A Narrow-Stencil Finite Difference Method for - - PowerPoint PPT Presentation

A Narrow-Stencil Finite Difference Method for Hamilton-Jacobi-Bellman Equations

Xiaobing Feng

Department of Mathematics The University of Tennessee, Knoxville, U.S.A.

Linz, November 23, 2016

Collaborators

Tom Lewis, North Carolina Stefan Schnake, Tennessee The work to be presented here has been partially supported by NSF

Outline

• Motivation and Background
• A Narrow-Stencil Finite Difference Method
• High Order Extensions
• Numerical Experiments
• Conclusion

Outline

• Motivation and Background
• A Narrow-Stencil Finite Difference Method
• High Order Extensions
• Numerical Experiments
• Conclusion

We consider second order fully nonlinear PDEs F(D2u, Du, u, x) = 0 Two best known classes of equations:

• Monge-Ampére equation:

det(D2u) = f

• HJB equations:

infν∈V(Lνu − fν) = 0, where Lνu := Aν(x) : D2u + bν(x) · ∇u + cν(x)u, ν ∈ V. Both equations arise from many applications such as differential geometry, optimal mass transfer, stochastic optimal control, mathematical finance etc. Remark: cν ≡ 0 in several applications (e.g., stochastic optimal control, Bellman reformulation of Monge-Ampère equation).

Example: (Stochastic Optimal Control) Suppose a stochastic process x(τ) is governed by the stochastic differential equation dx(τ) = f (τ, x(τ), u(τ)) dt + σ (τ, x(τ), u(τ)) dW(τ), τ ∈ (t, T] x(t) = x ∈ Ω ⊂ Rn, W : Wiener process u : control vector and let J (t, x, u) = E t x T

t

L (τ, x(τ), u(τ)) dτ + g (x(T))

• .

Stochastic optimal control problem involves minimizing J (t, x, u) over all u ∈ U for each (t, x) ∈ (0, T] × Ω.

Bellman Principle Suppose u∗ ∈ U such that u∗ ∈ argmin

u∈U

J (t, x, u) , and define the value function v (t, x) = J (t, x, u∗) . Then, v is the minimal cost achieved starting from the initial value x(t) = x, and u∗ is the optimal control that attains the minimum.

Bellman Principle (Continued) Let Ω ⊂ Rn, T > 0, and U ⊂ Rm. The Bellman Principle says v is the solution of vt = F(D2v, ∇v, v, x, t) in (0, T] × Ω, (1) for F(D2v, ∇v, v, x, t) = inf

u∈U (Luv − hu) ,

Luv =

n

• i=1

n

• j=1

au

i,j(t, x)vxixj + n

• i=1

bu

i (t, x)vxi + cu (t, x) v

with Au := 1 2σσT bu := f (t, x, u) cu := 0 hu := L (t, x, u)

Ellipticity

Definition: Let F[u] := F(D2u, ∇u, u, x) and A, B ∈ SL(n). (a) F is said to be uniformly elliptic if ∃Λ > λ > 0 such that λtr(A − B) ≤ F(A, p, r, x) − F(B, p, r, x) ≤ Λtr(A − B) ∀A ≥ B. (b) F is said to be proper elliptic if F(A, p, v, x) ≤ F(B, p, w, x) ∀A ≥ B; v, w ∈ Rd, v ≤ w. (c) F is said to be degenerate elliptic if F(A, p, r, x) ≤ F(B, p, r, x) ∀A ≥ B.

Viscosity solutions

Definitions: Assume F is elliptic in a function class A ⊂ B(Ω) (set of bounded functions), (i) u ∈ A is called a viscosity subsolution of F[u] = 0 if ∀ϕ ∈ C2, when u∗ − ϕ has a local maximum at x0 then F∗(D2ϕ(x0), Dϕ(x0), u∗(x0), x0) ≤ 0 (ii) u ∈ A is called a viscosity supersolution of F[u] = 0 if ∀ϕ ∈ C2, when u∗ − ϕ has a local minimum at x0 then F ∗(D2ϕ(x0), Dϕ(x0), u∗(x0), x0) ≥ 0 (iii) u ∈ A is called a viscosity solution of F[u] = 0 if u is both a sub- and supersolution of F[u] = 0 where u∗(x) := lim sup

x′→x

u(x′) and u∗(x) := lim inf

x′→x u(x′) are the

upper and lower semi-continuous envelops of u

Barles-Souganidis Framework I

For approximating viscosity solutions, we first recall the Barles-Souganidis framework.

