Computatonal Informaton Games Houman Owhad TexAMP 215
Man Queston Can we turn the process of dscovery of a scalable numercal method nto a UQ problem and, to some degree, solve t as such n an automated fashon? Can we use a computer, not only to mplement a numercal method but also to fnd the method tself?
Problem: Fnd a method for solvng (1) as fast as possble to a gven accuracy (1) dv(a u) =g, x Ω, u =, x Ω, Ω R d Ω s pec. Lp. a unf. ell. a,j L (Ω) log 1 (a)
Multgrd Methods Multgrd: [Fedorenko, 1961, Brandt, 1973, Hackbusch, 1978] Multresoluton/Wavelet based methods [Brewster and Beylkn, 1995, Beylkn and Coult, 1998, Averbuch et al., 1998] R m Lnear complexty wth smooth coeffcents Problem Severely affected by lack of smoothness
Robust/Algebrac multgrd [Mandel et al., 1999,Wan-Chan-Smth, 1999, Xu and Zkatanov, 24, Xu and Zhu, 28], [Ruge-Stüben, 1987] [Panayot - 21] Stablzed Herarchcal bases, Multlevel precondtoners [Vasslevsk - Wang, 1997, 1998] [Panayot - Vasslevsk, 1997] [Chow - Vasslevsk, 23] [Aksoylu- Holst, 21] Some degree of robustness but problem remans open wth rough coeffcents Why? Interpolaton operators are unknown Don t know how to brdge scales wth rough coeffcents!
Low Rank Matrx Decomposton methods Fast Multpole Method: [Greengard and Rokhln, 1987] Herarchcal Matrx Method: [Hackbusch et al., 22] [Bebendorf, 28]: N ln d+3 N complexty
Common theme between these methods Ther process of dscovery s based on ntuton, brllant nsght, and guesswork Can we turn ths process of dscovery nto an algorthm?
Answer: YES Compute fast Play adversaral Informaton game Identfy game Compute wth partal nformaton Fnd optmal strategy [Owhad 215, Mult-grd wth rough coeffcents and Multresoluton PDE decomposton from Herarchcal Informaton Games, arxv:153.3467] Resultng method: N ln 2 N complexty Ths s a theorem
( dv(a u) =g n Ω, Resultng method: u =on Ω, H 1 (Ω) =W (1) a W (2) a a W (k) a < ψ, χ > a := R Ω ( ψ)t a χ =for(ψ, χ) W () W (j), 6= j Theorem For v W (k) C 1 2 k kvk a k dv(a v)k L 2 (Ω) C 2 2 k kvk 2 a :=< v,v> a = R Ω ( v)t a v Looks lke an egenspace decomposton
u = w (1) + w (2) + + w (k) + w (k) =F.E.sol.ofPDEnW (k) Can be computed ndependently B (k) :Stffness matrx of PDE n W (k) Theorem λ max(b (k) ) λ mn (B (k) ) C Just relax n W (k) to fnd w (k) Quacks lke an egenspace decomposton
Multresoluton decomposton of soluton space u.14 w (1) w (2) w (3).3 8 1 3 + = 1.5 1 3 4 1 4 4 1 5 + w (4) w (5) w (6) + + Solve tme-dscretzed wave equaton (mplct tme steps) wth rough coeffcents n O(N ln 2 N)-complexty Swms lke an egenspace decomposton
V: F.E.spaceofH 1 (Ω) ofdm.n Theorem The decomposton V = W (1) a W (2) a a W (k) Canbeperformedandstoredn O(N ln 2 N)operatons Doesn t have the complexty of an egenspace decomposton
ψ (1) χ (2) χ (3) χ (4) χ (5) χ (6) Bass functons look lke and behave lke wavelets: Localzed and can be used to compress the operator and locally analyze the soluton space
H 1 (Ω) u dv(a ) Reduced operator H 1 (Ω) g Inverse Problem R m u m g m Numercal mplementaton requres R m computaton wth partal nformaton. φ 1,...,φ m L 2 (Ω) u m =( φ Ω 1 u,..., Ω φ mu) u m R m Mssng nformaton u H 1 (Ω)
Dscovery process ( dv(a u) =g n Ω, u =on Ω, Identfy underlyng nformaton game Measurement functons: φ 1,...,φ m L 2 (Ω) Player A Player B Chooses g L 2 (Ω) Sees Ω uφ 1,..., Ω uφ m kgk L 2 (Ω) 1 Chooses u L 2 (Ω) Max Mn u u a kfk 2 a := R Ω ( f)t a f
Determnstc zero sum game Player B Player A 3-2 -2 1 Player A s payoff Player A & B both have a blue and a red marble At the same tme, they show each other a marble How should A & B play the (repeated) game?
