qs r t rt - - PowerPoint PPT Presentation

❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s

❯♥✐✈❡rs✐t② ♦❢ P❛r✐s ❱■✱ ▲❛❜♦r❛t♦✐r❡ ❏✳✲▲✳ ▲✐♦♥s ❖❧✐✈✐❡r P✐r♦♥♥❡❛✉

✇✐t❤ ❨✈❡s ❆❝❤❞♦✉✱ ◆✳ ▲❛♥t♦s✱ ❛♥❞ ✐♥♣✉t ❢r♦♠ ❈✳ ◆❣✉②❡♥✱ ❊✳❲✳ ❙❛❝❤s ❛♥❞ ▼✳ ❙❝❤✉ ✫ ❆✳ ❈♦♥③❡ ✭◆❛t✐①✐s✮ ✰ ❛♥❞ ❩❡❧✐❛❞❡ ✇✇✇✳❛♥♥✳❥✉ss✐❡✉✳❢r✴♣✐r♦♥♥❡❛✉ ❘■❈❆▼ s♣❡❝✐❛❧ ❙❡♠❡st❡r ✷✵✵✽

❖❝t♦❜❡r ✸✵✱ ✷✵✵✽

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶ ✴ ✺✶

❖✉t❧✐♥❡

✶

✳ ■♥tr♦❞✉❝t✐♦♥

✷

✳ ❈❛❧✐❜r❛t✐♦♥

✸

✳ ❉✉♣✐r❡✬s ❡q✉❛t✐♦♥ ❘❡❞✉❝❡❞ ❇❛s✐s ❢♦r ❙♦❧✈❡rs

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷ ✴ ✺✶

▼♦♥t❡✲❈❛r❧♦ ❙✐♠✉❧❛t✐♦♥ ♦❢ t❤❡ ❇❧❛❝❦ ✫ ❙❝❤♦❧❡s ▼♦❞❡❧

❆ ✜♥❛♥❝✐❛❧ ❛ss❡t ✇✐t❤ t❡♥❞❡♥❝② µ ❛♥❞ ✈♦❧❛t✐❧✐t② σ ❞❙t = ❙t(µ❞t + σ❞❲t), ❙✵ ❦♥♦✇♥

• µ = r(t) t❤❡ ✐♥t❡r❡st r❛t❡ ✉♥❞❡r t❤❡ r✐s❦✲♥❡✉tr❛❧ ♣r♦❜❛❜✐❧✐t② ❧❛✇✳ (❲t) ✐s

❛ st❛♥❞❛r❞ ❇r♦✇♥✐❛♥ ♠♦t✐♦♥✳ ❊✉r♦♣❡❛♥ ❝❛❧❧s ❛♥❞ ♣✉ts ♦♥ ❙ ❛r❡ ✈❛❧✉❡❞ ❛t t ❜② t❤❡ ❡①♣❡❝t❡❞ ♣r♦✜t✱ ❞✐s❝♦✉♥t❡❞ ❛t t✿ ✸ ♥✉♠❡r✐❝❛❧ ♠❡t❤♦❞s ✶✳ ❚r❡❡ ♠❡t❤♦❞s ✷✳ ▼♦♥t❡❝❛r❧♦✿ ❈t = ❡−r(❚−t)❊

• (❙❚ − ❑)+

❞❙t = ❙t(µ❞t + σ❞❲t), ❙(✵) = ❙✵. ❞❲t ≈ √ ❞tN(✵, ✶) ✸✳ ■t♦ ❈❛❧❝✉❧✉s ✿ ∂t❈ + σ✷①✷ ✷ ∂✷

①①❈ + r①∂①❈ − r❈ = ✵

❈(❚, ①) = (① − ❑)+

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸ ✴ ✺✶

▼♦♥t❡✲❈❛r❧♦✿ ●♦♦❞s ❛♥❞ ❜❛❞s

◆❡❛r t♦ t❤❡ ♠♦❞❡❧✐s❛t✐♦♥

• ✐✈❡s ✉♣♣❡r ❛♥❞ ❧♦✇❡r ❜♦✉♥❞s ♦♥ t❤❡ ❡rr♦r

❈♦♥✈❡r❣❡s ✐♥ ✶/ √ ◆ ⇒ ◗✉❛s✐✲▼♦♥t❡ ❈❛r❧♦ ❈❛♥ ❝♦♠♣✉t❡ s❡✈❡r❛❧ ❞❡r✐✈❛t✐✈❡s ✇✐t❤ ❖( √ ◆) ♦♣❡r❛t✐♦♥s ❊❛s② t♦ ♣❛r❛❧❧❡❧✐③❡✳ ❈♦♠♣❧❡①✐t② ❣r♦✇s ✐♥ ❖(❞) ✇✐t❤ t❤❡ ❞✐♠❡♥s✐♦♥ ❞✳

• r❡❡❦s ❜② ▼❛❧❧✐❛✈✐♥ ❝❛❧❝✉❧✉s✳

❈❛❧✐❜r❛t✐♦♥ ✐s ✈❡r② ❤❛r❞✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹ ✴ ✺✶

P❉❊s✿ ●♦♦❞s ❛♥❞ ❜❛❞s

◆❡❛r❡r t♦ t❤❡ ❛♥❛❧②t✐❝❛❧ ❢♦r♠✉❧❛s ❯♣♣❡r ❛♥❞ ❧♦✇❡r ❡rr♦r ❜♦✉♥❞s ✇✐t❤ ❛ ♣♦st❡r✐♦r✐ ❡st✐♠❛t❡s ❈♦♥✈❡r❣❡s ✐♥ ❖(◆−♣) ✇✐t❤ ♦r❞❡r ♦❢ ❛♣♣r♦①✐♠❛t✐♦♥ ♣ ❈♦♠♣✉t❡ s❡✈❡r❛❧ ❞❡r✐✈❛t✐✈❡s ✐♥ ❖(◆ ❧♦❣ ◆) ♦♣❡r❛t✐♦♥s ✭❉✉♣✐r❡✮✳ ❈❛♥ ❜❡ ♣❛r❛❧❧❡❧✐③❡❞✳ ❈✉rs❡❞ ❜② ❞✐♠❡♥s✐♦♥ ❞ ✐♥ ❖(◆❞) ❡①❝❡♣t ✇✐t❤ s♣❛rs❡ ❣r✐❞✳

• r❡❡❦s ❛r❡ ✈❡r② ❡❛s②

❈❛❧✐❜r❛t✐♦♥ ✐s r❡❛s♦♥❛❜❧② ❡❛s②

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✺ ✴ ✺✶

❊①✐st❡♥❝❡✱ ❯♥✐q✉❡♥❡ss✱ ❘❡❣✉❧❛r✐t② ✭ ❆❝❤❞♦✉✱ ❈r❡♣❡②✮

∂tP − σ✷①✷ ✷ ∂✷

①①P − r①∂①P + rP = ✵

P(✵, ①) = (❑ − ①)+, ❧✐♠

①→∞ P(①, t) = ✵

❚❤❡♦r❡♠ ❆ss✉♠❡ ✵ < σ♠ ≤ σ(①, t) < σ▼, ①∂①σ ∈ ▲∞ ❛♥❞ r, σ, ①∂①σ ▲✐♣s❝❤✐t③ ✐♥ t✱ t❤❡♥ P ❡①✐sts✱ ✐s ❝♦♥t✐♥✉♦✉s ✐♥ t✐♠❡ ❛♥❞ ① ✷∂①①P ∈ ▲✷(R+)✳ ❋✉rt❤❡r♠♦r❡ ✐❢ ①✷∂①①σ ∈ ▲∞ t❤❡♥ P ✐s ❝♦♥✈❡① ❛♥❞ ♣♦s✐t✐✈❡✳ ❯s❡ ❛ ✈❛r✐❛t✐♦♥❛❧ ❢♦r♠ ✐♥ t❤❡ ✇❡✐❣❤t❡❞ ❙♦❜♦❧❡✈ ❱ ✐♥ ✇❤✐❝❤ ❛t(., .) ✐s

• ❛r❞✐♥❣✲❝♦❡r❝✐✈❡✿

❱ = {✉ ∈ ▲✷(R+) : ①∂①✉ ∈ ▲✷(R+)} (∂tP, ✇) + ❛t(P, ✇) = ✵, ∀✇ ∈ ❱ , P(✵) = (❑ − ①)+, P ∈ ▲✷(✵, ❚, ❱ ) ❛t(P, ✇) =

• ❘+ ∂①(✇ σ✷①✷

✷ )∂①P − r①✇∂①P + rP✇]

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✻ ✴ ✺✶

❚❤❡ ❋✐♥✐t❡ ❊❧❡♠❡♥t ▼❡t❤♦❞ ✐♥ ①

∂✉ ∂t − ①✷σ✷ ✷ ∂✷✉ ∂①✷ − r① ∂✉ ∂① + r✉ = ✵, ✉(①, ✵) = (❑ − ①)+, ✉(▲, t) = ✵ ∂✉ ∂t − ∂ ∂① (①✷σ✷ ✷ ∂✉ ∂① ) − ①ν ∂✉ ∂① + r✉ = ✵ ✇✐t❤ ν = r − ∂ ∂① (①✷σ✷ ✷ )

✶ ■♠♣❧✐❝✐t ✐♥ t✐♠❡✱ ❝❡♥t❡r❡❞ ✐♥ s♣❛❝❡ ✭✉♣✇✐♥❞ ✉s✉❛❧❧② ♥♦t ♥❡❝❡ss❛r②✮✿

✉♥+✶ − ✉♥ δt♥ − ∂ ∂① (①✷σ✷ ✷ ∂✉ ∂①

♥+✶

) − ①ν ∂✉ ∂①

♥+✶

+ r✉♥+✶ = ✵

✷ ❱❛r✐❛t✐♦♥❛❧ ❢♦r♠ ✐♥ ❱ = {✈ ∈ ▲✷(R+) : ①∂①✈ ∈ ▲✷(R+)} ✸ ❋✐♥✐t❡ ❡❧❡♠❡♥t ❜❛s✐s ❱❤ ≈ ❱ ✹ (❇ + δt❆)❯♥+✶ = ❈❯♥ s♦❧✈❡❞ ❜② ●❛✉ss ▲❯ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✼ ✴ ✺✶