Theorem (Barles-Souganidis (’91))

Suppose that the elliptic problem F[u](x) = 0 in Ω satisfies the comparison principle. Assume that the (approximation)

• perator S : R+ × Ω × R × B(Ω) → R is consistent, monotone

and stable (as well as admissible), then the solution uρ of problem: S(ρ, x, uρ(x), uρ) = 0 in Ω, converges locally uniformly to the unique viscosity of u.

Barles-Souganidis Framework II

(i) Admissibility and Stability. For all ρ > 0, there exists a solution uρ ∈ B(Ω) to the following problem: S(ρ, x, uρ(x), uρ) = 0 in Ω. Moreover, there exists a ρ-independent constant C > 0 such that uρL∞(Ω) ≤ C. (ii) Monotonicity. For all x ∈ Ω, t ∈ R and ρ > 0 S(ρ, x, t, u) ≤ S(ρ, x, t, v) ∀u, v ∈ B(Ω), u ≥ v. (iii) Consistency. For all x ∈ Ω and φ ∈ C∞(Ω) there hold lim sup

ρ→0 y→x ξ→0

S(ρ, y, φ(y) + ξ, φ + ξ) ρ ≤ F(D2φ(x), ∇φ(x), φ(x), x), lim inf

ρ→0 y→x ξ→0

S(ρ, y, φ(y) + ξ, φ + ξ) ρ ≥ F(D2φ(x), ∇φ(x), φ(x), x).

Wasow-Motzkin Theorem

Theorem (Wasow-Motzkin (’53))

Any monotone and consistent method has to be a wide stencil scheme.

Theorem (Bonnans and Zidani (’03))

If Aν in the HJB equation is not diagonally dominant, then wide stencils are required to preserve monotonicity. Remark: The difficulty is the directional resolution. Wider stencils are used to increase the resolution.

Remarks on Monotone Schemes and Wide-stencils

◮ Why monotonicity? For the numerical scheme to identify

the correct viscosity solution, it needs to respect ordering in some sense. The monotonicity provides such an

• rdering, which is the best known one so far.

◮ A “drawback" of monotonicity is that one must use wide

stencils according to Wasow-Motzkin (’53), which could be very problematic for anisotropic problems because very fine directional resolution is required, besides the difficulty from handling boundary conditions.

Remarks on Monotone Schemes and Wide-stencils

◮ Why monotonicity? For the numerical scheme to identify

the correct viscosity solution, it needs to respect ordering in some sense. The monotonicity provides such an

• rdering, which is the best known one so far.

◮ A “drawback" of monotonicity is that one must use wide

stencils according to Wasow-Motzkin (’53), which could be very problematic for anisotropic problems because very fine directional resolution is required, besides the difficulty from handling boundary conditions.

◮ Consequently, in order to avoid using wide stencils, one

must relax (or abandon) the concept of monotonicity (in the sense of Barles and Souganidis).

Outline

• Motivation and Background
• A Narrow-Stencil Finite Difference Method
• High Order Extensions
• Numerical Experiments
• Conclusion

Goals:

◮ To construct finite difference methods (FDMs) whose

solutions converge to viscosity solutions of the underlying fully nonlinear 2nd order PDE problems, especially, to go beyond the domain of Barles-Souganidis’ framework and to be more suitable for FDMs and DG methods. Remark: A few existing “narrow-stencil" methods are

◮ Glowinski et al. (’04-’12): Mixed FE for MA eqns (H2 solns). ◮ Brenner et al. (’09-’13): DG for MA eqns (classical solns). ◮ Jensen-Smears (’12): Linear FE for isotropic HJB eqns. ◮ Smears-Süli (’14): DG-FE for (Cordes-) HJB (H2 solns). ◮ F

.-Neilan (’07-’11): FE and DG based on the vanishing moment approach.

