ❝♦♠♣❛r❡❞ t♦ ❛♥ ❡①♣❧✐❝✐t ♠❛tr✐① ♠❡t❤♦❞❄

• ◆♦✳ ♦❢ ❣r✐❞ ♣♦✐♥ts✿ nx × ny✳
• ◆♦✳ ♦❢ ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐♥ δx✿ 3 × nx × ny✳
• ◆♦✳ ♦❢ ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✐♥ δχ✿ nx × ny✳
• ◆♦✳

♦❢ ✐♥❞❡♣❡♥❞❡♥t ❡❧❡♠❡♥ts ✐♥ ❡①♣❧✐❝✐t B✿ ∼

1 2 (3 × nx × ny)2 ∼ 9 2n4 x ✭❛ss✉♠✐♥❣ nx ∼ ny✮✳

• ◆♦✳

♦❢ ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ♥❡❡❞❡❞ ❢♦r ❈❱❚✿ ∼ ◆♦✳ ♦❢ t♦t❛❧ ✇❛✈❡♥✉♠❜❡rs ♥❡❡❞❡❞ t♦ ❦♥♦✇ λp(K) ∼ √ 2nx✳ ■❢ nx = 1000✱ t❤❡♥

• ◆♦✳ ♦❢ ✐♥❞❡♣❡♥❞❡♥t ❡❧❡♠❡♥ts ✐♥ ❡①♣❧✐❝✐t B✿ ∼ 5 × 1012✳
• ◆♦✳ ♦❢ ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ♥❡❡❞❡❞ ❢♦r ❈❱❚✿ ∼ 1500✳

SLIDE 43

✹✸ ❖♣❡r❛t✐♦♥❛❧ ❈❱❚s

• ❚❤❡ ▼❡t ❖✣❝❡ ✉s❡ ❛ s✐♠✐❧❛r ❛♣♣r♦❛❝❤ ✐♥ ✐ts ♦♣❡r❛t✐♦♥❛❧ ✹❉✲❱❆❘ ❛♥❞ ✸❉❋●❆❚ s②st❡♠s✳ ●❡♦str♦♣❤✐❝ ❜❛❧❛♥❝❡ ✭✐♠♣♦s❡❞

✇❡❛❦❧②✮ ❛♥❞ ❤②❞r♦st❛t✐❝ ❜❛❧❛♥❝❡ ❛r❡ ✉s❡❞✳ ❚❤❡ s♣❛t✐❛❧ ❝♦♠♣♦♥❡♥t ✐♥❝❧✉❞❡s ❛ s✐♠✐❧❛r ❛♣♣r♦❛❝❤ ❛s s❤♦✇♥ ❛❜♦✈❡ ✭s♣❡❝tr❛❧ s♣❛❝❡✮ ❢♦r t❤❡ ❤♦r✐③♦♥t❛❧ str✉❝t✉r❡ ♦❢ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s✱ ❛♥❞ ✈❡rt✐❝❛❧ ♠♦❞❡s ✭❡♠♣✐r✐❝❛❧ ♦rt❤♦❣♦♥❛❧ ❢✉♥❝t✐♦♥s✮ ❢♦r t❤❡ ✈❡rt✐❝❛❧ str✉❝t✉r❡✳ ▲♦r❡♥❝ ❆✳❈✳✱ ❇❛❧❧❛r❞ ❙✳P✳✱ ❇❡❧❧ ❘✳❙✳✱ ■♥❣❧❡❜② ◆✳❇✳✱ ❆♥❞r❡✇s P✳▲✳❋✳✱ ❇❛r❦❡r ❉✳▼✳✱ ❇r❛② ❏✳❘✳✱ ❈❧❛②t♦♥ ❆✳▼✳✱ ❉❛❧❜② ❚✳✱ ▲✐ ❉✳✱ P❛②♥❡ ❚✳❏✳✱

❙❛✉♥❞❡rs ❋✳❲✳✱ ❚❤❡ ▼❡t ❖✣❝❡ ❣❧♦❜❛❧ ✸✲❞✐♠❡♥s✐♦♥❛❧ ✈❛r✐❛t✐♦♥❛❧ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ s❝❤❡♠❡✱ ◗✳❏✳❘✳▼❡t❡♦r✳❙♦❝✳ ✶✷✻ ♣♣✳✷✾✾✶✲✸✵✶✷ ✭✷✵✵✵✮✳

• ❚❤❡ ❊❈▼❲❋ ✉s❡ s✐♠✐❧❛r ❜❛❧❛♥❝❡ r❡❧❛t✐♦♥s❤✐♣s✱ ❜✉t ✉s❡ ❛ s♣❛t✐❛❧ ❝♦♠♣♦♥❡♥t t❤❛t ♠❛❦❡s ✉s❡ ♦❢ ✇❛✈❡❧❡ts✳ ❋✐s❤❡r ▼✳✱ ❆♥❞❡rss♦♥ ❊✳✱

❉❡✈❡❧♦♣♠❡♥ts ✐♥ ✹❞✲❱❛r ❛♥❞ ❑❛❧♠❛♥ ✜❧t❡r✐♥❣✱ ❊❈▼❲❋ ❘❡s❡❛r❝❤ ❘❡♣♦rt ◆♦✳ ✸✹✼ ♣♣✳✸✻ ✭✷✵✵✶✮✳

• ❚❤❡ ❞✐✛✉s✐♦♥ ♦♣❡r❛t♦r ❛♣♣r♦❛❝❤ ✐s ✉s❡❞ ✐♥ ♦❝❡❛♥ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ s②st❡♠s✳