Optmal strateges are mxed strateges Optmal way to play s at random q Player B 1 q Game theory p Player A 3-2 John Von Neumann 1 p -2 1 John Nash A s expected payoff =3pq +(1 p)(1 q) 2p(1 q) 2q(1 p) =1 3q + p(8q 3) = 1 8 for q = 3 8
Player A Chooses g L 2 (Ω) kgk L 2 (Ω) 1 Player B Sees R Ω uφ 1,..., R Ω uφ m Chooses u L 2 (Ω) u u a Contnuous game but as n decson theory under compactness t can be approxmated by a fnte game Abraham Wald The best strategy for A s to play at random Player B s best strategy lve n the Bayesan class of estmators
Player B s class of mxed strateges Pretend that player A s choosng g at random g L 2 (Ω) ξ: Randomfeld ( dv(a u) =g n Ω, u =on Ω, ( dv(a v) =ξ n Ω, v =on Ω, Player B s u (x) :=E bet v(x) R Ω v(y)φ (y) dy = R Ω u(y)φ (y) dy, Player s B optmal strategy? Player B s best bet? mn max problem over dstrbuton of ξ
Computatonal effcency ξ N (, Γ) Theorem Gamblets Elementary gambles form determnstc bass functons for player B s bet u (x) = P m =1 ψ (x) R Ω u(y)φ (y) dy ψ : Elementary gambles/bets Player B s bet f R Ω uφ j = δ,j,j=1,...,m ψ (x) :=E ξ N (,Γ) hv(x) RΩ v(y)φ j(y) dy = δ,j,j {1,...,m} ψ
What are these gamblets? Depend on Γ: Covarance functon of ξ (Player B s decson) (φ ) m =1 : Measurements functons (rules of the game) Example [Owhad, 214] arxv:146.6668 x 1 Γ(x, y) =δ(x y) φ (x) =δ(x x ) Ω x x m a = I d a,j L (Ω) ψ : Polyharmonc splnes [Harder-Desmaras, 1972][Duchon 1976, 1977,1978] ψ : Rough Polyharmonc splnes [Owhad-Zhang-Berlyand 213]
What s Player B s best strategy? What s Player B s best choce for Γ(x, y) =E ξ(x)ξ(y)? Γ = L R Ω ξ(x)f(x) dx N (, kfk2 a) kfk 2 a := R Ω ( f)t a f L = dv(a ) Why? See algebrac generalzaton
The recovery s optmal (Galerkn projecton) Theorem If Γ = L then u (x) s the F.E. soluton of (1) n span{l 1 φ =1,...,m} ku u k a =nf ψ span{l 1 φ : {1,...,m}} ku ψk a L = dv(a ) ( dv(a u) =g, x Ω, (1) u =, x Ω,
Optmal varatonal propertes Theorem P m =1 w ψ mnmzes kψk a over all ψ such that R Ω φ jψ = w j for j =1,...,m Varatonal characterzaton ψ : Unque mnmzer of Theorem ( Mnmze kψk a Subject to ψ H 1(Ω) and R Ω φ jψ = δ,j, j =1,...,m
Selecton of measurement functons Example Indcator functons of a Partton of Ω of resoluton H φ =1 τ τ Ω τ j dam(τ ) H Theorem ku u k a H λ mn (a) kgk L 2 (Ω)
Elementary gamble ψ Your best bet on the value of u gven the nformaton that R τ u =1and R τ j u =forj 6= τ 1 ( dv(a u) =g, x Ω, (1) u =, x Ω, τ j Ω
Exponental decay of gamblets Ω τ r Theorem RΩ (B(τ,r)) c ( ψ ) T a ψ e r lh kψ k 2 a ψ ψ 4 x-axs slce 1 x-axs slce log 1 1 1 + ψ
