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

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 Outline methods Monotone numerical 2 estimates Error last order 3 eqhs 2nd order 4 eqlns convex order Fractional eqins 5 convex 2nd order 6 ' ns convex eq non - Fractional order New eqins 7 convex non : -

Bellman / equations T Isaacs . Controlled SDES % (b) dws + b%( XD { It dxs or = xex E { letftxs }=:u( g ( Xt ) , as ) x. t ) ds + map . Dynamic Programming Dutfd }=O + b9 sup { D ?u] tr[ oooo at + { , .BeUmaneg@ next ) gas =

/ Bellman equations T Isaacs . ti→T . t value initial value Terminal i ) > m

/ Bellman 1. equations Isaacs ti→T . t value value initial Terminal i ) > in Levy SDES ( jumps ) ii ) driven . tfz . ,z)N~( dzidt ) , ,oy%( Xs dXs = . . where at ))=v( A) t ECNC A. { [ ¥ y%,⇒Du]v(d⇒]= syp§ 0 tfz , > o[ulxty9xi⇒ ) ucx ) - - - - .

/ Bellman 1. equations Isaacs ti→T . t value value Terminal initial i ) > in Levy ( jumps ) SDES ii ) driven . tfz . ,z)N~( dzidt ) , ,oy%( Xs dXs = . . P ptoyssonmdmeas . jumptensthttrh where jump intensity ) [ o .tD=v( A) t ECNC A. § t measure Levy y%,⇒Du]v(d⇒]= syp§ tfz 0 , > o[ulxty9xi⇒ ) Ut + ucx ) - - - - . 4- - Jan = :

Bellman / equations T Isaacs . iii ) 0 games : sum - opposite interests with controllers 2 f Elliot Katten - - Souganidis Fleming satisfy and value functions Upper lower Isaaes like equations e. g . ' P }=O ' Pu + fd { LAB Jo inaf + Utt u sgp

/ Bellman equations T Isaacs . assumptions Typical : , p ' P oaip bo fo ' B , ya g , , , , uniformly a ,p bounded Lifschitz and in are + f gcz ) v ( dz ) S ,< MZPVHH IHSTT < is , , weight some unless explicitly statet No uniform elhipti city .

2. methods Monotone numerical an ] ) Sh ( t 0 [ = × ah , , , Monotone offense is 0 ffuhn so : , + parabolicity assumption F approx > at in . - eq iij Consistency , [ y ] ) / Sn ( n [ y ] I E E rr ( h ) ' : - - . - stability Hun K iii ) La t h < 1 £ : 11 La - Souganidv Convergence by Bates 's

2. numerical methods Monotone Examples : . ukthh.hr# . hcHyukhL btcx ) (a) ban b- ( × ) upwind ! e u× g. - 02k ) uKtH-2ulnytuK= ( b ) ok e u×× uCx*oh)-2u{HtuK-oh= { (c) SL e 1 oi.org u× × ; , . , on )T idvldz ( O= o , , ... ukthltugntuanhlg (d) l↳fyktP - y Du ] vldz ) ) UH = tzfygwcdz - linear interpolation f + Is a Putty ) ) • + - µ ,

2. numerical methods Monotone Examples : explicit method Time implicit 0 my - , , , IMEX , - - . f . stability Monotonicity La CFL ⇒ +

estimates Error 1st other equations 3 . Fractional 2nd 4 order equations , convex . order 5 equations , convex . 6 2nd equations , order non convex - . Fractional order New 7 equations , convex : non -

3. estimates Tst Error order eqins - of proof Modification comparison

3. estimates Tst Error order eqins - FkiDu)=0 of proof : at Modification comparison Fh ( × , Un ,[Un])=O Uht ' Ulh ( kg ) I. ij ( × ) uly ) at it y max - - . sotn Fty , Dyty ) ) def visa 30 + u . Uh FEE ,D×y ) Kh + consistency E + 1113×2411 a non . Hence awe proof in comparison as - - FCFDYTED , Dxytiyt ) FCI Unix ) Kh 1113×2411 µ + - - ' ) 0 0 ( he (e) = tgl y ( X , y ) y 12 X -

3. estimates Tst Error order eqins - , Du ) Flx 0 u of Proof : + = modification comparison Fhk , Un ,[un])=O Un + ' Uh C kg ) I. ij ( × ) u ( y ) at it y Max - - - ' ) 0 ( Units he at 5) e c + - ' ' ) 0 ( hk ) w ) sup I Un 0 ( eths Units uiy ) g) £ ye , E = - - - 0 ( e)

3. estimates 1st Error order eqins - Jhtufyl 2nd Does work for order NOT equations : ! Uh 0 F ( , D2u ) O = u + + × = . . . V - uig ) ? Ya ycxiy ) ) Un (E) , ) FCI Fly , D tkh 'HD4xyH s - - , . HM yexiyie Xi F =D I ' that Problem Need + ' Unix ) such . : f- conclude ! ts ( It ) %) ( xo to ←

4. Error estimates 2nd order ' ns eg convex - , Convex Regularization comparison + . coeff A. const , . B variable coeff Bellman . , . , general C variable Bellman coeff schemes . , . .