❆ ♣♦st❡r✐♦r✐ ❡st✐♠❛t❡s ✭❨✳❆❝❤❞♦✉✮

Pr♦♣♦s✐t✐♦♥

[[✉ − ✉❤,δt]](t♥) ≤ ❝(✉✵)δt + µ σ✷

♠✐♥

 

♥

• ♠=✶

η♠

✷ + δt♠

σ✷

♠✐♥

❣(ρδt)

♠−✶

• ✐=✶

(✶ − ✷λδt✐)

• ω∈T♠❤

η♠,ω

✷

 

✶ ✷

✇❤❡r❡ ▲, µ ❛r❡ t❤❡ t✐♠❡✲❝♦♥t✐♥✉✐t② ❝♦♥st❛♥ts ♦❢ σ✷, r, ①σ∂①σ ✐♥ ▲∞✱ ❝(✉✵) = (✉✵✷ + δt∇✉✵✷)✶/✷✱ ❣(ρδt) = (✶ + ρδt)✷ ♠❛①(✷, ✶ + ρδt) η✷

♠ = δt♠❡−✷λt♠−✶ σ✷ ♠✐♥

✷ |✉♠

❤ − ✉♠−✶ ❤

|✷

❱ ,

η♠,ω = ❤ω ①♠❛①(ω)✉♠

❤ − ✉♠−✶ ❤

δt♠ − r① ∂✉♠

❤

∂① + r✉♠

❤ ▲✷(ω)

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✽ ✴ ✺✶

❇❡st ◆✉♠❡r✐❝❛❧ ▼❡t❤♦❞

• ■♠♣❧✐❝✐t ✐♥ t✐♠❡✱ ❝❡♥t❡r❡❞ ✐♥ s♣❛❝❡ ✭✉♣✇✐♥❞ ✉s✉❛❧❧② ♥♦t ♥❡❝❡ss❛r②✮✿

P♥+✶ − P♥ δt − ①✷σ✷ ✷ ∂✷

①①P♥+ ✶

✷ + rP♥+ ✶ ✷ − ①r∂①P♥+ ✶ ✷ = ✵

• ❋❊▼✲P✶ ✰ ▲❯ ❢❛❝t♦r✐③❛t✐♦♥
• ♠❡s❤ ❛❞❛♣t✐✈✐t② ✰ ❛ ♣♦st❡r✐♦r✐ ❡st✐♠❛t❡s
• ❇❛♥❦s ♥❡❡❞ ✵✳✶✪ ♣r❡❝✐s✐♦♥ ✐♥ s♣❧✐t s❡❝♦♥❞s ✳✳✳✇✐t❤✐♥ ❊①❝❡❧

50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5

• 20

20 40 60 80 100 PDE solution Black-Scholes formula

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✾ ✴ ✺✶

◆✉♠❡r✐❝❛❧ ❊①❛♠♣❧❡

50 100 150 200 250 300 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 "u.txt"using 1:2:7 50 100 150 200 250 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.2 0.4 0.6 0.8 1 1.2 1.4 "u.txt"using 1:2:7 50 100 150 200 250 300 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

• 0.05
• 0.04
• 0.03
• 0.02
• 0.01

0.01 0.02 "u.txt"using 1:2:5 50 100 150 200 250 300 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.001 0.002 0.003 0.004 0.005 0.006 0.007 "u.txt"using 1:2:6

❚❖P✿ ■♥❞✐❝❛t♦r ✶ ✇✐t❤ ❛ ✜①❡❞ ♥✉♠❜❡r ♦❢ ♥♦❞❡s ✭❧❡❢t✮ ❛♥❞ ✈❛r②✐♥❣ ✭r✐❣❤t✮ ❇❖❚❚❖▼✿ ❆❝t✉❛❧ ❡rr♦r ✭❧❡❢t✮✳ ❙❡❝♦♥❞ ✐♥❞✐❝❛t♦r ✭r✐❣❤t✮✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✵ ✴ ✺✶

▼♦r❡ ❈♦♠♣❧❡① ❖♣t✐♦♥s

❏✉♠♣ Pr♦❝❡ss❡s ■♥ ❇✫❙ ✉s❡ ❛ ▲é✈② ✐♥st❡❛❞ ♦❢ ❛ ❲✐❡♥❡r ♣r♦❝❡ss ✿ ∂t❈ − σ✷①✷ ✷ ∂①①❈ − r①∂①❈ − r❈ + A[❈] = ✵ A[❈] =

• R

(❈(①❡②, t) − ❈(①, t) − ①(❡② − ✶)∂①❈) ❦(②)❞② Pr♦♣♦s✐t✐♦♥ ✭❊✳❱♦❧t❝❤❦♦✈❛✲❘✳❈♦♥t✮ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✈❡r❣❡s ✐❢ δt < ❈ ❛♥❞ ✭✰❋❊▼✮ ❈ ♠ − ❈ ♠−✶ δt − σ✷①✷ ✷ ∂①①❈ ♠ − r①∂①❈ ♠ − r❈ ♠ = −

• R
• ❈ ♠−✶(①❡②) − ❈ ♠−✶(①) − ①(❡② − ✶)∂①❈ ♠−✶

❦❞② ❘❡♠❛r❦✿

• R

❈(①, t)A[❈]❞① =

• R×R

|❈(①❡②, t) − ❈(①, t)|✷❦(②)❞①❞②

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✶ ✴ ✺✶

▼✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ♣r♦❜❧❡♠s

✶✳ ❇❛s❦❡t ❖♣t✐♦♥ ❞ < ❲✐❲❥ >= σ✐❥❞t ❞❙✐ = ❙✐(r❞t + ❞❲✐), ✐ = ✶..❞, ✉ = ❡−(❚−t)r❊(

• ❙✐ − ❑)+

■t♦ ❝❛❧❝✉❧✉s ❧❡❛❞s t♦ ❛ ♠✉❧t✐❞✐♠❡♥s✐♦♥❛❧ ❇❧❛❝❦✲❙❝❤♦❧❡s ❡q✉❛t✐♦♥ ∂τ✉ −

• ✐

( σ✷

✐❥①✐①❥

✷ ∂①✐①✐✉ − r①✐∂①✐✉) + r✉ = ✵ ✉(✵) = (

• ✐

①✐ − ❑)+ ❙t♦❝❤❛st✐❝ ❱♦❧❛t✐❧✐t② ♠♦❞❡❧s ✭❙t❡✐♥✲❙t❡✐♥✱ ❖rst❡✐♥✲❯❤❧❡♥❜❡❝❦✱ ❍❡st♦♥✮

❞❙t = ❙t(r❞t + σt❞❲t), σt =

• ❨t,

❞❨t = κ(θ − ❨t)❞t + β❞❩t, ❞ < ❲t, ❩t >= ρ❞t

∂t❯ + ❛❯ + ❜①∂①❯ − ①✷② ✷ ∂①①❯ − β✷② ✷ ∂②②❯ − ρβ②①∂✷

①②❯ + ❝∂②❯ = ✵

❛ = µ − κ − ρβ − ②, ❜ = (µ − ✷② − ρβ), ❝ = κ(θ − ②) − ρβ② − β✷

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✷ ✴ ✺✶

❆♠❡r✐❝❛♥ ❖♣t✐♦♥s

∂✉ ∂t − σ✷①✷ ✷ ∂✷✉ ∂①✷ − r①∂✉ ∂① + r✉ ≥ ✵ ✉ ≥ ✉◦, ♦♥❡ ✐s ❂ ✇✐t❤ ✉(t, ✵) = ❑❡−rt, ✉(✵, ①) = ✉◦(①) := (❑ − ①)+ ❱ =

• ✈ ∈ ▲✷(R+), ① ∂✈

∂① ∈ ▲✷(R+)

• ,

K = {✈ ∈ ▲✷(✵, ❚; ❱), ✈ ≥ ✉◦}. ❋✐♥❞ ✉ ∈ K ∩ ❈ ✵([✵, ❚]; ▲✷(R+)), ∂✉ ∂t ∈ ▲✷(✵, ❚; ❱ ′), s.t. (∂✉ ∂t , ✈ − ✉) + ❛(✉, ✈ − ✉) ≥ ✵, ∀✈ ∈ K, ✉(t = ✵) = ✉◦, ✇✐t❤ ❛(✈, ✇) =

• R+

σ✷ ✷ ①✷ ∂✈ ∂① ∂✇ ∂① + (σ✷ + ①∂σ ∂① − r)①∂✈ ∂①✇ + r✈✇

• ❞①.

❆ss✉♠❡ σ ∈ [σ, σ] ❛♥❞ ①|∂①σ✷| < ▼✱ t❤❡♥ t❤❡ s♦❧✉t✐♦♥ ❡①✐sts ❛♥❞ ✐s ✉♥✐q✉❡✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✸ ✴ ✺✶

❘❡❣✉❧❛r✐t② ❘❡s✉❧ts

❆ss✉♠❡ t❤❛t t❤❡r❡ ❡①✐sts ▼ > ✵ s✳t✳ |①✷ ∂✷σ ∂①✷ | + |∂σ ∂t | + |① ∂✷σ ∂①∂t | ≤ ▼, ❛.❡.