❲❡❛✈❡r ❆✳❚✳✱ ❉❡❧t❡❧ ❈✳✱ ▼❛❝❤✉ ❊✳✱ ❘✐❝❝✐ ❙✳✱ ❉❛❣❡t ◆✳✱ ❆ ♠✉❧t✐✈❛r✐❛t❡ ❜❛❧❛♥❝❡ ♦♣❡r❛t♦r ❢♦r ✈❛r✐❛t✐♦♥❛❧ ♦❝❡❛♥ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✱ ◗✳❏✳❘✳▼❡t❡♦r✳❙♦❝✳ ✶✸✶ ♣♣✳✸✻✵✺✲✸✻✷✻ ✭✷✵✵✺✮✳

SLIDE 44

✹✹ ✸✭❢✮ ❈♦♥❞✐t✐♦♥✐♥❣ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❚❤❡ r❛t❡ ♦❢ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✐s ❛✛❡❝t❡❞ str♦♥❣❧② ❜② t❤❡ ❝♦♥❞✐t✐♦♥✐♥❣ ♦❢ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠✳ ❈♦♥s✐❞❡r t❤❡ ❝❛s❡ ✇❤❡♥ δx ✐s t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡✳ ❆ ❚❛②❧♦r ❡①♣❛♥s✐♦♥ ♦❢ J(x) ✇✐t❤ r❡s♣❡❝t t♦ ♣❡rt✉r❜❛t✐♦♥s δx ❛❜♦✉t x ✐s✿ J(x + δx) = J(x) + ∂J ∂δx

• x

δx + 1 2δxT ∂2J ∂δx2

• x

δx. ❣r❛❞✐❡♥t ❍❡ss✐❛♥ ✈❡❝t♦r ♠❛tr✐① (1 × 1) (1 × 1) (1 × n)(n × 1) (1 × n)(n × n)(n × 1) ❚❤❡ ❍❡ss✐❛♥ ♠❛tr✐① ✐s ❛♥ n × n ♠❛tr✐① t❤❛t ❞❡s❝r✐❜❡s ❛❧❧ ♣♦ss✐❜❧❡ s❡❝♦♥❞ ❞❡r✐✈❛t✐✈❡s ♦❢ J ✇✐t❤ r❡s♣❡❝t t♦ t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ ❡❧❡♠❡♥ts✿ ∂2J ∂δx2 =      

∂2J ∂x2

1

∂2J ∂x1∂x2

· · ·

∂2J ∂x1∂xn ∂2J ∂x2∂x1 ∂2J ∂x2

2

· · ·

∂2J ∂x2∂xn

✳ ✳ ✳ ✳ ✳ ✳ ✳✳✳ ✳ ✳ ✳

∂2J ∂xn∂x1 ∂2J ∂xn∂x2

· · ·

∂2J ∂x2

n

      , ❛♥❞ ❞❡s❝r✐❜❡s t❤❡ ❡❝❝❡♥tr✐❝✐t② ❛♥❞ ♦r✐❡♥t❛t✐♦♥ ♦❢ t❤❡ ❡❧❧✐♣s♦✐❞s t❤❛t ❞❡s❝r✐❜❡ s✉r❢❛❝❡s ♦❢ ❝♦♥st❛♥t J ✐♥ ♣❤❛s❡ s♣❛❝❡✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ ❝♦♥❞✐t✐♦♥ ♥✉♠❜❡r ✐s ✐♠♣♦rt❛♥t✿ κ = ❝♦♥❞✐t✐♦♥ ♥✉♠❜❡r = ♠❛①✐♠✉♠ ❡✐❣❡♥✈❛❧✉❡ ♦❢ t❤❡ ❍❡ss✐❛♥ ♠✐♥✐♠✉♠ ❡✐❣❡♥✈❛❧✉❡ ♦❢ t❤❡ ❍❡ss✐❛♥ .

SLIDE 45

✹✺

• ■❢ κ ≈ 1✱ t❤❡♥ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✐s ✇❡❧❧ ❝♦♥❞✐t✐♦♥❡❞ ❛♥❞ ✐t ✇✐❧❧ ❜❡ ♣♦ss✐❜❧❡ ❢♦r t❤❡ s♦❧✉t✐♦♥ t♦ ❜❡ ❢♦✉♥❞ t♦ ❛ ❤✐❣❤

❛❝❝✉r❛❝②✳

• ■❢ κ ≫ 1✱ t❤❡♥ t❤❡ ✈❛r✐❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ✇✐❧❧ ❝♦♥✈❡r❣❡ s❧♦✇❧② ❛♥❞ ✐t ✐s ❤❛r❞ ❢♦r t❤❡ s♦❧✉t✐♦♥ t♦ ❜❡ ❢♦✉♥❞ t♦ ❛ ❤✐❣❤ ❛❝❝✉r❛❝②✳

❚❤❡ ❢♦❧❧♦✇✐♥❣ t❛❜❧❡ ❝♦♠♣❛r❡s ✇❡❛❦ ❝♦♥str❛✐♥t ✹❉✲❱❆❘ ✇✐t❤ δχ ❛♥❞ δx ❛s t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡✳ δχ δx ❍❡ss✐❛♥ I + T

t=0 UTMT t←0HT t R−T t

HtMt←0U B−1 + T

t=0 MT t←0HT t R−T t

HtMt←0 ♠✐♥ ❡✐❣❡♥✈❛❧✉❡ λχ

min 1

λx

min ≥ 0

♠❛① ❡✐❣❡♥✈❛❧✉❡ λχ

max

λx

max ≫ 1 ✐♥ ♣r❛❝t✐❝❡

❝♦♥❞✐t✐♦♥ ◆♦✳ λχ

max/1 ∼ λχ max

λx

max/0+ → ∞

SLIDE 46

✹✻

✹✳ ❖♣❡r❛t✐♦♥❛❧ ❛❧❣♦r✐t❤♠s

✬

SLIDE 47

✹✼

✺✳ ▼❡❛s✉r✐♥❣ t❤❡ ❇✲♠❛tr✐①

✺✭❛✮ ❆♥❛❧②s✐s ♦❢ ✐♥♥♦✈❛t✐♦♥s

(b) (a)