Localzaton of the computaton of gamblets ψ loc,r ( Mnmze : Mnmzer of Subject to kψk a ψ H 1 (S r )and R S r φ j ψ = δ,j τ S r r Ω No loss of accuracy f localzaton H ln 1 H u,loc (x) = P m for τ j S r =1 ψloc,r (x) R Ω u(y)φ (y) dy Theorem If r CH ln 1 H ku u,loc 1 k a Hkgk L λmn (a) 2 (Ω)
Formulaton of the herarchcal game
Herarchy of nested Measurement functons φ (k) 1,..., k φ (k) Example wth k {1,...,q} = P j c,jφ (k+1),j φ (1) 1 φ (2) 1,j 1 φ (2) 1,j 2 φ (2) 1,j 3 φ (2) 1,j 4 φ (3) 1,j 2,k 1 φ (3) 1,j 2,k 2 φ (3) 1,j 2,k 3 φ (3) 1,j 2,k 4 : Indcator functons of a herarchcal nested partton of Ω of resoluton H k =2 k φ (k) τ (1) 2 τ (2) 2,3 τ (3) 2,3,1 φ (1) 2 =1 τ (1) 2 φ (2) 2,3 =1 τ (2) 2,3 φ (3) 2,3,1 =1 τ (3) 2,3,1
In the dscrete settng smply aggregate elements (as n algebrac multgrd) Π 1,2 j Π 1,2 Π 2 I τ (1) τ (2) j Π 1,3 Π 2,3 j I 1 I 2 I 3 φ (1) φ (2) φ (3) φ (4) φ (5) φ (6)
Formulaton of the herarchy of games Player A Chooses g L 2 (Ω) kgk L2 (Ω) 1 ( dv(a u) =g n Ω, u =on Ω, Player B Sees { uφ (k), I Ω k } Must predct u and { uφ (k+1),j I Ω j k+1 }
Player B s best strategy ( dv(a u) =g n Ω, u =on Ω, ξ N (, L) ( dv(a v) =ξ n Ω, v =on Ω, Player B s bets u (k) (x) :=E v(x) R Ω v(y)φ(k) (y) dy = R Ω u(y)φ(k) (y) dy, I k The sequence of approxmatons forms a martngale under the mxed strategy emergng from the game F k = σ( R Ω vφ(k), I k ) v (k) (x) :=E v(x) F k Theorem F k F k+1 v (k) (x) :=E v (k+1) (x) F k
Accuracy of the recovery Theorem ku u (k) k a H k λ mn (a) kgk L 2 (Ω) τ (k) H k := max dam(τ (k) ) φ (k) =1 τ (k) dam(τ (k) ) H k
In a dscrete settng the last step of the game recovers the soluton to numercal precson log 1 ku u (k) k a kuk a log 1 ku u (k) k a kuk a k 3.5 12 k u (1) u(2) u (3) u (4) u (5) u (6)
Gamblets Elementary gambles form a herarchy of determnstc bass functons for player B s herarchy of bets Theorem u (k) (x) = P ψ(k) ψ (k) R (x) Ω u(y)φ(k) (y) dy : Elementary gambles/bets at resoluton H k =2 k h ψ (k) (x) :=E v(x) R Ω v(y)φ(k) j (y) dy = δ,j,j I k ψ (1) ψ (2) ψ (3) ψ (4) ψ (5) ψ (6)
Gamblets are nested V (k) := span{ψ (k), I k } ψ (1) 1 Theorem V (k) V (k+1) ψ (2) 1,j 1 ψ (2) 1,j 2 ψ (2) 1,j 3 ψ (2) 1,j 4 ψ (3) 1,j 2,k 1 ψ (3) 1,j 2,k 2 ψ (3) 1,j 2,k 3 ψ (3) 1,j 2,k 4 ψ (k) (x) = P j I k+1 R (k),j ψ(k+1) j (x)
Interpolaton/Prolongaton operator R (k),j = E R Ω v(y)φ(k+1) j (y) dy R Ω v(y)φ(k) l (y) dy = δ,l,l I k R (k),j τ (k) R Your best bet on the value of τ (k+1) j gven the nformaton that R τ (k) u =1and R τ l u =forl 6= 1 R (k),j u τ (k+1) j