• ❚❤❡ ❢r❡❡ ❜♦✉♥❞❛r② ✭❡①❡r❝✐s❡ ♣r✐③❡✮ ✐s ❛ ❝♦♥t✐♥✉♦✉s ❝✉r✈❡ t → γ(t)
• ① → ✉(t, ①) ✐s ❝♦♥✈❡①
• ■❢ t → γ(t) ✐s ▲✐♣s❝❤✐t③ ✐♥ [τ, ❚] t❤❡♥

(γ(σ′) − γ(σ))+✸

▲✸(τ,❚) ≤ ❝τσ − σ′✷ ▲∞((τ,❚)×R+)

. ✇❤✐❝❤ ✐♠♣❧✐❡s ❞✐✛❡r❡♥t✐❛❜✐❧✐t② ✐♥ σ ❛✇❛② ❢r♦♠ ③❡r♦✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✹ ✴ ✺✶

❙❡♠✐✲❙♠♦♦t❤ ◆❡✇t♦♥ ▼❡t❤♦❞

❑✉♥✐s❝❤ ❡t ❛❧ s✉❣❣❡st❡❞ t♦ r❡❢♦r♠✉❧❛t❡ t❤❡ ♣r♦❜❧❡♠ ❛s ❛(✉, ✈) − (λ, ✈) = (❢ , ✈) ∀✈ ∈ ❍✶(R+), ✐.❡.❆✉ − λ = ❢ λ − ♠✐♥{✵, λ + ❝(✉ − φ)} = ✵, ❚❤❡ ❧❛st ❡q✳ ✐s ❡q✉✐✈❛❧❡♥t t♦ λ ≤ ✵, λ ≤ λ + ❝(✉ − φ) ✐✳❡✳ ✉ ≥ φ, λ ≤ ✵, ✇✐t❤ ❡q✉❛❧✐t② ♦♥ ♦♥❡ ♦❢ t❤❡♠ ❢♦r ❡❛❝❤ ❙✳ ❚❤✐s ♣r♦❜❧❡♠ ✐s ❡q✉✐✈❛❧❡♥t ❢♦r ❛♥② r❡❛❧ ❝♦♥st❛♥t ❝ > ✵ ❜❡❝❛✉s❡ λ ✐s t❤❡ ▲❛❣r❛♥❣❡ ♠✉❧t✐♣❧✐❡r ♦❢ t❤❡ ❝♦♥str❛✐♥t✳ ◆❡✇t♦♥✬s ❛❧❣♦r✐t❤♠ ❛♣♣❧✐❡❞ t♦ ✭✶✮ ❣✐✈❡s

• ❈❤♦♦s❡ ❝ > ✵, , ✉✵, λ✵✱ s❡t ❦ = ✵✳
• ❉❡t❡r♠✐♥❡ ❆❦ := {❙ : λ❦(❙) + ❝(✉❦(❙) − φ(❙)) < ✵}
• ❙❡t ✉❦+✶ = ❛r❣ ♠✐♥✉∈❍✶(R+){✶

✷❛(✉, ✉) − (❢ , ✉) : ✉ = φ ♦♥ ❆❦}

• ❙❡t λ❦+✶ = ❢ − ❆✉❦+✶

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✺ ✴ ✺✶

❙✐♠♣❧✐❝✐t②

✈♦✐❞ ❖♣t✐♦♥✿✿❝❛❧❝✭ ❝♦♥st ❜♦♦❧ ❆▼❊❘■❈❆◆✮④ ❞♦✉❜❧❡ ❝❂✶✵✱ t❣✈ ❂ ✶❡✸✵❀ ✐♥t ❦♠❛① ❂ ❆▼❊❘■❈❆◆✯✸✰✶❀ ❢♦r✭✐♥t ✐❂✵❀✐❁♥❳❀✐✰✰✮ ✉❬✐❪ ❂ ♠❛①✭❑✲✐✯❞①✱✵✮❀ ❢♦r✭✐♥t ❥❂✵❀❥❁♥❚❀❥✰✰✮④ ✐♥t ❥❚❂✭♥❚✲✶✲❥✮✯❚❀ ❢♦r✭✐♥t ✐❂✶❀✐❁♥❳✲✶❀✐✰✰✮ ✴✴ r❤s ♦❢ ❊❉P ④ ✉♦❧❞❬✐❪ ❂ ✉❬✐❪ ✰ ❞t✯r✯✐✯✭✉❬✐✰✶❪✲✉❬✐✲✶❪✮✴✷❀ ❧❛♠❬✐❪ ❂ ✵❀ ⑥ ✉❬♥❳✲✶❪❂✵❀ ✉❬✵❪ ❂ ❡①♣✭✲r✯✭❥✰✵✳✺✮✯❞t✮❀ ✴✯✯✴ ❢♦r✭✐♥t ❦❂✵❀❦❁❦♠❛①❀❦✰✰✮④ ❢♦r✭✐♥t ✐❂✶❀✐❁♥❳✲✶❀✐✰✰✮④ ❞♦✉❜❧❡ ❛✉①❂✐✯s✐❣♠❛❬❥❚❪❬✐❪✯✐✯s✐❣♠❛❬❥❚❪❬✐❪✯❞t✴✷❀ ❜♠❬✐❪ ❂ ✭✶✰ r✯❞t ✰ ✷✯❛✉①✮❀ ❛♠❬✐❪ ❂ ✲❛✉①❀ ❝♠❬✐❪ ❂✲❛✉①❀ ✴✯✯✴ ✐❢✭❆▼❊❘■❈❆◆ ✫✫ ❧❛♠❬✐❪✰❝✯✭✉❬✐❪✲♠❛①✭❑✲✐✯❞①✱✵✮✮❁✵✮ ④ ✐♥❞✐❝❬✐❪❂✶❀ ❜♠❬✐❪ ❂ t❣✈❀⑥ ❡❧s❡ ✐♥❞✐❝❬✐❪❂✵❀ ✴✯✯✴ ⑥ ❢❛❝t▲❯✭✮❀ ❢♦r✭✐♥t ✐❂✶❀✐❁♥❳✲✶❀✐✰✰✮ ✴✯✯✴ ✐❢✭✐♥❞✐❝❬✐❪✮ ✉❬✐❪ ❂ t❣✈✯♠❛①✭❑✲✐✯❞①✱✵✮❀ ❡❧s❡ ✉❬✐❪❂✉♦❧❞❬✐❪❀ s♦❧✈❡▲❯✭✉✮❀ ❢♦r✭✐♥t ✐❂✶❀✐❁♥❳✲✶❀✐✰✰✮④ ❞♦✉❜❧❡ ❛✉①❂✐✯s✐❣♠❛❬❥❚❪❬✐❪✯✐✯s✐❣♠❛❬❥❚❪❬✐❪✯❞t✴✷❀ ✴✯✯✴ ❧❛♠❬✐❪❂✉♦❧❞❬✐❪✲✭✶✰r✯❞t✰ ✷✯❛✉①✮✯✉❬✐❪✰❛✉①✯✭✉❬✐✰✶❪✰✉❬✐✲✶❪✮❀⑥⑥⑥⑥

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✻ ✴ ✺✶

❘❡s✉❧ts

0 10 20 30 40 50 60 70 80 90 100 10 20 30 40 50 60 70 80 90 100 "exercise_250"

❇❡st ♦❢ ♣✉t ❜❛s❦❡t ♦♣t✐♦♥✱ σ✶ = ✵.✷, σ✷ = ✵.✶, ρ = −✵.✽

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✼ ✴ ✺✶

❖✉t❧✐♥❡

✶

✳ ■♥tr♦❞✉❝t✐♦♥

✷

✳ ❈❛❧✐❜r❛t✐♦♥

✸

✳ ❉✉♣✐r❡✬s ❡q✉❛t✐♦♥ ❘❡❞✉❝❡❞ ❇❛s✐s ❢♦r ❙♦❧✈❡rs

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✽ ✴ ✺✶

❈❛❧✐❜r❛t✐♦♥✿ ❍♦✇ t♦ ❝❤♦♦s❡ σ(❙, t)❄

✳✳✳ ❜② tr②✐♥❣ t♦ r❡♣r♦❞✉❝❡ t❤❡ ♣r✐❝❡s ♦❜s❡r✈❛❜❧❡ ♦♥ t❤❡ ♠❛r❦❡t✿ ❊✈❡r② ❞❛②✱ ♦♥❡ ❝❛♥ ♦❜s❡r✈❡

• ❚❤❡ ✉♥❞❡r❧②✐♥❣ ❛ss❡t ❙♦
• ❚❤❡ ✈❛❧✉❡s (❝✐)✐∈■ ♦❢ ❝❛❧❧s (❈❑✐,❚✐(❙✵, ✵))✐∈■✳

■♥✈❡rs❡ Pr♦❜❧❡♠ ✿ ✜♥❞ σ(①, t) s✳t✳ ❢♦r ❛❧❧ ✱ {❈✐}✐∈■ ❣✐✈❡♥ ❜② ∂t❈✐ + σ✷①✷ ✷ ∂✷

①①❈✐ + r①∂①❈✐ − r❈✐ = ✵,

❈✐(①, ❚✐) = (① − ❑✐)+, ❢♦r ❛❧❧ t ∈ [✵, ❚✐[, ① ∈ R+, s✉❝❤ t❤❛t ❈✐(①✵, ✵) = ❝✐✳✳✳✐t ✐♥✈♦❧✈❡s ❛s ♠❛♥② ❇✫❙ ❛s ❞✐✛❡r❡♥t ❑✐, ✐ ∈ ■✳ ♠✐♥