• ❚❤❡ ❍✰▲ ♠❡t❤♦❞ ✇❛s ♣♦♣✉❧❛r ✐♥ t❤❡ ✶✾✽✵s ❛♥❞ ✶✾✾✵s✳
• ■t r❡♣❧✐❡s ♦♥ ❛ ❤✉❣❡ ♥✉♠❜❡r ♦❢ ❞✐r❡❝t ✭✐♥✲s✐t✉✮ ♦❜s❡r✈❛t✐♦♥s✳
• ◆♦t ✉s❡❢✉❧ ✐♥ ♣r❛❝t✐❝❡ t♦ ♣r♦❜❡ ✢♦✇ ❞❡♣❡♥❞❡♥❝❡ ♦❢ B✱ ♦r B ✐♥ ✉♥♦❜s❡r✈❡❞ r❡❣✐♦♥s✳
• ❍♦❧❧✐♥❣s✇♦rt❤ ❆✳✱ ▲♦♥♥❜❡r❣ P✳✱ ❚❤❡ st❛t✐st✐❝❛❧ str✉❝t✉r❡ ♦❢ s❤♦rt✲r❛♥❣❡ ❢♦r❡❝❛st ❡rr♦rs ❛s ❞❡t❡r♠✐♥❡❞ ❢r♦♠ r❛❞✐♦s♦♥❞❡ ❞❛t❛✳ P❛rt ■✿ ❚❤❡ ✇✐♥❞ ✜❡❧❞✱ ❚❡❧❧✉s ✸✽❆ ♣♣✳✶✶✶✲✶✸✻

✭✶✾✽✻✮✳ ▲♦♥♥❜❡r❣ P✳✱ ❍♦❧❧✐♥❣s✇♦rt❤ ❆✳✱ ❚❤❡ st❛t✐st✐❝❛❧ str✉❝t✉r❡ ♦❢ s❤♦rt✲r❛♥❣❡ ❢♦r❡❝❛st ❡rr♦rs ❛s ❞❡t❡r♠✐♥❡❞ ❢r♦♠ r❛❞✐♦s♦♥❞❡ ❞❛t❛✳ P❛rt ■■✿ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♦❢ ❤❡✐❣❤t ❛♥❞ ✇✐♥❞ ❡rr♦rs✱ ❚❡❧❧✉s ✸✽❆ ♣♣✳✶✸✼✲✶✻✶ ✭✶✾✽✻✮✳

SLIDE 48

✹✽

✺✭❜✮ ❚❤❡ ◆▼❈ ♠❡t❤♦❞

Pr♦♣♦s❡ ❛ ♣r♦①② ❢♦r ❢♦r❡❝❛st ❡rr♦r✿ ηNMC ≈ x48

f (0) − x24 f (0).

SLIDE 49

✹✾

✺✭❝✮ ▼♦♥t❡✲❈❛r❧♦ ✭❡♥s❡♠❜❧❡✮ ♠❡t❤♦❞

• ❡♥❡r❛t❡ ❛♥ ❡♥s❡♠❜❧❡ t❤❛t ✐❞❡❛❧❧② s✐♠✉❧❛t❡s ❛❧❧ ❦♥♦✇♥

s♦✉r❝❡s ♦❢ ❢♦r❡❝❛st ❡rr♦r✳ ❋♦r t❤❡ it❤ ❡♥s❡♠❜❧❡ ♠❡♠❜❡r ✭1 ≤ i ≤ N✮✿ x(i)(t + δt) = Mt+δt←t

• x(i)(t)
• + e(i)(t),

✐♥t❡❣r❛t❡❞ ❢r♦♠ t = −T t♦ t = 0✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ s♦✉r❝❡s ♦❢ ❡rr♦r ❛r❡ ❝♦♥s✐❞❡r❡❞✿

• ■♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❡rr♦r✱ δx(i)

A (−T)✱ ❡✳❣✳✿

x(i)(−T) = xA(−T) + δx(i)

A (−T),

✇❤❡r❡ 1 N − 1

N

• i=1

δx(i)

A (−T)δx(i)T A (−T) ≈ PA(−T).

❆❧❧ ❡rr♦rs ✐♥❤❡r✐t❡❞ ❢r♦♠ ♣r❡✈✐♦✉s ❉❆ ❝②❝❧❡s ❛r❡ r❡♣r❡s❡♥t❡❞ ❛s ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥ ❡rr♦rs✳