At ths stage you can fnsh wth classcal multgrd But we want multresoluton decomposton
Elementary gamble χ (k) R τ (k) Your best bet on the value of u gven the nformaton that u =1, R τ (k) u = 1 and R τ (k) j τ (k) u =forj 6= τ (k) -1 1 τ (k) j Ω
χ (k) = ψ (k) ψ (k) ψ (1) 1 =( 1,..., k 1, k ) =( 1,..., k 1, k 1) ψ (2) 1,j 1 ψ (2) 1,j 2 ψ (2) 1,j 3 ψ (2) 1,j 4 1 +1 1 +1 1 +1 +1 1
χ (k) = ψ (k) ψ (k) ψ (1) χ (2) χ (3) χ (4) χ (5) χ (6)
Multresoluton decomposton of the soluton space V (k) := span{ψ (k), I k } W (k) := span{χ (k),} W (k+1) : Orthogonal complement of V (k) n V (k+1) wth respect to < ψ, χ > a := R Ω ( ψ)t a χ Theorem H 1 (Ω) =V (1) a W (2) a a W (k) a
Multresoluton decomposton of the soluton Theorem u (k+1) u (k) = F.E. sol. of PDE n W (k+1) u =.14 u (1).3 u (2) u (1) 8 1 3 u (3) u (2) + + u (4) u (3) u (5) u (4) u (6) u (5) 1.5 1 3 4 1 4 4 1 5 + + Subband solutons u (k+1) u (k) can be computed ndependently
A (k),j Unformly bounded condton numbers := ψ (k), ψ (k) j a B (k),j := χ (k), χ (k) j a Theorem λ max (B (k) ) λ mn (B (k) ) C 4.5 log1 ( λ max(a (k) ) λ mn (A (k) ) ) log 1 ( λ max(b (k) ) λ mn (B (k) ) ) Just relax! In v W (k) to get u (k) u (k 1)
c (1) c (2) j c (3) j 4 x 1-4 2 4 x 1-5 2 c (6) j c (4) j u = P c(1) -2-4 c (5) -2 j c (6) -6 2 4 6 8 ψ (1) kψ (1) k a + P q k=2 Pj c(k) j -4 1 2 3 4 χ (k) j kχ (k) j k a Coeffcents of the soluton n the gamblet bass j
Operator Compresson Gamblets behave lke wavelets but they are adapted to the PDE and can compress ts soluton space u Compresson rato = 15 Energy norm relatve error =.7 Gamblet compresson Throw 99% of the coeffcents
Fast gamblet transform O(N ln 2 N)complexty Nestng A (k) =(R (k,k+1) ) T A (k+1) R (k,k+1) Level(k) gamblets and stffness matrces can be computed from level(k+1) gamblets and stffness matrces Well condtoned lnear systems Underlyng lnear systems have unformly bounded condton numbers ψ (k) = ψ (k+1) (,1) + P j C(k+1),χ,j χ (k+1) j C (k+1),χ =(B (k+1) ) 1 Z (k+1) Localzaton Z (k+1) j, := (e (k+1) j e (k+1) j ) T A (k+1) e (k+1) (,1) The nested computaton can be localzed wthout compromsng accuracy or condton numbers
ϕ,a h,m h ψ (q) χ (q),b (q),a (q) u (q) u (q 1) Parallel operatng dagram both n space and n frequency ψ (3) ψ (q 1),A (q 1) χ (q 1),B (q 1) u (q 1) u (q 2) ψ (3)...... χ (3),B (3) u (1) 8 1 3...,A (3) u (3) u (2) ψ (2),A (2) χ (2),B (2) u (2) u (1) ψ (1),A (1) χ (3) u (3) u (2) ψ (2) χ (2).3 u (2) u (1) ψ (1) ψ (1).14 u (1)