• |❈✐(①✵, ✵) − ❝✐|✷ : s✉❜❥❡❝t t♦ ❛❧❧ P❉❊s✱ ✐❂✶✳✳■

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✶✾ ✴ ✺✶

❇r✉t❡ ❋♦r❝❡ ❖♣t✐♠❛❧ ❈♦♥tr♦❧

❆❞❞ ❛ ❚✐❦❤♦♥♦✈ r❡❣✉❧❛r✐③❛t✐♦♥ ✇✐t❤ ♥♦r♠s ❣✐✈❡♥ ❜② t❤❡ ❡①✐st❡♥❝❡ t❤❡♦r❡♠s✿ ∂①σ✷

✵ + ∂tσ✷ ✵ + ∂①tσ✷ ✵

P❛r❛♠❡tr✐③❡ t❤❡ ✈♦❧❛t✐❧✐t② ❜② ❛ ❜✐❝✉❜✐❝ s♣❧✐♥❡ ✇✐t❤ ✭✶✵ × ✶✺✮ ❝♦♥tr♦❧ ♣♦✐♥ts ❯s❡ ❈♦♥❥✉❣❛t❡ ●r❛❞✐❡♥t ♦r ❇❋●❙ ❛❧❣♦r✐t❤♠ ❢♦r t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ❈♦♠♣✉t❡ t❤❡ ❣r❛❞✐❡♥ts ✇✴r ♣❛r❛♠❡t❡rs ❜② ✉s✐♥❣ ❛♥ ❛❞❥♦✐♥t ❙❡❡ ❆❝❤❞♦✉ ❡t ❛❧ ✭❊❈❈❖▼❆❙✮ ❢♦r ❛ ❝♦♠♣❧❡t❡ ♠❛t❤❡♠❛t✐❝❛❧ ❥✉st✐✜❝❛t✐♦♥ ❲❡ ❝♦♥s✐❞❡r ❆♠❡r✐❝❛♥ ♣✉ts ♦♥ t❤❡ ❢♦♦ts✐❡ ✐♥❞❡① ♦♥ ❏✉♥❡ ✻✱ ✷✵✵✶✳ ❚❤❡ s♣♦t ♣r✐❝❡ ✐s ✺✽✾✵✳ ❚❤❡r❡ ❛r❡ ❢♦✉r ❞✐✛❡r❡♥t ♠❛t✉r✐t✐❡s✿ ✵✳✶✷✷✱ ✵✳✶✾✾✱ ✵✳✷✾✺✱ ✵✳✺✺ ②❡❛rs✳ ❚❤❡ ✐♥t❡r❡st r❛t❡ r ✈❛r✐❡s ✇✐t❤ t✐♠❡ ❜✉t ✐s ❦♥♦✇♥✳ ❲❡ ❤❛✈❡ ❡♥♦✉❣❤ ❞❛t❛ t♦ r❡❝♦♥str✉❝t t❤❡ ♣r✐❝❡s ❢♦r ❛❧❧ str✐❦❡s ❜❡t✇❡❡♥ ✹✵✵✵ ❛♥❞ ✽✵✵✵ ❢♦r t❤❡s❡ ✹ ♠❛t✉r✐t✐❡s✳ ❚❤❡ ❣r✐❞ ❢♦r ✉ ✐s ♥♦♥ ✉♥✐❢♦r♠ ✇✐t❤ ✼✹✺ ♥♦❞❡s ✐♥ t❤❡ ❙✲❞✐r❡❝t✐♦♥ ❛♥❞ ✷✶✵ ♥♦❞❡s ✐♥ t❤❡ t ❞✐r❡❝t✐♦♥✳ ❋♦r s✐♠♣❧✐❝✐t②✱ t❤❡ ❣r✐❞ ✐s ❝❤♦s❡♥ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ♣♦✐♥ts (❚✐, ❑✐)■

✶ ❝♦✐♥❝✐❞❡ ✇✐t❤ s♦♠❡ ❣r✐❞ ♥♦❞❡s✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✵ ✴ ✺✶

❘❡s✉❧ts ■

❘❡❧❛t✐✈❡ ❡rr♦rs ❜❡t✇❡❡♥ t❤❡ ♦❜s❡r✈❡❞ ♣r✐❝❡s ❛♥❞ t❤♦s❡ ♦❜t❛✐♥❡❞ ✇✐t❤ σ ❢♦✉♥❞ ❛❢t❡r r✉♥♥✐♥❣ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ♣r♦❣r❛♠✳ ❆ ❝✉r✈❡ ❝♦rr❡s♣♦♥❞s t♦ ❛ ❣✐✈❡♥ ♠❛t✉r✐t②✳ ❘❡❧❛t✐✈❡ r♦✉♥❞✲♦✛ ❡rr♦r ♦♥ ♦❜s❡r✈❡❞ ♣r✐❝❡s✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✶ ✴ ✺✶

❘❡s✉❧ts ■■

■t ✇♦r❦s ❜✉t✳✳✳ ■t ✐s ✈❡r② ❡①♣❡♥s✐✈❡✱ ❤❡♥❝❡ ✉♥✉s❛❜❧❡ ■♠♣r♦✈❡♠❡♥ts

• ❘❡❞✉❝❡ t❤❡ ♥✉♠❜❡r ♦❢ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ s♣❧✐♥❡
• ❘❡✜♥❡ t❤❡ ♠❡s❤ ❛❧♦♥❣ t❤❡ ♦♣t✐♠✐③❛t✐♦♥
• ❖♣t✐♠✐③❡ σ(①, tt✐) ✇✐t❤ |

❥ ✉(①❥, t✐) − ❝❥|✷ ✦✦

• ❯s❡ ♦t❤❡r tr✐❝❦s✿ ❉✉♣✐r❡✬s✱ ✐♠♣❧✐❡❞ ✈♦❧✱ r❡❞✉❝❡❞ ❜❛s✐s✳✳✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✷ ✴ ✺✶

❖✉t❧✐♥❡

✶

✳ ■♥tr♦❞✉❝t✐♦♥

✷

✳ ❈❛❧✐❜r❛t✐♦♥

✸

✳ ❉✉♣✐r❡✬s ❡q✉❛t✐♦♥ ❘❡❞✉❝❡❞ ❇❛s✐s ❢♦r ❙♦❧✈❡rs

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✸ ✴ ✺✶

❉✉♣✐r❡✬s ❊q✉❛t✐♦♥

❚❤❡♦r❡♠ C❙,t(❚, ❑) = ❈❚,❑(❙, t) ✇❤❡r❡ C s♦❧✈❡s✿ ∂tC − σ✷(t, ①)①✷ ✷ ∂✷

①①C + r①∂①C = ✵,

C(✵, ①) = ( ❙ − ① )+

Pr♦♦❢ ✭r❂✵✮ ▲❡t ∂t✉ + µ(①, t)∂①①✉ = ✵; ▲❡t ∂t♣ − ∂①①(µ♣) = ✵ ❚❤❡♥✱ ✇✐t❤ ❛♣♣r♦♣r✐❛t❡ ❜♦✉♥❞❛r② ❝♦♥❞✐t✐♦♥s .

• R+ ✉(✵)♣✵ =
• R+ ✉❚♣(❚)

= ✉(①✵, ✵) ✇❤❡♥ ♣(①, ✵) = δ(① − ①✵) ▲❡t ∂①①✈ = ♣ t❤❡♥ ∂t✈ − µ∂①①✈ = ❛① + ❜, ✈(✵) = ❝ + ❞(① − ①✵)+ ❋✐♥❛❧❧② ✉❚ = (❑ − ①)+, ⇒ ∂①①✉❚ = −δ(① − ❑) ⇒ ✉(①✵, ✵) = −✈(❑, ❚)❞

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✹ ✴ ✺✶

❈❛❧✐❜r❛t✐♦♥ ✇✐t❤ ❉✉♣✐r❡✬s ❡q✉❛t✐♦♥

■♥✈❡rs❡ Pr♦❜❧❡♠ ✿ ✜♥❞ σ(❙, t) s✳t✳ ♠✐♥

σ

• |✉(❑✐, ❚✐) − ❝✐|✷

s✉❜❥❡❝t t♦ ∂t✉ − σ✷①✷ ✷ ∂✷

①①✉ + r①∂①✉ = ✵,

. ✉(①, ✵) = (❙✵ − ①)+ ❖♥❧② ♦♥❡ P❉❊ ❘❡♠❛r❦s ❖♥❧② ♦♥❡ P❉❊ ▲❡t ˜ ✉ ❜❡ ❛ ❈ ✷ ❡①tr❛♣♦❧❛t✐♦♥ ♦❢ {❑✐, ❚✐, ❝✐}■✱ t❤❡♥ σ✷ = ∂t˜

✉+r①∂①

✶ ✷∂✷ ①① ˜

✉

✐s s♦❧✉t✐♦♥ ❊①t❡♥s✐♦♥ t♦ ❇❧❛❝❦✲❙❝❤♦❧❡s ✇✐t❤ ❥✉♠♣s ❝❛♥ ❜❡ ❞♦♥❡ ❜② t❤❡ s❛♠❡ ♠❡t❤♦❞✳ ❚❤❡ r❡s✉❧t ✐s ❛ s✐♠✐❧❛r ♦♣t✐♠❛❧ ❝♦♥tr♦❧ ♣r♦❜❧❡♠ ❜✉t ✇✐t❤ ❛ P■❉❊✳ ❲♦r❦s ♦♥❧② ✇❤❡♥ ❉✉♣✐r❡✬s ❛♣♣❧②❄

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✺ ✴ ✺✶

❉✉❛❧✐t② ❛t t❤❡ ❉✐s❝r❡t❡ ▲❡✈❡❧

❋❊▼ ✰ ❊✉❧❡r ✐♠♣❧✐❝✐t ✇✐t❤ t✐♠❡ st❡♣ δt ⇒ (❇ + ❆)❈ ♥ − ❇❈ ♥+✶ = ✵ ❈ ◆ = ❈❚ ✇❤❡r❡ ❈ ♥ ✐s t❤❡ ✈❡❝t♦r ♦❢ ✈❛❧✉❡s ♦❢ ❈❤(q✐, ♥δt) ❛♥❞ ❆✱❇ ❛r❡ ❇✐❥ = ✶ δt