• ▼♦❞❡❧ ❡rr♦r✱ t❤❡ ✐♥t❡❣r❛t❡❞ ❡✛❡❝t ♦❢ e(i)(t)✳ ❚❤❡ ♠♦❞❡❧

❡rr♦r ✐s ✉♥❦♥♦✇♥✱ ❜✉t ❝❛♥ ❜❡ ✐♥❝❧✉❞❡❞ st♦❝❤❛st✐❝❛❧❧② ❞✉r✲ ✐♥❣ t❤❡ ✐♥t❡❣r❛t✐♦♥ ♦❢ t❤❡ ♠♦❞❡❧✳ Pr❛❝t✐❝❛❧ ♠❡t❤♦❞s ♦❢ ✐♠♣❧✐❝✐t❧② ❛♣♣r♦①✐♠❛t✐♥❣ ♠♦❞❡❧ ❡rr♦r ✐♥❝❧✉❞❡✿ ✕ ▼✉❧t✐✲♠♦❞❡❧✴♠✉❧t✐✲♣❤②s✐❝s ♠❡t❤♦❞s ✭t❤❡s❡ ✉s❡ ❞✐❢✲ ❢❡r❡♥t ♠♦❞❡❧s✱ ❞✐✛❡r❡♥t ♣❛r❛♠❡t❡r✐③❛t✐♦♥s ♦r ❞✐✛❡r✲ ❡♥t ♣❛r❛♠❡t❡r ✈❛❧✉❡s ♦❢ t❤❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥s ❢♦r ❡❛❝❤ ❡♥s❡♠❜❧❡ ♠❡t❤♦❞s t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❡✛❡❝t ♦❢ e(i)(t)✮✳ ✕ ❙t♦❝❤❛st✐❝ ❦✐♥❡t✐❝ ❡♥❡r❣② ❜❛❝❦s❝❛tt❡r ✭❙❑❊❇✮ ♠❡t❤✲ ♦❞s ✭❢♦r❡❝❛st ♠♦❞❡❧s ❞♦ ♥♦t r❡♣r❡s❡♥t t❤❡ ❡♥❡r❣② ✇❡❧❧ ❛t s❝❛❧❡s ❝❧♦s❡ t♦ t❤❡ ❣r✐❞✲s❝❛❧❡ ✲ ❧❡❛❞✐♥❣ t♦ s✐❣♥✐✜✲ ❝❛♥t ♠♦❞❡❧ ❡rr♦rs❀ ❙❑❊❇ ✐♥❥❡❝ts ❦✐♥❡t✐❝ ❡♥❡r❣② ✐♥t♦ t❤❡ ♠♦❞❡❧ t♦ tr② t♦ ♠❛❦❡ ✉♣ ❢♦r t❤✐s✮✳ ✕ ❙t♦❝❤❛st✐❝❛❧❧② ♣❡rt✉r❜❡❞ t❡♥❞❡♥❝✐❡s ✭❙P❚✮ ✭t❡♥❞❡♥✲ ❝✐❡s ❢r♦♠ t❤❡ ✲ ✐♠♣❡r❢❡❝t ✲ ♣❛r❛♠❡tr✐③❛t✐♦♥ s❝❤❡♠❡s ❛r❡ s❝❛❧❡❞ ❛♥❞ ❛❞❞❡❞ ❛s ♣♦ss✐❜❧❡ ♠♦❞❡❧ ❡rr♦rs✮✳

• ❖t❤❡r ❡rr♦rs ✭❡✳❣✳ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥ ♣❡rt✉r❜❛t✐♦♥s ❢♦r

❧✐♠✐t❡❞ ❛r❡❛ ♠♦❞❡❧s✱ ♣❡rt✉r❜❛t✐♦♥s t♦ t❤❡ ✉♥❦♥♦✇♥ ❢♦r❝✲ ✐♥❣s ♦❢ t❤❡ ♠♦❞❡❧✮✳

SLIDE 50

✺✵

✻✳ ❍②❜r✐❞ ✭✈❛r✴❡♥s❡♠❜❧❡✮ ❢♦r♠❛t✐♦♥s

✻✭❛✮ ❇❛s✐❝ ✐❞❡❛s

▲❡t ✉s ❝♦♥s✐❞❡r t❤❡ ♣r♦s ❛♥❞ ❝♦♥s ♦❢ ✈❛r✐❛t✐♦♥❛❧ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❛♥❞ ❡♥s❡♠❜❧❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✭s✉❝❤ ❛s t❤❡ ❡♥s❡♠❜❧❡ ❑❛❧♠❛♥ ✜❧t❡r ❞✐s❝✉ss❡❞ ✐♥ ♣❛rt ■■ ♦❢ t❤✐s ❝♦✉rs❡✮✳ ❱❆❘■❆❚■❖◆❆▲ ❉❆❚❆ ❆❙❙■▼■▲❆❚■❖◆ ❊◆❙❊▼❇▲❊ ❑❆▲▼❆◆ ❋■▲❚❊❘ ✶✳ ❊✣❝✐❡♥❝②

• ♦♦❞
• ♦♦❞

✷✳ ❉❛t❛ ✈♦✐❞s ❘❡✈❡rts t♦ t❤❡ ❜❛❝❦❣r♦✉♥❞ st❛t❡✱ xB ❘❡✈❡rts t♦ t❤❡ ❜❛❝❦❣r♦✉♥❞ st❛t❡✱ xB ✸✳ Pr♦❝❡ss✐♥❣ ❈♦♥t✐♥✉♦✉s ✭✇✐t❤✐♥ ❛ss✐♠✐❧❛t✐♦♥ ✇✐♥❞♦✇✮ ■♥t❡r♠✐tt❡♥t ✹✳ ❙❝❛❧✐♥❣ ❢♦r ♣❛r❛❧❧❡❧ ❝♦♠♣✉t✐♥❣ ▲✐♠✐ts t♦ s❝❛❧✐♥❣ ◆♦ ❧✐♠✐ts t♦ s❝❛❧✐♥❣ ✺✳ ❊rr♦rs ✐♥ ✐♥♣✉ts ❆❧❧♦✇s ❢♦r ❡rr♦rs ✐♥ xB ❛♥❞ y ❆❧❧♦✇s ❢♦r ❡rr♦rs ✐♥ xB ❛♥❞ y ✻✳ ❊rr♦rs ✐♥ ♠♦❞❡❧ ❆❝❝♦✉♥t❡❞ ❢♦r ✐♥ ❲❈ ✹❉✲❱❆❘ ❆❝❝♦✉♥t❡❞ ❢♦r ✼✳ ■♥❞✐r❡❝t ♦❜s❡r✈❛t✐♦♥s ❨❡s ❨❡s ✽✳ ❇❛❧❛♥❝❡ ❛♥❞ s♠♦♦t❤♥❡ss ♦❢ ❛♥❛❧②s✐s ❨❡s ◆♦✱ ✉♥❧❡ss N ✐s s✉✣❝✐❡♥t❧② ❧❛r❣❡ ✯ ✾✳ ❋❧♦✇ ❞❡♣❡♥❞❡♥t ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ◆♦✱ Pf ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② B ❨❡s✱ Pf ✐s ❛♣♣r♦①✐♠❛t❡❞ ❜② Pf

