An approximation scheme for an Eikonal Equation with discontinuous - PowerPoint PPT Presentation

An approximation scheme for an Eikonal Equation with discontinuous coefficient Adriano Festa 1 Maurizio Falcone 2 1 EEE Department, IC London. 2 Dipartimento di Matematica, SAPIENZA - Universita’ di Roma. HYP 2012 Padova, 25th June - 29th June, 2012 Festa-Falcone Discontinuous Eikonal Equation

Outline of the talk Degenerate eikonal equation with discontinuous 1 coefficient Theoretical background 2 A semiLagrangian scheme with good properties 3 Tests and Applications 4 Concluding Remarks 5 Festa-Falcone Discontinuous Eikonal Equation

Degenerate Eikonal Equation with discontinuous coefficient Ω ⊂ R n open bounded domain with regular boundary N  � b ij ( x ) U x i U x j = [ f ( x )] 2  x ∈ Ω  (EK) i , j = 1   U ( x ) = G ( x ) x ∈ ∂ Ω G : ∂ Ω → [ 0 , + ∞ [ is continuous b is symmetric, positive semidefinite kj ) with σ ( · ) : Ω → R NM is L-Lipschitz ( b i , j ) = ( σ ik ) · ( σ t continuous but possibly degenerate f : R N → [ ρ, + ∞ [ , ρ > 0 is Borel measureable but possibly discontinuous Festa-Falcone Discontinuous Eikonal Equation

Optimal Control InterprGammation Rewriting the differential operator in the following form N M ( p · σ k ( x )) 2 = | p · σ ( x ) | 2 , � � b ij ( x ) p i p j = i , j = 1 k = 1 where ( σ ik ) k := σ k : Ω → R N , k = 1 , ... M . Festa-Falcone Discontinuous Eikonal Equation

Optimal Control InterprGammation Rewriting the differential operator in the following form N M ( p · σ k ( x )) 2 = | p · σ ( x ) | 2 , � � b ij ( x ) p i p j = i , j = 1 k = 1 where ( σ ik ) k := σ k : Ω → R N , k = 1 , ... M . We get the equivalent Bellman equation � � � a k σ k ( x ) max − DU ( x ) · = f ( x ) (BL) | a |≤ 1 k = 1 associated to the symmetric optimal control system M � a k σ k ( y ) , y = ˙ y ( 0 ) = x , k = 1 where a : [ 0 , + ∞ [ → { a ∈ R M : | a | ≤ 1 } and y ( · ) ≡ y x ( · , a ) is a solution. Festa-Falcone Discontinuous Eikonal Equation

Viscosity solution via semicontinuous envelopes f ∗ ( x ) = lim r → 0 + inf { f ( y ) : | y − x | ≤ r } , f ∗ ( x ) = lim r → 0 + sup { f ( y ) : | y − x | ≤ r } Definition (Discontinuous Viscosity Solution [Ishii 1985]) A lower semicontinuous (resp. upper) function U is a viscosity super- solution (resp. sub-) of the equation (EK) if for all φ ∈ C 1 (Ω) , and x 0 ∈ argmin x ∈ Ω ( U − φ ) , (resp. x 0 ∈ argmax x ∈ Ω ( U − φ ) ), we have N b ij ( x 0 ) φ x i ( x 0 ) φ x j ( x 0 ) ≥ [ f ∗ ( x 0 )] 2 , � i , j = 1 N b ij ( x 0 ) φ x i ( x 0 ) φ x j ( x 0 ) ≤ [ f ∗ ( x 0 )] 2 ). � (resp. i , j = 1 A function U is a discontinuous viscosity solution of the equation (EK) if U ∗ is a subsolution and U ∗ is a supersolution . Festa-Falcone Discontinuous Eikonal Equation

Dirichelet Conditions Definition (DC in weaker sense) We say that an upper semicontinuous function U , subsolution of the equation in (EK), satisfies weakly the DC if for all φ ∈ C 1 and ˆ x ∈ ∂ Ω , ˆ x ∈ argmax x ∈ Ω ( U − φ ) such that U (ˆ x ) > G (ˆ x ) , then we have N x )] 2 . � b ij (ˆ x ) φ x i φ x j ≤ [ f ∗ (ˆ i , j = 1 Lower semicontinuous functions that satisfy weakly the DC are defined accordingly. Festa-Falcone Discontinuous Eikonal Equation