• Ω❤

❲ ✐❲ ❥, ❆✐❥ = ❛(❲ ✐, ❲ ❥) t❤❡ ❜✐❧✐♥❡❛r ❢♦r♠ ♦❢ t❤❡ P❉❊✳ ▲❡t P♥+✶ ❜❡ (❆ + ❇)❚P♥+✶ − ❇❚P♥ = ✵ P✵ = P✵ ❚❤❡♥ ✵ = P♥❚(❆ + ❇)❯♥ − P♥❚❇❯♥+✶ = P♥−✶❚❇❯♥ − P♥❚❇❯♥+✶ ❙✉♠♠✐♥❣ ✉♣ ♦✈❡r ❛❧❧ ♥ ❣✐✈❡s P✵❚❇❯✶ = P◆−✶❚❇❯◆ ❈❤♦♦s✐♥❣ P✵

❥ = δ✐❥ ❣✐✈❡s ( ■

• ❥=✶

❜✐❥)❯✶

✐ ≈ (❇❯✶)✐ = P◆−✶❚❇❯◆

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✻ ✴ ✺✶

❙♠✐❧❡ ✇✐t❤ ❛ ❈❤❡❛♣ ❙♣❧✐♥❡

❊①✿ σ(①, t) ❈ ✶✱ ❜✐❧✐♥❡❛r ✐♥ ❛♥ ✭①✱t✮ ♣❛r❛❧❧❡❧♦❣r❛♠✱ ❣r♦✇s ❧❛r❣❡ ♦✉ts✐❞❡ ⇒ ✽ ♣❛r❛♠❡t❡rs✿ ❙✐❥, σ✐❥ ❜✉t ✵ < ❙✶(t) < ❙✷(t) ✐s ♥❡❡❞❡❞ s♦ s❡t ❙✶✶ = ③✷

✵, ❙✷✶ = ③✷ ✶, ❙✶✷ = ❙✶✶ + ③✷ ✷, ❙✷✷ = ❙✷✶ + ③✷ ✸,

σ✶❥ = ③✷

✸+❥

✶ + ③✷

✸+❥

σ✷❥ = ③✷

✺+❥

✶ + ③✷

✺+❥

, ❥ = ✶, ✷ s♦ ❛s t♦ ✇♦r❦ ✇✐t❤ ❛♥ ✉♥❝♦♥str❛✐♥❡❞ s❡t ♦❢ ♣❛r❛♠❡t❡rs {③✐}✼

✵

▲❡t ❙✐, σ✐, ✐ = ✶, ✷ ❜❡ ❧✐♥❡❛r ✐♥ t✿ ❙✐ = ❙✐✶(✶ − t ❚ ) + ❙✐✷ t ❚ , σ✐ = σ✐✶(✶ − t ❚ ) + σ✐✷ t ❚ ❚❤❡ ✈♦❧ s✉r❢❛❝❡ ✐s ✭❛ = σ(✵, t)✮✿ σ(❙, t) =        ❛ + (✷σ✶−❛

❙✶

− σ✷−σ✶

❙✷−❙✶ )❙ + (σ✷−σ✶ ❙✷−❙✶ − σ✶−❛ ❙✶ )❙✷ ❙✶

✐❢ (❙ < ❙✶) σ✷ ❙−❙✶

❙✷−❙✶ + σ✶ ❙✷−❙ ❙✷−❙✶

✐❢ ❙✶ ≤ ❙ ≤ ❙✷ σ✷ + (❙ − ❙✷)σ✷−σ✶

❙✷−❙✶ +

• (❙ − ❙✷)σ✷−σ✶

❙✷−❙✶

✷ ✐❢ ❙ > ❙✷

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✼ ✴ ✺✶

❖❜s❡r✈❛t✐♦♥ ❉❛t❛

❙tr✐❦❡ ✶ ▼♦♥t❤ ✷ ▼♦♥t❤s ✻ ▼♦♥t❤s ✶✷ ▼♦♥t❤s ✷✹ ▼♦♥t❤s ✸✻ ▼♦♥t❤s ✼✵✵ ✼✸✸ ✽✵✵ ✻✺✵✳✻ ✾✵✵ ✺✻✾✳✽ ✶✵✵✵ ✹✻✼✳✽ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✶✷✶✺ ✷✺✸✳✹ ✶✷✷✺ ✷✶✾ ✷✹✺ ✶✷✺✵ ✶✾✻✳✻ ✷✷✹✳✷ ✷✻✾✳✷ ✶✷✼✺ ✶✼✹✳✺ ✷✵✸✳✾ ✷✺✶ ✶✸✵✵ ✶✺✷✳✾ ✶✽✹✳✶ ✷✸✸✳✷ ✶✸✷✺ ✶✸✶✳✾ ✶✻✹✳✾ ✷✶✺✳✽ ✶✸✺✵ ✶✶✶✳✼ ✶✹✻✳✸ ✶✾✽✳✾ ✶✸✻✺ ✶✵✵ ✶✸✼✺ ✺✵✳✻ ✻✵ ✾✷✳✺ ✶✸✾ ✶✽✷✳✻ ✶✸✽✵ ✹✻✳✶ ✺✺✳✽ ✶✸✽✺ ✹✶✳✽ ✺✶✳✽ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ ✶✼✵✵ ✸✷✳✼ ✼✺✱✼ ✶✽✵✵ ✶✺✳✺ ✶✾✵✵ ✺✳✷ ✐♥❞❡① ❙P❳ ♦♥ ✷✶✳✶✷✳✷✵✵✻✮ ❛t s♣♦t ♣r✐❝❡ ✶✹✶✽✳✸✱ r❂✸✴✶✵✵ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✽ ✴ ✺✶

❘❡s✉❧ts ■

500 1000 1500 2000 2500 3000 3500 4000 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 "usigma.txt"using 1:2:4 500 1000 1500 2000 2500 3000 3500 4000 0.5 1 1.5 2 2.5 3

• 200

200 400 600 800 1000 1200 1400 "usigma.txt"using 1:2:3

10 20 30 40 50 60 70 5 10 15 20 25 30 35 40 45 50 "converge.h"using 1:2 "converge.h"using 1:3

600 800 1000 1200 1400 1600 1800 0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800 "uduo.txt"using 1:2:3 "uduo.txt"using 1:2:4

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✷✾ ✴ ✺✶

■♠♣❧✐❡❞ ❱♦❧❛t✐❧✐t✐❡s

• ✐✈❡♥ ❛ ❢❡❛s✐❜❧❡ ♣r✐❝❡ s✉r❢❛❝❡ ✉(❑, ❚) ✭ ♥♦t❡ ✉(❑, ✵) = (❙✵ − ❑)+ ✐s

♥❡❡❞❡❞ ❢♦r ❛ ❝❛❧❧✮ ✇❡ ❞❡✜♥❡ t❤❡ ❧♦❝❛❧ ✭❉✉♣✐r❡✮ ✈♦❧❛t✐❧✐t② ❜② σ✷

▲

✷ = ∂t✉ + r①∂①✉ ∂①①✉ ❛♥❞ t❤❡ ✐♠♣❧✐❡❞ ✈♦❧❛t✐❧✐t② ❜② t❤❡ ❇❧❛❝❦✲❙❝❤♦❧❡s ❢♦r♠✉❧❛ ✭s❡❡ ❍♦❢♠❛♥♥✮ ❇❙(❙✵, ✵, ❑, ❚, σ■) = ✉(❑, ❚) ◆♦✇ t♦ ✜t ❛ s❡t ♦❢ ❞✐s❝r❡t❡ ♦❜s❡r✈❛t✐♦♥s ✇❡ s♦❧✈❡ ♠✐♥

σ■ ∈B

• |❇❙(❙✵, ✵, ❑✐, ❚✐, σ■) − ✉✐|✷

❚❤❡ ❧♦❝❛❧ ✈♦❧ σ▲ ✐s r❡❝♦✈❡r❡❞ ✐ts ❞❡✜♥✐t✐♦♥ ✐♥ t❡r♠s ♦❢ ✉(❑, ❚) := ❇❙(❙✵, ✵, ❑, ❚, σ■) ❥✉st s♦ ❝♦♠♣✉t❡❞✳ ❱❡r② ❢❛st ❜✉t ❤❛r❞ t♦ ❡①t❡♥❞✳ ❯s❡❞ t♦ s♣❡❡❞✲✉♣ ❙t♦❝❤✲❱♦❧ ♠♦❞❡❧s ◗❄✿ ❋✐♥❞ σ■✿ ❇❙(❙, t, ❑, ❚, σ■) = ✉(❙, t)❙t♦❝❤✲❱♦❧ ▼♦❞❡❧❑,❚ ❄

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✵ ✴ ✺✶

❚❤❡ s❛♠❡ ✇✐t❤ ■♠♣❧✐❡❞ ❱♦❧❛t✐❧✐t✐❡s

500 1000 1500 2000 2500 3000 3500 4000 0.5 1 1.5 2 2.5 3 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700

10 20 30 40 50 60 70 5 10 15 20 25 30 35 40 45 50 Cost/8000 2*log(gradient)

600 800 1000 1200 1400 1600 1800 2000 0.5 1 1.5 2 2.5 3 100 200 300 400 500 600 700 800