(N)

✯ ✶✵✳ ❘❛♥❦ ♦❢ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❋✉❧❧ r❛♥❦ rank ≤ N ✯ ✯ ❚❤❡s❡ ✐ss✉❡s ❛r❡ r❡❧❛t❡❞✳ ❚❤❡ ❛✐♠ ♦❢ ❤②❜r✐❞ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ✐s t♦ ❝♦♠❜✐♥❡ ❱❆❘ ✇✐t❤ ❛♥ ❡♥s❡♠❜❧❡ t♦ ❣❡t t❤❡ ❜❡st ❜✐ts ♦❢ ❡❛❝❤ ❛♣♣r♦❛❝❤✳

SLIDE 51

✺✶ ❱❛r✐❛t✐♦♥❛❧ ❛ss✐♠✐❧❛t✐♦♥ str✉❝t✉r❡ ❢✉♥❝t✐♦♥

• ❋✉❧❧ r❛♥❦✱ ❜✉t ♥♦t ✢♦✇ ❞❡♣❡♥❞❡♥t✳

❊♥s❡♠❜❧❡✲❞❡r✐✈❡❞ str✉❝t✉r❡ ❢✉♥❝t✐♦♥ ✭N = 24✮

correlation (c) v−p correlation (NAE) longitude 24−members 15−members 05−members theoretical

• ❋❧♦✇ ❞❡♣❡♥❞❡♥t✱ ❜✉t r❛♥❦ ❞❡✜❝✐❡♥t✳

■♥ t❤❡ ❤②❜r✐❞ s♦❧✉t✐♦♥✱ ✇❡ s♦❧✈❡ ❛ ❱❆❘✲❧✐❦❡ ♣r♦❜❧❡♠ ❜✉t B → PH✿ PH = αB + (1 − α)Pf

(N)✱ ✇❤❡r❡ 0 ≤ α ≤ 1.

SLIDE 52

✺✷

✻✭❜✮ ■♥❝♦r♣♦r❛t✐♥❣ ❛ s✐♠♣❧❡ ❤②❜r✐❞ s❝❤❡♠❡ ✐♥ ❱❆❘

■♥ ♦r❞❡r t♦ ✉s❡ PH = αB + (1 − α)Pf

(N) ✐♥ ✈❛r✐❛t✐♦♥❛❧ ❛ss✐♠✐❧❛t✐♦♥✱ PH ♥❡❡❞s t♦ ❜❡ ♠❛❞❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡

tr❛♥s❢♦r♠ ✭❈❱❚✮✳ ❘❡❝❛❧❧ ❢r♦♠ ✸✭❡✮✱ B ✐s ♠♦❞❡❧❧❡❞ ❜② ♠✐♥✐♠✐③✐♥❣ t❤❡ ❝♦st ❢✉♥❝✲ t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ ❛ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ δχ✿ J[δχ] = 1 2δχTδχ + 1 2

T

• t=0

[δy(t) − HtMt←0Uδχ]T R−1

t

× [δy(t) − HtMt←0Uδχ] , ✇❤❡r❡ δx = Uδχ, ❛♥❞

• δχδχT

= I, ❛♥❞ t❤❡ ✐♠♣❧✐❡❞ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✐s✿ Bimp = UUT. ◆♦✇ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦st ❢✉♥❝t✐♦♥ ❛♥❞ ♠♦❞✐✜❝❛t✐♦♥ t♦ t❤❡ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ ❛♥❞ ✐ts ❈❱❚✿ JH[δχH] = 1 2δχT

varδχvar + 1

2δχT

ensδχens +

1 2

T

• t=0
• δy(t) − HtMt←0UHδχHT R−1

t

×

• δy(t) − HtMt←0UHδχH

, ✇❤❡r❡ δx = UHδχH, ❛♥❞

• δχHδχHT

= I, ❜✉t ♥♦✇ δχH =

• δχvar

δχens

• ,

δχvar ∈ Rn, δχens ∈ RN, ❛♥❞ UH = √αU

• 1−α

N−1X

• .

❲❤❛t ✐s t❤❡ ✐♠♣❧✐❡❞ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤✐s s❝❤❡♠❡❄ BH

imp =

• δxδxT

= UH δχHδχHT UHT = UHUHT, = √αU

• 1−α

N−1X

√αUT

• 1−α

N−1XT

• = αUUT + 1 − α

N − 1XXT, = αB + (1 − α)Pf

(N).

❚❤❡ ✜rst t❡r♠ ❝♦♥t❛✐♥s UUT✱ ✇❤✐❝❤ ✐s t❤❡ ✐♠♣❧✐❡❞ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ❢r♦♠ t❤❡ ♣✉r❡ ✈❛r✐❛t✐♦♥❛❧ s❝❤❡♠❡✱ ❛♥❞ t❤❡ s❡❝♦♥❞ t❡r♠ ❝♦♥t❛✐♥s XXT/(N − 1)✱ ✇❤✐❝❤ ✐s t❤❡ ❡♥s❡♠❜❧❡✲❞❡r✐✈❡❞ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✭✇❡ ✉s❡❞ t❤✐s ♥♦t❛t✐♦♥ ✐♥ s❡❝t✐♦♥ ✸✭❜✮✱ ❛♥❞ ✐♥ ♣r♦❜❧❡♠ ✶✶✮✳