◮ · · ·

Finite Difference Operators

Let {ej}d

j=1 denote the canonical basis of Rd. Define

δ+

xk,hk v(x) ≡ v(x + hkek) − v(x)

hk , δ−

xk,hk v(x) ≡ v(x) − v(x − hkek)

hk δµν

xk,hk v(x) ≡ δν xk,hk δµ xk,hk v(x),

δµν

xk,hk;xℓ,hℓv(x) ≡ δν xℓ,hℓδµ xk,hk v(x)

Discrete Gradients: Two natural "sided" choices

• ∇±

h

• k ≡ δ±

xk,hk

Discrete Hessians: Four natural "sided" choices

• Dµν

h

• k,ℓ ≡ δµν

xk,hk;xℓ,hℓ,

µ, ν ∈ {−, +} Remark: Low-regularity can be resolved by using “sided" gradient and Hessian approximations.

Ideas Used for 1st Order Hamilton-Jacobi Equations

(Crandall and Lions, ’84) FD schemes with the form

• H(∇−

h Uα, ∇+ h Uα, Uα, xα) = 0

converge to the viscosity solution of a Hamilton-Jacobi equation assuming

◮ Consistency:

H(q, q, u, x) = H(q, u, x),

◮ Monotonicity:

H(↑, ↓, u, x).

• H is called a numerical Hamiltonian.

◮

H is a function of both ∇−

h Uα and ∇+ h Uα.

◮ The monotonicity requirement is compatible with a discrete first

derivative test.

Vanishing Viscosity and Numerical Viscosity

H(∇u, u, x) = 0 − → −ǫ∆uǫ + H(∇uǫ, uǫ, x) = 0

(E. Tadmor, ’97) Every convergent monotone finite difference scheme for HJ equations implicitly approximates the differential equation −βh“∆u” + H(∇u, x) = 0 for sufficiently large and possibly nonlinear β > 0, where −βh“∆u” is called a numerical viscosity. Note: If hk ≡ h, then 1 ·

• ∇+

h Uα − ∇− h Uα

• = h∆hUα ≡ h

d

• k=1

δ2

xk,hUα.

Lax-Friedrichs numerical Hamiltonian

• H(q−, q+, u, x) ≡ H

q− + q+ 2 , u, x

• − b · (q+ − q−)
• Numerical Viscosity

FD Approximations of F(D2u, ∇u, u, x) = 0

◮ Since uxixj may be discontinuous at xα, it needs be

approximated from multiple directions.

FD Approximations of F(D2u, ∇u, u, x) = 0

◮ Since uxixj may be discontinuous at xα, it needs be

approximated from multiple directions.

◮ There are 4 possible FD approximations of uxi,xj(xα):

uxi,xj(xα) ≈ δµ

xiδν xju(xα)

µ, ν ∈ {+, −}. and D2u(xα) ≈ Dµν

h u(xα)

µ, ν ∈ {+, −}.

FD Approximations of F(D2u, ∇u, u, x) = 0

Inspired by the above observations, we propose the following form of FD methods for F(D2u, ∇u, u, x) = 0:

• F(D++

h

Uα, D+−

h

Uα, D−+

h

Uα, D−−

h

Uα, ∇−

h Uα, ∇+ h Uα, Uα, xα) = 0

• F is called a numerical operator, which needs to satisfy

FD Approximations of F(D2u, ∇u, u, x) = 0

Inspired by the above observations, we propose the following form of FD methods for F(D2u, ∇u, u, x) = 0:

• F(D++

h

Uα, D+−

h

Uα, D−+

h

Uα, D−−

h

Uα, ∇−

h Uα, ∇+ h Uα, Uα, xα) = 0

• F is called a numerical operator, which needs to satisfy

◮ Consistency:

F(P, P, P, P, q, q, u, x) = F(P, q, u, x),

◮ Generalized Monotonicity:

F(↑, ↓, ↓, ↑, ↑, ↓, ↑, x), uses the natural (partial) orderings for symmetric matrices, vectors, and scalars

◮ Solvability/Admissibility and Stability: ∃h0 > 0 and

C0 > 0, which is independent h, such that F[Uα, xα] = 0 has a (unique) solution U and Uℓ∞(Th) < C0 for h < h0.