Hypotheses on the discontinuity interface Assumptions on discontinuities Γ = { x ∈ R N : f is discontinuous at x } is a disjoint union of finite Lipschitz hypersurfaces ( ∃ η + transversal vector) f is continuous in each component Ω ± � � if x ∈ Γ , f ( x ) ∈ Ω − ∋ y → x f ( y ) , lim Ω + ∋ y → x f ( y ) lim . x ∈ Γ ∩ ∂ Ω we can choose η + , η − both inward for Ω . if ˆ Festa-Falcone Discontinuous Eikonal Equation

Comparison Results [Soravia 2006] Theorem Let Ω be an open domain with Lipschitz boundary. Let U , V : Ω → R be upper and a lower- semicontinuous function, a subsolution and a supersolution of (EK) with weak Dirichlet conditions. Suppose that V is nontangentially continuous on ∂ Ω \ Γ in the inward direction η Ω and on Γ ∩ Ω in the direction of η + . Then U ≤ V in Ω . Corollary Let U : Ω → R be a continuous, bounded viscosity solution of the problem (EK) . Then U is unique. Festa-Falcone Discontinuous Eikonal Equation

Example I � � 2 = [ f ( x , y )] 2 x 2 ( u x ( x , y )) 2 + � ] − 1 , 1 [ × ] − 1 , 1 [ u y ( x , y ) u ( ± 1 , y ) = u ( x , ± 1 ) = 0 x , y ∈ [ − 1 , 1 ] where f ( x , y ) = 2, for x > 0, and f ( x , y ) = 1 for x ≤ 0. � x 2 � x � � 0 0 b i , j = , σ ( x ) = , 0 1 0 1 therefore the Bellman’s equation in this case is � − Du ( x , y ) · a 1 ( x , 0 ) T − Du ( x , y ) · a 2 ( 0 , 1 ) T � max = f ( x , y ) . | a |≤ 1 Festa-Falcone Discontinuous Eikonal Equation

Example II  2 ( 1 − | y | ) x > 0 , | y | > 1 + ln x  − 2 ln ( x ) x > 0 , | y | ≤ 1 + ln x u ( x , y ) = (1) u ( − x , y )  x ≤ 0 . 2 is a viscosity solution of the problem. Festa-Falcone Discontinuous Eikonal Equation

A semiLagrangian Approx.: time discretization (Kruzkov’s transform). V ( x ) = 1 − e − U ( x ) 1 | DV ( x ) · σ ( x ) | = f ( x )( 1 − V ( x )) (1 / f as velocity) 2 | DV ( x ) · σ ( x ) | = 1 − V ( x ) f ( x ) (Bellman-type equation) 3 k a k σ k ( x ) �� � sup · DV ( x ) = 1 − V ( x ) f ( x ) a ∈ B ( 0 , 1 ) (discretize as diretional derivative) 4  k a k σ k ( x ) � � �� � 1 h V h ( x ) = inf V h x − h + x ∈ Ω  1 + h f ( x ) 1 + h a ∈ B ( 0 , 1 ) V h ( x ) = 1 − e − G ( x ) x ∈ ∂ Ω  (SDE) Festa-Falcone Discontinuous Eikonal Equation

The set A(x,h) � a k σ k ( x ) � � Figure: The set A ( x , h ) := x − h ; x ∈ B ( 0 , 1 ) in dimension f ( x ) 2. In grey A ( x , h ) := Ω ∩ A ( x , h ) Festa-Falcone Discontinuous Eikonal Equation

A semiLagrangian Approx.: space discretization Let us assume that Ω = Π n i = 1 ( a i , b i ) , Ω ∆ x := Z n ∆ x ∩ Ω and that the grind size ∆ x > 0. We look for a solution of  k a k σ k ( x α ) � 1 h a ∈ B ( 0 , 1 ) I [ W ]( x α − h x α ∈ Ω ∆ x W ( x α ) = min ) +  1 + h f ( x α ) 1 + h W ( x α ) = 1 − e − φ ( x α ) x α ∈ ∂ Ω ∆ x  (SL) where I [ W ]( x ) is a linear interpolation , in the space W ∆ x := � W : Ω → R | W ∈ C (Ω) and DW ( x ) = c α for any x ∈ ( x α , x α + 1 ) } . Festa-Falcone Discontinuous Eikonal Equation