SLIDE 53

✺✸

✻✭❝✮ ■♥❝♦r♣♦r❛t✐♥❣ ❛ ❧♦❝❛❧✐③❡❞ ❤②❜r✐❞ s❝❤❡♠❡ ✐♥ ❱❆❘

❚❤❡ ❡♥s❡♠❜❧❡ ❝♦♥tr✐❜✉t✐♦♥ t♦ t❤❡ ❤②❜r✐❞ ❝♦✈❛r✐❛♥❝❡ ✐s ♥♦✐s② ✇❤❡♥ N ✐s s♠❛❧❧✳ ❍♦✇ ❝❛♥ ✇❡ ♠✐t✐❣❛t❡ t❤✐s ♥♦✐s❡❄

• ❆ st❛t✐st✐❝❛❧ r❡s✉❧t t❡❧❧s ✉s t❤❛t t❤❡ ❡rr♦r ✐♥ t❤❡ s❛♠♣❧❡ ❝♦rr❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t✇♦ ✈❛r✐❛❜❧❡s x ❛♥❞ y ❤❛s ❡①♣❡❝t❛t✐♦♥

(1 − cor2(x, y))/ √ N − 1✳

• ❋♦r ❛ ❣✐✈❡♥ N✱ s❛♠♣❧✐♥❣ ❡rr♦rs ❛r❡ ❡①♣❡❝t❡❞ t♦ ❜❡ ❧❛r❣❡st ✇❤❡♥ t❤❡ ❝♦rr❡❧❛t✐♦♥s ❛r❡ ❝❧♦s❡ t♦ ③❡r♦✳
• ❈♦rr❡❧❛t✐♦♥s ❛r❡ ❡①♣❡❝t❡❞ t♦ ❜❡ s♠❛❧❧❡r ❛t ❧❛r❣❡r s❡♣❛r❛t✐♦♥s✳
• ✬▲♦❝❛❧✐③❛t✐♦♥✬ ❛rt✐✜❝✐❛❧❧② r❡❞✉❝❡s ❝♦✈❛r✐❛♥❝❡s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s s❡♣❛r❛t❡❞ ❜② ❧❛r❣❡ ❞✐st❛♥❝❡s✳

▲❡t x = ηB(r1) ❛♥❞ y = ηB(r2)✳ ❚❤❡ r❛✇ ❝♦✈❛r✐❛♥❝❡ ❜❡t✇❡❡♥ x ❛♥❞ y ✐s✿ Pf

(N)(r1, r2) =

1 N − 1

N

• i=1

η(i)

B (r1)η(i) B (r2).

❋♦r t❤❡ ❝♦✈❛r✐❛♥❝❡ ❛❝t✉❛❧❧② ✉s❡❞ ✐♥ t❤❡ ❤②❜r✐❞ s❝❤❡♠❡✱ ✇❡ ✇✐s❤ t♦ ♠✉❧t✐♣❧② t❤✐s ❜② ❛ ♠♦❞❡r❛t✐♦♥ ❢✉♥❝t✐♦♥ t❤❛t ❞❡❝r❡❛s❡s ✇✐t❤ s❡♣❛r❛t✐♦♥ ❜❡t✇❡❡♥ r1 ❛♥❞ r2✿ Ω(r1, r2) = ♣r❡s❝r✐❜❡❞ ❢✉♥❝t✐♦♥ ♦❢ |r1 − r2|✱ 0 ≤ Ω(r1, r2) ≤ 1✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ✉s❡❞ ✐s t❤❡♥✿ Pf,l

(N)(r1, r2) = Pf (N)(r1, r2)Ω(r1, r2).

❚❤✐s ✐s ❢♦r ❛ ♣❛rt✐❝✉❧❛r ♠❛tr✐① ❡❧❡♠❡♥t✳ ❋♦r t❤❡ ✇❤♦❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✱ ✐♥tr♦❞✉❝❡ t❤❡ ❙❝❤✉r ♣r♦❞✉❝t ♦❢ ♠❛tr✐❝❡s✿ Pf,l

(N) = Pf (N) ◦ Ω,

Ω ∈ Rn×n.

SLIDE 54

✺✹ ❍♦✇ ❞♦ ✇❡ ✐♥❝♦r♣♦r❛t❡ t❤✐s ✐♥t♦ t❤❡ ❈❱❚❄ ❚❤✐s s❡❝t✐♦♥ ✐s ♣r♦✈✐❞❡❞ ❢♦r ✐♥❢♦r♠❛t✐♦♥ ♦♥❧②✳ ■♥ ♦✉t❧✐♥❡✿

• ❲❡ ❦♥♦✇ t❤❛t Pf

(N) = 1 N−1XXT✱ Pf (N) ∈ Rn×n✱ X ∈ Rn×N✳

• ◆♦✇ s✉♣♣♦s❡ t❤❛t ✇❡ ❝❛♥ ❞❡❝♦♠♣♦s❡ Ω ✐♥ t❡r♠s ♦❢ M ♠❡♠❜❡rs ✐♥ Y✿ Ω =

1 M−1YYT✱ Ω ∈ Rn×n✱ Y ∈ Rn×M✳

• ❚❤❡♥ t❤❡ ❧♦❝❛❧✐③❡❞ ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ✐s✿

Pf,l

(N) = Pf (N) ◦ Ω,

=

• 1

N − 1XXT

• 1

M − 1YYT

• ,

= 1 (M − 1)(N − 1)

• XXT
• YYT

.