◆♦ ♠♦r❡ P❉❊✱ σ■ ✐s t❤❡ s♣❧✐♥❡ ❛♥❞ ✉ ✐s ❣✐✈❡♥ ❜② ❇❙ ❢♦r♠✉❧❛

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✶ ✴ ✺✶

◗✉❛s✐✲◆❡✇t♦♥ ▼❡t❤♦❞ ✭▲❡✈❡♥❜❡r❣✲▼❛rq✉❛r❞t✮

▲❡t ❈ ❜❡ s♦❧ ♦❢ ❉✉♣✐r❡✬s ✇✐t❤ ✈♦❧ σ(❛)✱ ❛ ∈ ❘♥❛ t❤❡ s♣❧✐♥❡ ♣❛r❛♠❡t❡rs ♠✐♥

σ ❏(❛) = ✶

✷

♥❊

• ❥=✶

|❊❥|✷ ❊❥ = ❈(❑❥, t❥) − ❈❥ ⇒ ∂❏ ∂❛✐ =

• ❥

∂❊❥ ∂❛✐ ❊❥ ❏” = ❊”❊ + ❊ ′❚❊ ′≈ ❊ ′❚❊ ′ + α■ ❆♣♣r♦①✐♠❛t❡ ◆❡✇t♦♥ st❡♣✿ (❊ ′❚❊ ′ + α■)(❛♠+✶ − ❛♠) = −❊ ′❊

200 400 600 800 1000 1200 1400 1600 1800 0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 "s.txt" 200 400 600 800 1000 1200 1400 1600 1800 0.5 1 1.5 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 "s.txt"

❱♦❧❛t✐❧✐t② s✉r❢❛❝❡ ❝♦♠♣✉t❡❞ ✇✐t❤ t❤❡ q✉❛s✐✲◆❡✇t♦♥ ♠❡t❤♦❞ ✭❧❡❢t✮ ❛♥❞ ✇✐t❤ t❤❡ ❈● ♠❡t❤♦❞ ✭r✐❣❤t✮✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✷ ✴ ✺✶

❖✉t❧✐♥❡

✶

✳ ■♥tr♦❞✉❝t✐♦♥

✷

✳ ❈❛❧✐❜r❛t✐♦♥

✸

✳ ❉✉♣✐r❡✬s ❡q✉❛t✐♦♥ ❘❡❞✉❝❡❞ ❇❛s✐s ❢♦r ❙♦❧✈❡rs

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✸ ✴ ✺✶

❘❡❞✉❝❡❞ ❇❛s✐s ✭❆✳ P❛t❡r❛✮

❘❡♣❧❛❝❡ t❤❡ ❤❛t ❢✉♥❝t✐♦♥s ✇✐(①) ✐♥ t❤❡ ❢♦r♠✉❧❛ ✉❤(①, t) =

◆

• ✶

✉✐(t)✇✐(①) ❜② ❛ ●❛❧❡r❦✐♥ ♠❡t❤♦❞ ✇✐t❤ ▼ << ◆✿ ✉❘❇(①, t) =

▼

• ✶

❛✐(t)✉✐(①, t) ✇✐t❤ ❛✐ ❣✐✈❡♥ ❜② ✭✇❤❡♥ r = ✵✮

• R+ ✉❘❇(❚)✉✐(❚) −
• R+×(t✵,❚)

(✉❘❇∂t✉✐ + ✉✐ σ✷①✷ ✷ ∂①①✉❘❇) =

• R+(❑ − ①)+✉✐(t✵)

❯♥❧❡ss ✉✐(①, ✵) = (❑ − ①)+✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✹ ✴ ✺✶

P❖❉ ♦r ❙❱❉ t♦ ❈❤♦♦s❡ t❤❡ ❜❛s✐s

❙♥❛♣ s❤♦ts ✭✉s❡❞ ✇✐t❤ t❤❡ ◆❛✈✐❡r✲❙t♦❦❡s ❡q✉❛t✐♦♥s✮ ❈♦♠♣✉t❡ t❤❡ s♦❧✉t✐♦♥ ♦❢ t❤❡ ♣❛r❛❜♦❧✐❝ P❉❊ ✉(①, t) = ▼

✶ ❛✐(t)✉✐(①) ✇❤❡r❡

{✉✐} ✐s ♦❜t❛✐♥❡❞ ❜② ❙✐♥❣✉❧❛r ❱❛❧✉❡ ❉❡❝♦♠♣♦s✐t✐♦♥ ✭❙❱❉✮ ♦❢ ❆✐❥ =

• Ω

✉(t✐) · ✉(t❥)❞①, ❆ =

• ❯

▼ × ▼

• 

   λ✶ λ✷ λ✸ ▼ × ◆     ❱ ❚ ◆ × ◆

• .

♦r t❤❡ ♦rt❤♦❣♦♥❛❧ ❞❡❝♦♠♣♦s✐t✐♦♥ ✐❢ s②♠♠❡tr② ❆ = PΛ◗ ✇✐t❤ λ✶ ≥ λ✷...✱ ❯ ♦rt❤♦❣♦♥❛❧ ✉♥✐t❛r②✳ ❍❡r❡ ✈✐(①) ✐s t❤❡ ✈❡❝t♦r ❛ss♦❝✐❛t❡❞ t♦ λ✐ ❛♥❞ ✐ts ❝♦♠♣♦♥❡♥ts ♦♥ t❤❡ ✉(①, t❥) ❛r❡ ✐♥ ❯✿ ✉✐(①) = ✉✐❥✉(①, t❥) ❊✳❲✳ ❙❛❝❤s ❛♥❞ ▼✳ ❙❝❤✉ ❞✐❞ ✐t ❢♦r ❇✫ ❙ ❛♥❞ P■❉❊ ❛♥❞ r❡♣♦rt ❛ r❡❛❧ ✐♠♣r♦✈❡♠❡♥t✳

POD approximations, PIDE model POD Basis El. Deviation smallest sing. val. 3 8.36e-001 1.63e+003 4 1.36e-001 1.76e+002 5 2.17e-002 2.28e+001 6 3.31e-003 3.11e+000 7 4.86e-004 4.25e-001 8 6.82e-005 5.71e-002 9 9.17e-006 7.46e-003 10 1.18e-006 9.46e-004

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✺ ✴ ✺✶

❘❡❞✉❝❡❞ ❇❛s✐s ❢♦r t❤❡ P❉❊ s♦✈❡r

❈✳ ◆❣✉②❡♥ ❛♥❞ ❚✳ P❛t❡r❛ ❝♦♠♣✉t❡ ❛ ✷✲❜❛s❦❡t ♦♣t✐♦♥ ❜② ❛ r❡❞✉❝❡❞ ❜❛s✐s ♠❡t❤♦❞ ✇✐t❤ s♥❛♣s❤♦ts ✇✐t❤ ♣♦ss✐❜✐❧② ❞✐✛❡r❡♥t ❝♦♥st❛♥t ✈♦❧❛t✐❧✐t✐❡s ✳ ❚❤❡② ❤❛✈❡ ❛ s♦♣❤✐st✐❝❛t❡❞ s❡❛r❝❤ ♠❡t❤♦❞ ❢♦r t❤❡ ❜❡st ❜❛s✐s ❜❛s❡❞ ♦♥ ❛ ♣♦st❡r✐♦r✐ ❡st✐♠❛t❡s ❛♥❞ ♣r♦♦✈❡ ❡rr♦r ❜♦✉♥❞s✳ ❚❤❡ s♦❢t✇❛r❡ ✐s ♦♣❡♥ s♦✉r❝❡

5 10 15 20 25 30 10

2

10

1

10

N ∗

N,max

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✻ ✴ ✺✶

❙♣❡❡❞✐♥❣ ✉♣ ❝♦♠♣✉t❛t✐♦♥s ✇✐t❤ ❞✐✛❡r❡♥t ✈♦❧❛t✐❧✐t✐❡s

❲❤❡♥ σ, r ❛r❡ ❝♦♥t❛♥t ❝♦♠♣✉t✐♥❣ t❤❡ ♦♣t✐♦♥ ♣r✐❝❡ ✐s ✈❡r② ❢❛st✿

❞♦✉❜❧❡ ❇❙P✉t✭❞♦✉❜❧❡ ❙✱ ❞♦✉❜❧❡ t✱ ❞♦✉❜❧❡ r✱ ❞♦✉❜❧❡ ❑✱ ❞♦✉❜❧❡ s✐❣♠❛✮ ④ st❛t✐❝ ❝♦♥st ❞♦✉❜❧❡ sqrt✷❂sqrt✭✷✳✮✱ s✐❣✷ ❂ s✐❣♠❛✯s✐❣♠❛✴✷❀ ✐❢✭t❁✶❡✲✺✮ r❡t✉r♥ ❑❃❙❄ ❑✲❙ ✿ ✵❀ ❞♦✉❜❧❡ s✐❣st ❂ s✐❣♠❛✯sqrt✭t✮ ❀ ❞♦✉❜❧❡ ❞✶ ❂ ✭❧♦❣✭❙✴❑✮✰✭r✰s✐❣✷✮✯t✮✴s✐❣st❀ ❞♦✉❜❧❡ ❞✷ ❂ ✭❧♦❣✭❙✴❑✮✰✭r✲s✐❣✷✮✯t✮✴s✐❣st❀ ❞♦✉❜❧❡ ◆♠❞✶ ❂ ✶✲❡r❢❝✭✲❞✶✴sqrt✷✮✴✷❀ ❞♦✉❜❧❡ ◆♠❞✷ ❂ ✶✲❡r❢❝✭✲❞✷✴sqrt✷✮✴✷❀ r❡t✉r♥ ❑✯❡①♣✭✲r✯t✮✯◆♠❞✷ ✲ ❙✯◆♠❞✶❀ ⑥

P❡r❢♦r♠ ❛♥ ❙❱❉ ❞❡❝♦♠♣♦s✐t✐♦♥ ♦❢ ❆✐❥ =

• R×(✵,❚) ✉σ✐(①, t)✉σ❥(①, t)

❚❤❡ ❡✐❣❡♥ ✈❛❧✉❡s ⇒

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10 12 5 10 15 20 25 30 35 40 "eigen.txt"

