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2016 FOR LINZ estimates On monotone for approximations error - - PowerPoint PPT Presentation

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SLIDES 2016 FOR LINZ estimates On monotone for approximations error of Bellman / Isaacs equations Jakobsen R Espen . University of NTNU Norwegian Technology sience and RKAM 24.11.2016 , : / Bellman Isaacs T ecfns

slide-1
SLIDE 1

SLIDES

FOR LINZ

2016

slide-2
SLIDE 2 On error estimates for monotone approximations
  • f
Bellman / Isaacs equations Espen R . Jakobsen Norwegian University of sience and Technology NTNU RKAM , 24.11.2016
slide-3
SLIDE 3

Outline

: T Bellman / Isaacs ecfns 2 Monotone numerical methods Error estimates 3 last
  • rder
eqhs 4 2nd
  • rder
convex eqlns 5 Fractional
  • rder
convex eqins 6 2nd
  • rder
non
  • convex
eq ' ns 7 New : Fractional
  • rder
non
  • convex
eqins
slide-4
SLIDE 4 T . Bellman / Isaacs

equations

Controlled SDES dxs =
  • r
% (b) dws + b%(XD It

{

xex map . E {

letftxs

, as ) ds + g ( Xt )

}=:u(

  • x. t)

Dynamic Programming

at + sup { tr[
  • D?u]
+ b9 Dutfd }=O ,

{

next ) = gas .BeUmaneg@
slide-5
SLIDE 5 T . Bellman / Isaacs

equations

ti→T . t i) Terminal value m > initial value
slide-6
SLIDE 6 1. Bellman / Isaacs

equations

ti→T . t i) Terminal value in > initial value ii) Levy driven SDES ( jumps ) dXs = . . . tfz ,,oy%(Xs . ,z)N~( dzidt) where { ECNC A. [ at ))=v( A) t

¥

syp§

  • tfz
, > o[ulxty9xi⇒ ) . ucx )
  • y%,⇒Du]v(d⇒]=
slide-7
SLIDE 7 1. Bellman / Isaacs

equations

ti→T . t i) Terminal value in > initial value ii) Levy driven SDES ( jumps ) dXs = . . . tfz ,,oy%(Xs . ,z)N~( dzidt) P ptoyssonmdmeas . where jumptensthttrh jump intensity) ECNC A. [ o .tD=v( A) t

§

t Levy measure Ut +

syp§

  • tfz
, > o[ulxty9xi⇒ ) . ucx )
  • y%,⇒Du]v(d⇒]=
4-
  • =
: Jan
slide-8
SLIDE 8 T . Bellman / Isaacs

equations

iii )
  • sum
games : 2 controllers with
  • pposite interests

f

Elliot
  • Katten
Fleming
  • Souganidis
Upper and lower value functions satisfy

Isaaes

equations

like
  • e. g
. Utt

inaf

sgp

{ LAB

u + Jo ' Pu + fd ' P }=O
slide-9
SLIDE 9 T . Bellman / Isaacs

equations

Typical

assumptions

: g , fo ' B , bo ' P ,
  • aip
, ya , p are bounded and Lifschitz , uniformly in a ,p S , , ,< MZPVHH + f gcz) v ( dz) < is

IHSTT

some weight No uniform elhipti city unless explicitly statet .
slide-10
SLIDE 10 2. Monotone numerical methods Sh ( t , × , ah , [ an ] ) = is Monotone : ffuhn so ,
  • ffense
+ parabolicity assumption in > F approx . at iij Consistency : / Sn ( .
  • , [ y ] )
  • eq
' n [ y ] I E E rr (h) iii ) La
  • stability
: Hun 11 La £ K t h < 1 Convergence by Bates
  • Souganidv
's
slide-11
SLIDE 11 2. Monotone numerical methods Examples : (a) ban u× e btcx ) . ukthh.hr#
  • b- ( × )
. hcHyukhL upwind ! ( b ) 02k ) u×× e
  • k

g.

uKtH-2ulnytuK= (c) {
  • i.org
u× , × ; e

uCx*oh)-2u{HtuK-oh=

SL 1 . O= (
  • ,
, ... , on )T (d)

l↳fyktP

  • UH
  • y Du ]
vldz ) = tzfygwcdz )

ukthltugntuanhlg

f linear interpolation + Is , µ a Putty )
  • idvldz
) +
slide-12
SLIDE 12 2. Monotone numerical methods Examples : Time my implicit , explicit ,
  • method
, IMEX ,
  • .

f

CFL ⇒ Monotonicity + La . stability
slide-13
SLIDE 13 Error estimates 3 . 1st
  • ther
equations 4 . 2nd
  • rder
equations , convex 5 .