• ■t ✐s ♣♦ss✐❜❧❡ t♦ ❝♦♥str✉❝t ❛ ♥❡✇ ♠❛tr✐① XΩ s✉❝❤ t❤❛t Pf,l

(N) = 1 (N−1)(M−1)XΩXT Ω✱ XΩ ∈ Rn×NM✳

• ❚❤✐s ♥❡✇ ♠❛tr✐① ❤❛s t❤❡ ❢♦r♠✿

XΩ =   ↑ ↑ ↑ ↑ ↑ ↑ η(1)

B ◦ y(1)

η(1)

B ◦ y(2) · · · η(1) B ◦ y(M)

η(2)

B ◦ y(1) · · · η(2) B ◦ y(M) · · · · · · η(N) B

• y(M)

↓ ↓ ↓ ↓ ↓ ↓   , ✇❤❡r❡ η(i)

B ✐s t❤❡ it❤ ❝♦❧✉♠♥ ♦❢ X ❛♥❞ y(j) ✐s t❤❡ jt❤ ❝♦❧✉♠♥ ♦❢ XΩ✳ ❚❤❡r❡ ❛r❡ ♦t❤❡r ❝♦♠♣❛❝t ✇❛②s t♦ ✇r✐t❡ t❤✐s ♠❛tr✐①✿ ❇✉❡❤♥❡r ▼✳✱ ❊♥s❡♠❜❧❡ ❞❡r✐✈❡❞ st❛t✐♦♥❛r② ❛♥❞ ✢♦✇ ❞❡♣❡♥❞❡♥t ❜❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s✿ ❊✈❛❧✉❛t✐♦♥ ✐♥ ❛ q✉❛s✐✲♦♣❡r❛t✐♦♥❛❧ ◆❲P s❡tt✐♥❣✱ ◗✳❏✳❘✳▼❡t❡♦r✳❙♦❝✳ ✶✸✶ ♣♣✳✶✵✶✸✲✶✵✹✸ ✭✷✵✵✺✮✳

• ❚❤❡ ❧♦❝❛❧✐③❡❞ ❤②❜r✐❞ s❝❤❡♠❡ ✐s t❤❡♥ t❤❡ s❛♠❡ ❛s t❤❡ ✉♥❧♦❝❛❧✐③❡❞ ♦♥❡✱ ❜✉t ✇✐t❤

✕ t❤❡ N✲❡❧❡♠❡♥t ♣❛rt ♦❢ t❤❡ ❝♦♥tr♦❧ ✈❡❝t♦r δχH r❡❧❛❝❡❞ ✇✐t❤ ❛♥ NM✲❡❧❡♠❡♥t ❝♦♥tr♦❧ ✈❡❝t♦r✱ ❛♥❞ ✕

• 1−α

N−1X ✐♥ t❤❡ ❈❱❚ r❡♣❧❛❝❡❞ ✇✐t❤

• 1−α

(N−1)(M−1)XΩ✳

◆✳❇✳ ❚❤❡r❡ ❛r❡ ♦t❤❡r ✇❛②s ♦❢ r❡♣r❡s❡♥t✐♥❣ ❛ ❤②r✐❞ s②st❡♠ ✐♥ t❡r♠s ♦❢ ❝♦♥tr♦❧ ✈❛r✐❛❜❧❡s✿ ▲♦r❡♥❝ ❆✳❈✳✱ ❚❤❡ ♣♦t❡♥t✐❛❧ ♦❢ t❤❡ ❡♥s❡♠❜❧❡ ❑❛❧♠❛♥ ✜❧t❡r

❢♦r ◆❲P ✲ ❛ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ ✹❞✲❱❛r✱ ◗✳❏✳❘✳▼❡t❡♦r✳❙♦❝✳ ✶✷✾ ♣♣✳✸✶✽✸✲✸✷✵✸ ✭✷✵✵✸✮✳

SLIDE 55

✺✺

✼✳ ❉❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❞✐❛❣♥♦st✐❝s

• ❲❤❛t ❝❛♥ ❣♦ ✇r♦♥❣ ✇✐t❤ ❛ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ s❝❤❡♠❡❄ ❋♦r ❛ str♦♥❣ ❝♦♥str❛✐♥t ✹❉✲❱❆❘✱ ❡✳❣✳✿

✕ ■♥❝♦rr❡❝t ❡rr♦r ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐❝❡s✳ ✕ ◆♦♥✲●❛✐ss✐❛♥ ♦r ❜✐❛s❡❞ ❡rr♦rs ✐♥ t❤❡ ❜❛❝❦❣r♦✉♥❞ ♦r t❤❡ ♦❜s❡r✈❛t✐♦♥s✳ ✕ ❊rr♦rs ✐♥ M✱ h✱ M ♦r H✳ ✕ ❙tr♦♥❣ ♥♦♥✲❧✐♥❡❛r✐t✐❡s ✐♥ M ♦r h✳ ✕ ❱❛r✐❛t✐♦♥❛❧ ♣r♦❝❡❞✉r❡ ♥♦t ❝♦♥✈❡r❣❡❞ t♦ t❤❡ ♠✐♥✐♠✉♠✳ ✕ ❇❛❝❦❣r♦✉♥❞ ❛♥❞ ♦❜s❡r✈❛t✐♦♥ ❡rr♦rs ❛r❡ ❝♦rr❡❧❛t❡❞✳

• ❍♦✇ ❝❛♥ ✇❡ ❛ss❡ss ✐❢ ❛ ❣✐✈❡♥ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ s❝❤❡♠❡ ✐s s✉❜✲♦♣t✐♠❛❧❄ ❊✳❣✳ ❢♦r ✈❛r✐❛t✐♦♥❛❧ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✿