. coeff 4A 2nd const order convex , . , . F ( D2u ) fcx ) 0 + u + = ↳ . gs ( y f .dx x ) - . . , * gs F(D2u)*g[ f 0 * gs = + + u in ZF(D2u*g{) [ Jensen ] I + fe , ) F ( D2u E C) Ue + - Kh2HD4u [ Consistency ] 3 ud ) , His Fn ( us , [ -

. coeff 4 A 2nd const order convex . , , . Hence + Fn ( Us [ ue ] ) - fella Kh2HD4usH + f £ + Hf Us , , - + e) 0 ( h2i3 lip u no . of approximate subsolln scheme → comparison : , 0 ( h 's Min + e) Nn An -3 < Us - - 3+4=0 ( h± ) ( h2e £0 U < U ue Us + - - -

. coeff 4 A 2nd const order convex , . , . 0 ( h± ) In An E U - Symmetric argument : ( h± ) 0 < u - OBS Need Un , uniformly h to be Lipschutz in : Duo

4 B 2nd coeff order Bellman variable , . , . Shaking coefficients regularization + Krylov 1997 1999 2000 , , Bowles ERJ SINUM 2005 MZAN 2002 MCOMP 2007 - , ,

4 B 2nd coeff order Bellman variable , . , . Shaking coefficients regularization + + fetch } { aocx ) 0 U = uxx + sup a / approximate + f0Cx+e ) } 0 ( × sup { + e) uE× 0 ' = + u a 0 let < { ,

4 B 2nd coeff order Bellman variable . , , . Shaking coefficients regularization + fon } { aocx ) 0 U + = + sup uxx a + f0Cx+e ) } 0 ( × + e) uI× sup { 0 ' = + u a 0 let's I , * i→ × e - f 9× ) ' V ao ( × ) uE×× ( x 0 - e) lel U e) E 0 < E C x + - + ,

4 B 2nd coeff order Bellman variable . , , . Shaking coefficients regularization + focx ) } { aocx ) 0 U + = + sup uxx a + f0Cx+e ) } sup { + e) uI× • 0 ' = ( × + u a 0 let < { , ' H ao ( × ) uE×× ( x f Ocx ) - e) G lel U e) E 0 < E C x + - + , f. . g{ ( e) 5 de . , . . * gd for ) F ' at ( × ) ( G us u < 0 + + * g[ ××

4 B 2nd coeff € order Bellman variable , . , . Shaking coefficients regularization + Hence ' Us u * g : = [ , fok ) } [ smooth subsoil sup { at C x ) @ e) 0 + ue + ×× ; 4 A in as + HE 3) 0 ( htz ) t ( u 4 ud Uh ii. eh 06 £ s + u . . - - . confide pendents )

4 B 2nd coeff order Bellman variable . , , . Shaking coefficients regularization + Difficulty . dep Need Lipschutz cent for Uh : + . h uniformly in . in general , Not known ok for but SL schemes Bales ERJ MZAN 2002 - , " Symmetric " FDM Krylov 2005 , ...

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

2nd 4C . coeff Bellman order , general schemes var . , , . " of linearization Lower bnd " eq ' .via : n Lin ui ui ii Lo + f0 = M , ( ) uent 1 ui uj ' ' + + f 03=0 { up { Lou + u ¥ Lonusntfontflnci ) + Switching control { E } min - jt N Ce 's Lions trick ' ul t £ i > m -

2nd 4C . coeff Bellman order , general schemes var . , , . " of linearization Lower bnd " eq ' .via : n Mwai Long by ui ui ii Lo + fo = M , ( ) ui ' ' → { + + f 03=0 up { Lou + u + fon ) + = et ti ul C £ 1 - + regularize ( ! ) coeff shake . 0 ( h 's ) 4253 ) { ± - Uh 0 ( > e u < - + ~ ... . optimize

estimates 5. nonkralffractional Error order convex - |µ n%,⇒Du]v(d⇒↳=O syp§ Ut , > olulxtrfexiz ) + + u - - - . - GOFU fractional order Ju ex = - . , - quadrature Monotone difference schemes Linear equations : book Mollification Tanker cont arg - . Bellman equations : 2008 ) Namer !YToad La Karlsen Chioma Math ERJ 2010 , , { rylov " Karlsen Bis SINUM ERJ theory was , , a • •

- 2nd estimates 6 order eefns non convex nor - . ) ERJ BIT 2004 : } Remonstrations Erewjuhtnfyu obstacle Bellman ND + : 2006 Anal ERJ Asymptotic Bellman . - 2006 Zidani Bonnans Maroso , , cage Uniform elliptic eqlns : - Souganidis Caffarelli 2008 papers ) ] " ( 2 Turanova 2015 Krylov 2015

Error estimates 7 / fractional nonlocal order convex non - - . preprint New results soon - . First result for nonlocal and convex eqhs non Chowdhury First of general result for order Isaac T > eqhs . degenerate ecfns ! with Joint : Bangalore ) B is CTIFR Imran was , , Bangalore ) Indrani CTIFR