Fractional

  • rder
equations , convex 6 . 2nd
  • rder
equations , non
  • convex
7 New :

Fractional

  • rder
equations , non
  • convex
slide-14
SLIDE 14 3. Error estimates
  • Tst
  • rder
eqins Modification
  • f
comparison proof
slide-15
SLIDE 15 3. Error estimates
  • Tst
  • rder
eqins Modification
  • f
comparison proof : at FkiDu)=0 Uht Fh ( × , Un ,[Un])=O '

Ulh

( × )
  • uly )
  • y
( kg ) it max at
  • I. ij
u + Fty , Dyty )) 30 def . visa . sotn Uh + FEE ,D×y ) E Kh 1113×2411 a non . + consistency Hence as in comparison proof
  • Unix )
  • awe
FCI , Dxytiyt )
  • FCFDYTED
+ Kh 1113×2411 µ (e) 0 ( he
  • ')
y ( X , y ) = tgl X
  • y 12
slide-16
SLIDE 16 3. Error estimates
  • Tst
  • rder
eqins u + Flx , Du ) = modification
  • f
comparison Proof : Un + Fhk , Un ,[un])=O ' Uh ( × )
  • u ( y)
  • y
C kg ) it Max at
  • I. ij
Units
  • at
5) e 0 ( c + he
  • ' )
sup I Un
  • w)
£ Units
  • uiy)
  • ye,
g) E 0 ( eths ' ' ) = 0 ( hk) 0 ( e)
slide-17
SLIDE 17 3. Error estimates
  • 1st
  • rder
eqins Does NOT work for 2nd
  • rder
equations : ! u + F ( × , D2u ) = O Uh + . . . = V Un (E)
  • uig)
s Fly ,
  • Ya
, )
  • FCI
, D

?

ycxiy )) tkh 'HD4xyH . HM

Jhtufyl

Problem : Need

Xi

=D I

yexiyie

F ' ' + Unix ) such that . f-

( xo %)

← ts ( It ) to conclude !
slide-18
SLIDE 18 4. Error estimates
  • 2nd
  • rder
eg ' ns , convex

Regularization

+ comparison A.

Convex

, const . coeff . B . Bellman , variable coeff . C . Bellman , variable coeff . , general schemes .
slide-19
SLIDE 19 4A . 2nd
  • rder
, convex , const . coeff . u + F ( D2u) + fcx ) = ↳ . gs ( y
  • x )
, f . . .dx u * gs + F(D2u)*g[ + f * gs = in ZF(D2u*g{) [ Jensen ] I Ue + F ( D2u , ) + fe EC)
  • 3
Fn ( us , [ ud )
  • Kh2HD4u
, His [ Consistency]
slide-20
SLIDE 20 4 A . 2nd
  • rder
, convex , const . coeff . Hence Us + Fn ( Us , [ ue ]) + f £ Kh2HD4usH , + Hf
  • fella
  • u
lip . no 0 ( h2i3 + e) → approximate subsolln
  • f
scheme , comparison : Us
  • Min
< 0 ( h 's
  • 3
+ e) U
  • Nn
< U
  • Us
+ ue
  • An
£0 (h2e
  • 3+4=0 ( h±)
slide-21
SLIDE 21 4 A . 2nd
  • rder
, convex , const . coeff . U
  • In
E 0 ( h± ) Symmetric argument :

An

  • u
< ( h± ) OBS : Need Un to be Lipschutz , uniformly in h Duo
slide-22
SLIDE 22 4 B . 2nd
  • rder
, Bellman , variable coeff . Shaking coefficients + regularization Krylov 1997 , 1999 , 2000 Bowles
  • ERJ
MZAN 2002 , SINUM 2005 , MCOMP 2007
slide-23
SLIDE 23 4 B . 2nd
  • rder
, Bellman , variable coeff . Shaking coefficients + regularization U + sup { aocx ) uxx + fetch } = a

/

approximate u ' + sup { a 0 ( × + e) uE× + f0Cx+e) } = , let < {
slide-24
SLIDE 24 4 B . 2nd
  • rder
, Bellman , variable coeff . Shaking coefficients + regularization U + sup { aocx ) uxx + fon } = a u ' + sup { a 0 ( × + e) uI× + f0Cx+e) } = , let's I * i→ ×
  • e
U ' C x
  • e)
+ ao ( ×) uE×× ( x
  • e)
+ f 9× ) E V , lel < E
slide-25
SLIDE 25 4 B . 2nd
  • rder
, Bellman , variable coeff . Shaking coefficients + regularization U + sup { aocx ) uxx + focx) } = a u ' + sup { a
  • ( ×
+ e) uI× + f0Cx+e) } = , let < { U ' C x
  • e)
+ ao ( ×) uE×× ( x
  • e)
+ f Ocx) E H G , lel < E f. . g{ ( e) , 5 . . . de u ' * g[ + at ( × ) ( us * gd ×× + for ) < F G
slide-26
SLIDE 26 4 B . 2nd
  • rder
, Bellman , variable coeff . Shaking coefficients + regularization Hence , Us : = u ' * g [ ue + sup { at Cx ) @ e) ×× + fok ) }