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✼ ✴ ✺✶

❙♦♠❡ ❇❛s✐s ❋✉♥❝t✐♦♥s

"basis.txt" matrix 10 20 30 40 50 60 5 10 15 20 25 30 35 40

• 20

20 40 60 80 100 120 140 160 "basis.txt" matrix 10 20 30 40 50 60 5 10 15 20 25 30 35 40

• 5

5 10 15 20 25 30 35 40 "basis.txt" matrix 10 20 30 40 50 60 5 10 15 20 25 30 35 40

• 16
• 14
• 12
• 10
• 8
• 6
• 4
• 2

2

❋✐rst✱ s❡❝♦♥❞ ❛♥❞ ❢♦✉rt❤ ❜❛s✐s ❢✉♥❝t✐♦♥ ✭❘❡❝❛❧❧ t❤❛t t❤❡s❡ ❛r❡ ♥♦♥✲❝♦♥✈❡① ❝♦♠❜✐♥❛t✐♦♥s ♦❢ ❡❧❡♠❡♥t❛r② ❇❧❛❝❦✲❙❝❤♦❧❡s s♦❧✉t✐♦♥s✮✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✽ ✴ ✺✶

❘❡❞✉❝❡❞ ❇❛s✐s ❜❛s❡❞ ♦♥ ✈❛r②✐♥❣ ✈♦❧❛t✐❧✐t✐❡s

❆ ✜♥✐t❡ s❡t ♦❢ ✐♥❞❡♣❡♥❞❡♥t µ✐ = σ✷

✐ /✷ ✐s ❝❤♦s❡♥ ❛♥❞ ✉✐ ❞❡♥♦t❡s t❤❡ s♦❧✉t✐♦♥

♦❢ ❉✉♣✐r❡✬s ✇✐t❤ σ✐ ❢♦r ✈♦❧❛t✐❧✐t② ❤♦✇❡✈❡r ✉✐ ✐s ♥♦t r❡q✉✐r❡❞ t♦ s❛t✐s❢② t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥✳ ❚❤❡♥ ❢♦r ❛♥② σ t❤❡ ♦♣t✐♦♥ ✐s ❝♦♠♣✉t❡❞ ❜② ✉σ(①, t) =

■

• ✶

❛✐(t)✉✐(①, t) ✇✐t❤ ❛✐ ❞❡t❡r♠✐♥❡❞ ❜②

• R+ ✉σ(❚)✉✐(❚) −
• R+×(t✵,❚)

(✉σ∂t✉✐ + ✉✐ σ✷①✷ ✷ ∂①①✉σ) =

• R+(❑ − ①)+✉✐(t✵)

❈❛❧✐❜r❛t✐♦♥ ✐s t❤❡♥✿ ♠✐♥

❜ { ❑

• ❦=✶

|✉σ(①❦, t❦) − ✉❦|✷ : σ✷ = ✷

■

• ✶

❜✐µ✐(①, t), ✉σ ❜② ❛❜♦✈❡}

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✸✾ ✴ ✺✶

❆♥ ✐t❡r❛t✐✈❡ s♦❧✉t✐♦♥ ✇♦✉❧❞ t❤❡♥ ❜❡

✶ ❈❤♦♦s❡ ❛♥ ✐♥✐t✐❛❧ s❡t ❜ := {❜✵

✐ }■ ✶✳

✷ ❈♦♠♣✉t❡ σ ❜② t❤❡ ♠✐♥✐♠✐③❛t✐♦♥ ♣r♦❜❧❡♠ ✸ ❈♦♠♣✉t❡ {❛✐}■

✶ ❜② t❤❡ P❉❊ ♣r♦❥❡❝t❡❞ ♦♥ t❤❡ ❜❛s✐s✮

✹ ❈♦♥str✉❝t ❜♠+✶ ✇❤✐❝❤ ❞✐♠✐♥✐s❤❡s t❤❡ ❡rr♦r ✐♥ ✭✶❛✮ ✺ ●♦ ❜❛❝❦ t♦ ❙t❡♣ ✷ ✇✐t❤ ♠ ← ♠ + ✶ ✉♥t✐❧ ❝♦♥✈❡r❣❡♥❝❡✳ ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✵ ✴ ✺✶

◆✉♠❡r✐❝❛❧ ❚❡sts

❲❡ ❤❛✈❡ s♦❧✈❡❞ s❡✈❡r❛❧ ✐♥✈❡rs❡ ♣r♦❜❧❡♠s ❢♦r ✇❤✐❝❤ t❤❡ s♦❧✉t✐♦♥s ❛r❡ ❦♥♦✇♥✳ ❚❤❡s❡ ❛r❡ ❝♦♥str✉❝t❡❞ ❜② ❝❤♦♦s✐♥❣ ❛ σ❞✱ t❤❡♥ ❝♦♠♣✉t❡ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❝❛❧❧ ✉❞ ❛♥❞ s❛♠♣❧❡ ✶✵ ✈❛❧✉❡s ♦❢ ✉❞ ❛s ♦❜s❡r✈❛t✐♦♥s ❛t t✐♠❡ ❚ = ✶✳ ■♥ (✵.✺❙✵, ✶.✺❙✵) × (✵, ❚) t❤❡ t❛r❣❡t ✈♦❧❛t✐❧✐t② σ❞ ✐s ❛ ❜✐✲❝✉❜✐❝ s♣❧✐♥❡ ✐♥t❡r♣♦❧❛t✐♦♥ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ✈❛❧✉❡s ❢♦r ❥ = ✶...◆①, ❦ = ✶...◆t✿ ✐ = (❥ − ✶)◆t + ❦, ①✐ = ❙✵(✵.✺ + ❥ − ✶ ◆① − ✶), t✐ = ❚ ❦ − ✶ ◆t − ✶, σ❞(①✐, t✐) = σ✵ ✐ ■ (✷ − ✐ − ✶ ■ − ✶) ✇✐t❤ ■ = ◆① × ◆t✳ ❚❤❡ ❝♦♥tr♦❧ ✈♦❧❛t✐❧✐t② ✐s ❛ s✐♠✐❧❛r s♣❧✐♥❡✳ ❍♦✇❡✈❡r t♦ ♠❛❦❡ t❤❡ ♣r♦❜❧❡♠ ❤❛r❞❡r ✇❡ ❤❛✈❡ ❛❞❞❡❞ ❛ ♣❡rt✉r❜❛t✐♦♥ t♦ t❤✐s s♣❧✐♥❡ ✐♥t❡r♣♦❧❛t✐♦♥ ✐♥ s♦♠❡ ❝❛s❡s❀

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✶ ✴ ✺✶

❚❤❡ ❘❡❞✉❝❡❞ ❇❛s✐s

❚❤❡ ❜❛s✐s µ✐ ✐s t❤❡ s✐♠✐❧❛r ❝✉❜✐❝ s♣❧✐♥❡ ✐♥t❡r♣♦❧❛t✐♦♥ ✇❤❡♥ ❛❧❧ ♣♦✐♥t ✈❛❧✉❡s ❛r❡ σ✵ ❡①❝❡♣t t❤❡ ✐✲t❤ ♦♥❡ ✇❤❡r❡ t❤❡ ✈❛❧✉❡ ✐s ✶✳ ❚❤❡♥ ✉✐ ✐s t❤❡ s♦❧✉t✐♦♥ ♦❢ ❉✉♣✐r❡✬s ✇✐t❤ σ = ✷√µ✐✳

200 400 600 800 1000 1200 1400 1600 1800 0.2 0.4 0.6 0.8 1 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 "u.txt"

❖♥❡ ♦❢ t❤❡ ✹✵ × ✹✵ s♣❧✐♥❡✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✷ ✴ ✺✶

❚❤❡ ❖♣t✐♠✐③❛t✐♦♥ Pr♦❜❧❡♠

❙♦❧✈❡❞ ❜② ❈● ✰ ❆❉ ✇✐t❤ ◆① = ✽, ◆t = ✺ ♦r ◆① = ✺, ◆t = ✺ ✳ ❚❤❡ ❝♦st ❢✉♥❝t✐♦♥ ✐♥ ✭✶✮ ✐s r❡❞✉❝❡❞ ❢r♦♠ ✼✵ t♦ ✶✵−✺ ✐♥ ✷✵ ✐t❡r❛t✐♦♥s❀ t❤❡ ❣r❛❞✐❡♥t ♥♦r♠ ✐s t❤❡♥ ✶✵−✸✳

200 400 600 800 1000 1200 1400 1600 1800 1 2 3 4 5 6 7 8 9 100 200 300 400 500 600 700 800 "u.txt" ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✸ ✴ ✺✶

❱♦❧❛t✐❧✐t② ❜② t❤❡ ❘❇▼

200 400 600 800 1000 1200 1400 1600 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 8
• 6
• 4
• 2

2 4 "s.txt" "s0.txt" 200 400 600 800 1000 1200 1400 1600 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 "s.txt" "s0.txt" 200 400 600 800 1000 1200 1400 1600 1800 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 "s.txt" "s0.txt"

❚❤❡ r❡❝♦✈❡r② ♦❢ σ ❛s ❛ s♦❧✉t✐♦♥ ♦❢ t❤❡ ❧✐♥❡❛r s②st❡♠ ❞♦❡s ♥♦t ✇♦r❦ ✐❢ ✇❡ ❞♦ ♥♦t ❛❞❞ t❤❡ ❝♦♥str❛✐♥ts ■