✕ ❇❡♥♥❡tt✲❚❛❧❛❣r❛♥❞ ❞✐❛❣♥♦st✐❝✳ ✕ ❉❡sr♦③✐❡r✬s ❞✐❛❣♥♦st✐❝s✳

SLIDE 56

✺✻

✼✭❛✮ ❚❤❡ ❇❡♥♥❡tt✲❚❛❧❛❣r❛♥❞ t❤❡♦r❡♠

❚✇✐❝❡ t❤❡ ❝♦st ❢✉♥❝t✐♦♥ ✈❛❧✉❡ ❛t t❤❡ ♠✐♥✐♠✉♠ ✭✐✳❡✳ ❛t t❤❡ ❛♥❛❧②s✐s✮ ❢♦r ❛♥ ♦♣t✐♠❛❧ ❛ss✐♠✐❧❛t✐♦♥ s②st❡♠ ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ t❤❛t ♦❜❡②s χ2 st❛t✐st✐❝s ❛♥❞ t❤❡r❡❢♦r❡ ❤❛s ❛ ♣❛rt✐❝✉❧❛r ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡✳ ❙t❛t✐st✐❝s t❡❧❧s ✉s t❤❛t t❤❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡ ♦❢ ❛ χ2 ❞✐str✐❜✉t✐♦♥ t❤❛t r❡s✉❧ts ❢r♦♠ ❛ ✜t ♦❢ ν ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ t♦ q ♣✐❡❝❡s ♦❢ ❞❛t❛ ✐s E(2Jmin) = q − ν✳ ❚❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ♣r♦❜❧❡♠ tr✐❡s t♦ ✜t ν = n ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ t♦ q = n + p ♣✐❡❝❡s ♦❢ ✐♥❢♦r♠❛t✐♦♥ ✭t❤❡ ❜❛❝❦❣r♦✉♥❞ st❛t❡ ❛♥❞ t❤❡ ♦❜s❡r✈❛t✐♦♥s✮✳ ❚❤❡♥✱ E(2Jmin) = n + p − n = p✳ ❚❤❡r❡❢♦r❡ t❤❡ ❡①♣❡❝t❡❞ ✈❛❧✉❡ ♦❢ Jmin ✐s E(Jmin) = p 2. ■❢ ❛ ❣✐✈❡♥ ❛ss✐♠✐❧❛t✐♦♥ r✉♥ ❞♦❡s ♥♦t ❣✐✈❡ ❛ ✈❛❧✉❡ ♦❢ Jmin ❝❧♦s❡ t♦ t❤✐s ✈❛❧✉❡ t❤❡♥ ✐t ✐s ❛♥ ✐♥❞✐❝❛t✐♦♥ t❤❛t s♦♠❡t❤✐♥❣ ✐s ✇r♦♥❣ ✇✐t❤ t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✳ ❚❤✐s ❝❛♥ ❛❧s♦ ❜❡ ♣r♦✈❡❞ ❞✐r❡❝t❧② ❢♦r t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ♣r♦❜❧❡♠ ✭t❤❡ ❇❡♥♥❡t✲❚❛❧❛❣r❛♥❞ t❤❡♦r❡♠ ✲ s❡❡ ♥♦t❡s ♦♥ ❤❛♥❞♦✉t ❢♦r ❛ ♣r♦♦❢✮✳

SLIDE 57

✺✼

❉❡sr♦③✐❡r✬s ❉✐❛❣♥♦st✐❝s

❉❡sr♦③✐❡r ❞✐❛❣♥♦st✐❝s ✉s❡ t❤❡ ❢♦❧❧♦✇✐♥❣ q✉❛♥t✐t✐❡s ❝❛❧❝✉❧❛t❡❞ ❢♦r ❛ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ r✉♥ ✭❛❧❧ ✐♥ ♦❜s❡r✈❛t✐♦♥ s♣❛❝❡✮✿

• ■♥♥♦✈❛t✐♦♥s ✭♦❜s❡r✈❛t✐♦♥ ♠✐♥✉s ❜❛❝❦❣r♦✉♥❞✮✿ do

b = y − Hxb✳

• ❆♥❛❧②s✐s ✐♥❝r❡♠❡♥t ✭❛♥❛❧②s✐s ♠✐♥✉s ❜❛❝❦❣r♦✉♥❞✮✿da

b = Hδxa✳

• ❘❡s✐❞✉❛❧s ✭♦❜s❡r✈❛t✐♦♥ ♠✐♥✉s ❛♥❛❧②s✐s✮✿ do

a = y − Hxa✳

❚❤❡ ❝♦✈❛r✐❛♥❝❡s ♦❢ t❤❡s❡ q✉❛♥t✐t✐❡s r❡✈❡❛❧s t❤❡ ❝♦♥s✐st❡♥❝② ✭♦r ✐♥❝♦♥s✐st❡♥❝②✮ ♦❢ t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✳ ❊✳❣✳ ❢♦r ✸❉✲❱❆❘✿ ❈♦✈❛r✐❛♥❝❡ ❆❝t✉❛❧ r❡s✉❧t ✭s✉❜✲♦♣t✐♠❛❧✮ ❘❡s✉❧t ✐❢ ♦♣t✐♠❛❧ E{do

bdo b T}

R + HBHT R + HBHT E{da

bdo b T}

H ˆ BHT(H ˆ BHT + ˆ R)−1(R + HBHT) HBHT E{do

ado b T}

(I − H ˆ BHT(H ˆ BHT + ˆ R)−1)(R + HBHT) R E{da

bdo a T}

H ˆ BHT(H ˆ BHT + ˆ R)−1(R + HBHT)(I − H ˆ BHT(H ˆ BHT + ˆ R)−1)T HAHT ❍❡r❡ B ❛♥❞ R ❛r❡ t❤❡ tr✉❡ ❜❛❝❦❣r♦✉♥❞ ❛♥❞ ♦❜s❡r✈❛t✐♦♥ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s ♠❛tr✐❝❡s✱ ❛♥❞ ˆ B ❛♥❞ ˆ R ❛r❡ t❤❡ ♦♥❡s ❛ss✉♠❡❞ ❢♦r t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥✳ H ✐s ❛ss✉♠❡❞ ♣❡r❢❡❝t✳ ❙❡❡ ♥♦t❡s ♦♥ ❤❛♥❞♦✉t ❢♦r ❛ ♣r♦♦❢✳