€

[smooth subsoil

;

as in 4 A u
  • Uh
eh
  • u 4
t ( ii. ud + 06 + HE 3) s . . . £ 0 ( htz) confide pendents )
slide-27
SLIDE 27 4 B . 2nd
  • rder
, Bellman , variable coeff . Shaking coefficients + regularization

Difficulty

: Need Lipschutz + cent . dep . for Uh uniformly in h . Not known in general , but
  • k
for SL schemes Bales
  • ERJ
MZAN 2002 , " Symmetric " FDM Krylov 2005 , ...
slide-28
SLIDE 28

4C

. 2nd
  • rder
, Bellman , var . coeff . , general schemes NO Lipschutz / cent . dip .
  • n
scheme ! Kvylov 1999 , 2000
  • Barthes
  • ERJ
SINUM 2005 , MC0M2007P Best results Upper bound as before ( 4 B ) . Lower bnd .via " linearization " of ecf ' n .
slide-29
SLIDE 29

4C

. 2nd
  • rder
, Bellman , var . coeff . , general schemes Lower bnd .via " linearization " of eq ' n :

ui

+ Lo '

ui

+ f0 ' = M , (

ii

) u + up { Lou + f 03=0

¥

{

Lin

+

Lonusntfontflnci

) Switching control min {

uj

  • uent
E} jt N Lions ' trick m > 1 ui
  • ul
£ Ce 's t i
slide-30
SLIDE 30

4C

. 2nd
  • rder
, Bellman , var . coeff . , general schemes Lower bnd .via " linearization " of eq ' n :

ui

+ Lo '

ui

+ fo ' = M , (

ii

) u + up { Lou + f 03=0 → {

by

+

Long

+ fon =

Mwai

) 1

ui

  • ul
£ C et ti shake coeff . + regularize ( ! )
  • >
u
  • Uh
e . ... < 0 ( { ± + 4253) ~ 0 ( h 's)
  • ptimize
slide-31
SLIDE 31

5.

Error estimates
  • nonkralffractional
  • rder
convex Ut +

syp§

  • +

|µ

, > olulxtrfexiz) . u
  • n%,⇒Du]v(d⇒↳=O
fractional
  • rder
, ex . Ju =
  • GOFU
Monotone difference
  • quadrature
schemes Linear equations : cont
  • Tanker
book Mollification arg . Bellman equations : La Chioma , ERJ , Karlsen Namer Math

2008 )

!YToad { rylov " Bis was , ERJ , Karlsen SINUM

2010

theory a
slide-32
SLIDE 32 6 . nor estimates
  • 2nd
  • rder
non
  • convex
eefns ) : ERJ BIT 2004 ND Bellman +
  • bstacle
:

}

Remonstrations ERJ Asymptotic Anal . 2006 Bellman

cage

Bonnans , Maroso , Zidani 2006
  • Uniform
elliptic eqlns : Caffarelli
  • Souganidis
2008 Turanova 2015 ( 2 papers )] "

Erewjuhtnfyu

Krylov 2015
slide-33
SLIDE 33 7 . Error estimates
  • nonlocal
/ fractional
  • rder
non
  • convex
New results
  • preprint
soon . First result for nonlocal and non convex eqhs First general result for Isaac eqhs
  • f
  • rder
> T . degenerate ecfns ! Joint with : Imran B is was CTIFR , Bangalore ) Indrani

Chowdhury

CTIFR , Bangalore)
slide-34
SLIDE 34 7. Error estimates
  • nonlocal
/ fractional
  • rder
non
  • convex
Ut ,
  • inafsgp { To 'Pu
+ b4P . Du + caput

f9P}=0

Monotone difference .quadrature scheme : Jo ' Pu a f. a , , g[ Pulxtyap )
  • u] Von ( dz )
P
  • a
Assumption : Oevocdz ) I #
  • f

Yzd,÷m+I

,
  • e[
0,2) P Max .
  • rder
  • f operator
It
  • f
C- OF In
  • Uhl
£ C . ( 0×+2 Tot 's ,
  • r
econ ) Cox +
  • t )±
  • ±
, sect , 2)
slide-35
SLIDE 35 7 . Error estimates
  • nonlocal
/ fractional
  • rder
non
  • convex
In
  • Uh
1 £ C . ( 0×+2 + ottz ,
  • eeoa
) Cox +
  • t )
to
  • ±
,

rear

) Remarks : i ) Preliminary
  • expect
some improvement ... ii ) BUT , formal rate
  • f
scheme is dt +0×2-0 , so NO rate as
  • n
→ 2 ! iii ) Similar bounds holds for " ANY " monotone scheme .
  • .
II ( .. . hence NOT
  • ptimal
in general ) iv ) Proof : Extension
  • f
Tst
  • rder
proof !