✶ ❜✐ = ✶❀ ✇✐t❤ t❤✐s ❝♦♥str❛✐♥t t❤❡ r❡s✉❧ts ❛r❡

r❛t❤❡r ❣♦♦❞ ❡✈❡♥ ✐❢ t❤❡ t❛r❣❡t ✈♦❧❛t✐❧✐t② s✉r❢❛❝❡ ❞♦❡s ♥♦t ❜❡❧♦♥❣ t♦ t❤❡ ❛♣♣r♦①✐♠❛t✐♦♥ s♣❛❝❡✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✹ ✴ ✺✶

❈♦♥✈❡r❣❡♥❝❡ ❙t✉❞②

❚❤❡ ❡rr♦r s❡❡♠s ♥♦t t♦ ❞❡♣❡♥❞ ♠✉❝❤ ♦♥ t❤❡ ♥✉♠❜❡r ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s✳ ❜❛s✐s s✐③❡ ❡rr♦r ❣r❛❞✐❡♥t ✾ ✸✳✽✹✼✶✷❡✲✵✺ ✻✳✵✾✹✹✸❡✲✵✻ ✷✺ ✾✳✻✽✼✸✶❡✲✵✻ ✷✳✶✹✺✸✸❡✲✵✼ ✻✹ ✹✳✸✺✶✾✸❡✲✵✼ ✼✳✽✼✵✾✺❡✲✶✶ ✶✹✹ ✶✳✺✼✵✼✷❡✲✵✻ ✺✳✹✹✵✽✾❡✲✶✶ ❈♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ❡rr♦r ✈❡rs✉s ♥✉♠❜❡r ♦❢ ❜❛s✐s ❢✉♥❝t✐♦♥s✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✺ ✴ ✺✶

P❛rt ■❱ ✿ ❆ P❛r❛♠❡tr✐❝ ❋r❛♠❡✇♦r❦ ❢♦r ❈❛❧✐❜r❛t✐♦♥

♠✐♥

σ

• (✉(❑✐, τ✐) − ❝✐)✷ : ∂t✉ − σ✷①✷

✷ ∂①①✉ + r①✉ = ✵, ✉(✵) = (❙✵ − ①)+ ❋♦r s✐♠♣❧✐❝✐t② r = ✵✳ ❈❛♥ ♦♥❡ ✉s❡ σ✷ = ✷∂t✉ ①✷∂①①✉ ? ❚r② ✉(①, t) =

■

• ✶

❛✐✉✐(①, t) ✇❤❡r❡ ✉✐ ✐s s♦❧✉t✐♦♥ ♦❢ ❉✉♣✐r❡✬s ✇✐t❤ σ✐ ❛♥❞ ❛ = {❛✐}■

✶ ✐s s♦❧✉t✐♦♥ ♦❢

♠✐♥

❛∈R■{ ❑

• ❦=✶

|

■

• ✐=✶

❛✐✉✐(①❦, t❦) − ✉❦|✷ :

■

• ✶

❛✐ = ✶ } ◆♦t✐❝❡ t❤❛t ✉ s❛t✐s✜❡s ❉✉♣✐r❡✬s ✇✐t❤ σ✷ =

■

✶ ❛✐ σ✷ ✐ ∂①①✉✐

❏

✶ ❛✐ ∂①①✉✐

• ❈❤❡❝❦ t❤❛t σ✷ ✐s ♣♦s✐t✐✈❡ ❡✈❡r②✇❤❡r❡✳
• ❙❤♦✇ t❤❛t t❤❡ ♠❡t❤♦❞ ✐s st❛❜❧❡ ✇✐t❤ r❡s♣❡❝t t♦ ❞❛t❛ ✈❛r✐❛t✐♦♥s✳
• ❙❤♦✇ t❤❛t r❡s✉❧ts ❛r❡ ❣♦♦❞ ❡✈❡♥ ✇❤❡♥ ■ ✐s s♠❛❧❧
• ❲♦r❦s ♦♥❧② ❢♦r ♦♥❡ ✈♦❧ s✉r❢❛❝❡ ✭♥♦ ❢✉t✉r❡✮

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✻ ✴ ✺✶

❈♦♠♣❛r✐s♦♥

◆♦t✐❝❡ t❤❛t ❜♦t❤ ❛♣♣r♦❛❝❤❡s ✇♦✉❧❞ ❜❡ ♣r♦✈❡❞ ✐❞❡♥t✐❝❛❧ ✐❢ ♦♥❡ ❝♦✉❧❞ s❤♦✇ t❤❛t ❢♦r ❛♥② {❛✐}■

✶ t❤❡r❡ ❡①✐sts {❜❥}■ ✶ s✉❝❤ t❤❛t ✇✐t❤ ✉(①, t) = ■ ✶ ❛✐✉✐(①, t)

• R+ ✉(❚)✉✐(❚) −
• R+×(t✵,❚)

(✉∂t✉✐ + (

■

• ✶

❜❥µ❥)①✷ ✷ ∂①①✉ ✉✐)❞①❞t =

• R+(❑ − ①)+✉✐(t✵) ∀✐ = ✶..■

❚❤✐s ✐s ❛ ❧✐♥❡❛r s②st❡♠ ❇ ❜ = ❋ ✇✐t❤ ❇✐❥ =

• R+×(t✵,❚)

µ❥ ①✷ ✷ ∂①①✉ ✉✐ ❋✐ =

• R+×(t✵,❚)

✉✐∂t✉ ■❢ ❇ ✐s ♥♦t ♦❢ ❢✉❧❧ r❛♥❦ ♦♥❡ ♠❛② ❛❞❞ ■

✶ ❛✐ = ✶ ❢♦r ❡q✉✐✈❛❧❡♥❝❡✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✼ ✴ ✺✶

❱♦❧❛t✐❧✐t② r❡❝♦♥str✉❝t❡❞ ❢r♦♠ t❤❡ P❉❊

200 400 600 800 1000 1200 1400 1600 1800 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 "s.txt" "s0.txt" 200 400 600 800 1000 1200 1400 1600 1800 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 "s.txt" "s0.txt"

❈♦♠♣✉t❡❞ ✈♦❧❛t✐❧✐t② s✉r❢❛❝❡ ✇✐t❤ ✽ × ✺ ✉♥❦♥♦✇♥s ❛♥❞ t❤❡ t❛r❣❡t ✈♦❧❛t✐❧✐t② ✇❤✐❝❤ ❜❡❧♦♥❣ t♦ t❤❡ ❞✐s❝r❡t❡ s♣❛❝❡✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✽ ✴ ✺✶

❆♥❛❧②s✐s ♦❢ t❤❡ ▼❛tr✐① ❇

❆❧❧ ❝♦♠♣✉t❡❞ s♣❡❝tr❛ ❤❛✈❡ ✈❡r② s♠❛❧❧ ❡✐❣❡♥✈❛❧✉❡s✳ ❇❡❧♦✇ t❤❡ ✺ × ✺ ❝❛s❡✳ ❚❤❡ ❧❛r❣❡st ❡✐❣❡♥❢✉♥❝t✐♦♥✱ ✸✹.✽✸✽✽✺✶✸ + ✵✐ ✐s ♥♦t ❞✐s♣❧❛②❡❞✳

• 0.02

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

• 0.02
• 0.015
• 0.01
• 0.005

0.005 0.01 0.015 0.02 ❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✹✾ ✴ ✺✶

❘❡s✉❧ts ♦♥ r❡❛❧ ❞❛t❛

600 800 1000 1200 1400 1600 1800 2000 500 1000 1500 2000 2500 3000 3500 4000

• 100

100 200 300 400 500 600 700 800 "ud.txt" "od.txt" 800 1000 1200 1400 1600 1800 2000 2200 2400 500 1000 1500 2000 2500 3000 3500 4000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 "s.txt"

❈❛♣❛❝✐t② t♦ r❡♣r♦❞✉❝❡ t❤❡ ❞❛t❛ ✐♥❞❡① ❙P❳ ♦♥ ✷✶✳✶✷✳✷✵✵✻ ❛t s♣♦t ♣r✐❝❡ ✶✹✶✽✳✸✱ r❂✸✴✶✵✵ ❚❤❡ ✈♦❧❛t✐❧✐t② ❝❛♥♥♦t ❜❡ r❡❝♦♥str✉❝t❡❞ ♥❡❛r t❤❡ ❜♦✉♥❞❛r✐❡s ◆❡①t ✐❞❡❛ ✐s t♦ r❡str✐❝t t❤❡ ❞♦♠❛✐♥ ✇❤❡r❡ t❤❡ ❝❛❧✐❜r❛t✐♦♥ ✐s ❞♦♥❡✳

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✺✵ ✴ ✺✶

❈♦♥❝❧✉s✐♦♥

• P❖❉✲❙❱❉ ❝❛♥ ♣♦t❡♥t✐❛❧❧② ❛❝❝❡❧❡r❛t❡ ❣r❡❛t❧② t❤❡ ❝♦♠♣✉t❛t✐♦♥s
• ■t ❝❛♥ ❜❡ ♠✐①❡❞ ✇✐t❤ s♣❛rs❡ ❣r✐❞s
• ■t s❡❡♠s t♦ ❜❡ ❛ ❣♦♦❞ ✐❞❡❛ ❢♦r ❝❛❧✐❜r❛t✐♦♥
• ❲✐❧❧ ✇♦r❦ ♦♥❧② ✇✐t❤ ❧✐♥❡❛r ♠♦❞❡❧s
• ❍♦✇ t♦ ❜✉✐❧t ✐♥ t❤❡ ❝♦♥str❛✐♥ts❄

❖✳ P✐r♦♥♥❡❛✉ ✭▲❏▲▲✲❯P▼❈✮ ❚❡❝❤♥✐q✉❡s ❢♦r t❤❡ ❈❛❧✐❜r❛t✐♦♥ ❋✐♥❛♥❝✐❛❧ ❖♣t✐♦♥ ▼♦❞❡❧s ❖❝t♦❜❡r ✸✵✱ ✷✵✵✽ ✺✶ ✴ ✺✶