Properties of the scheme (There exists a fixed point ) Let A ( x α , h ) ∈ Ω , for every x α ∈ Ω ∆ x , for any a ∈ B ( 0 , 1 ) , so there exists a unique solution W of (SL) in W ∆ x (Consistency) Developing with che usual Taylor expansion, like in the DF case, we find consistency ( Monotonicity ) The following estimate holds true: � n � 1 || W n − W || ∞ ≤ || W 0 − W || ∞ . 1 + h Festa-Falcone Discontinuous Eikonal Equation

Error estimates Theorem Under the introduced Hypotheses, we have that √ h + C ′ ∆ x || V ( x ) − W ( x ) || L 1 (Ω) ≤ C for all h > 0 h for some constant C , C ′ > 0 independent from h. Moreover, if v ( x ) ∈ C (Ω) we have √ h + 1 + h ( C ′ ∆ x ) || V ( x ) − W ( x ) || L ∞ (Ω) ≤ C for all h > 0 h for some constant C , C ′ > 0 independent from h. Festa-Falcone Discontinuous Eikonal Equation

Sketch of the proof I We start introducing, for a ǫ > 0, the set Λ 2 ǫ := { x ∈ Ω | B ( x , 2 ǫ ) ∩ Λ � = ∅} . We observe that � || V ( x ) − W ( x ) || L 1 (Ω) ≤ | V ( x ) − W ( x ) | dx Ω \ Λ 2 ǫ � � + | V ( x ) − W ( x ) | dx ≤ | V ( x ) − W ( x ) | dx + m (Λ 2 ǫ ) Λ 2 ǫ Ω \ Λ 2 ǫ from the fact that | V ( x ) − W ( x ) | ≤ 1 for all x ∈ Ω . If x ∈ ∂ Ω the assumption is trivially verified because of Dirichlet boundary conditions. Festa-Falcone Discontinuous Eikonal Equation

Sketch of the proof II Let ˆ x ∈ Ω \ Λ 2 ǫ . Consider for ǫ > 0 the auxiliary function ψ ( x , y ) := V ( x ) − W ( y ) − | x − y | 2 x | 2 − | x − ˆ 2 ǫ 2 It is not hard to check that the boundness of v , v ∗ and the upper semicontinuity of ψ , implies the existence of some ( x , y ) in Ω ± (depending on ǫ ) such that for all x , y ∈ Ω ± . ψ ( x , y ) ≥ ψ ( x , y ) √ After some standard calculations, choosing ǫ = h and the boundness of f and σ , we obtain √ V ( x ) − W ( y ) ≤ C h For C suitable positive constants. Festa-Falcone Discontinuous Eikonal Equation

Test 1 Let Ω := ( − 1 , 1 ) × ( 0 , 2 ) and f : Ω → R be defined by  1 x 1 < 0 ,  f ( x 1 , x 2 ) := 3 / 4 x 1 = 0 1 / 2 x 1 > 0  It is not difficult to see that f satisfies our Hypotheses. We can verify that the function 1  2 x 2 , x 1 ≥ 0 , √   2 x 1 + 1 3 − 1 − 2 x 2 , 3 x 2 ≤ x 1 ≤ 0 , u ( x 1 , x 2 ) := √ x 1 < − 1  x 2 , 3 x 2 . √  is a viscosity solution of | Du | = f ( x ) in the sense of our definition. Festa-Falcone Discontinuous Eikonal Equation

Test 1 - Results || · || ∞ || · || 1 ∆ x = h Ord ( L ∞ ) Ord ( L 1 ) 0.2 3.812 e-1 1.821 e-1 0.1 1.734e-1 1.1364 8.112e-2 1.1666 0.05 8.039e-2 1.1095 3.261e-2 1.3148 0.025 4.359e-2 0.8830 1.616e-2 1.0178 0.0125 2.255e-2 0.9509 7.985e-3 1.0271 Festa-Falcone Discontinuous Eikonal Equation