Remarks on Numerical Operator F

◮

F depends on all four "sided" Hessians and both "sided" gradients.

◮ The generalized monotonicity requirement is an extension

• f the Crandall and Lions framework that enforces the

elliptic structure of the PDE with respect to the mixed discrete Hessians D±∓

h

.

◮ If F is not continuous,

F may not be continuous either. In this case the consistency definition should be replaced by

lim inf

Pk →P,r→q s→t,y→x

• F(P1, P2, P3, P4, r, s, y) ≥ F∗(P, q, t, x),

lim sup

Pk →P,r→qn s→t,y→x

• F(P1, P2, P3, P4, r, s, y) ≤ F∗(P, q, t, x).

◮ Above consistency and monotonicity are different from

Barles-Souganidis’ definitions.

Examples of Numerical Operators F I

Lax-Friedrichs-like schemes: 1-D (F-Lewis-Kao, ’14)

• F1(p1, p2, p3, x) := F

p1 + p2 + p3 3 , x

• + α(p1 − 2p2 + p3)
• F2(p1, p2, p3, x) := F(p2, x) + α(p1 − 2p2 + p3)
• F3(p1, p2, p3, x) := F

p1 + p3 2 , x

• + α(p1 − 2p2 + p3)

Examples of Numerical Operators F II

Godunov-like schemes: 1-D (F-Lewis-Kao, ’14):

• F4(p1, p2, p3, x) :=

ext

p∈I(p1,p2,p3) F(p, x)

where I(p1, p2, p3) := [p1 ∧ p2 ∧ p3, p1 ∨ p2 ∨ p3] and ext

p∈I(p1,p2,p3) :=

                 min

p∈I(p1,p2,p3)

if p2 ≥ max{p1, p3}, max

p∈I(p1,p2,p3)

if p2 ≤ min{p1, p3}, min

p1≤p≤p2

if p1 < p2 < p3, min

p3≤p≤p2

if p3 < p2 < p1.

Examples of Numerical Operators F III

Godunov-like schemes: 1-D (F-Lewis-Kao, ’14) (continued):

• F5(p1, p2, p3, x) :=

extr

p∈I(p1,p2,p3) F(p, x)

where I(p1, p2, p3) := [p1 ∧ p2 ∧ p3, p1 ∨ p2 ∨ p3] and extr

p∈I(p1,p2,p3) :=

                 min

p∈I(p1,p2,p3)

if p2 ≥ max{p1, p3}, max

p∈I(p1,p2,p3)

if p2 ≤ min{p1, p3}, max

p2≤p≤p3

if p1 < p2 < p3, max

p2≤p≤p1

if p3 < p2 < p1.

Higher Dimensional Lax-Friedrichs-like Schemes

Central Discrete Hessian: D

2 hUα ≡ 1

4

• D++

h

+ D+−

h

+ D−+

h

+ D−−

h

• Uα

Central Discrete Gradient: ∇hUα ≡ 1 2

• ∇+

h + ∇− h

• Uα

Lax-Friedrichs-like Numerical Operator:

• F[Uα, xα] ≡ F
• D

2 hUα, ∇2 hUα, Uα, xα

• + A(Uα, xα) :
• D++

h

Uα − D+−

h

Uα − D−+

h

Uα + D−−

h

Uα

• − b(Uα, xα) ·
• ∇+

h Uα − ∇− h Uα

Numerical Moment I

The discrete operator A(Uα, xα) :

• D++

h

Uα − D+−

h

Uα − D−+

h

Uα + D−−

h

Uα

• is called a numerical moment.

Observation:

• δ+

xk,hkδ+ xℓ,hℓ − δ+ xk,hkδ− xℓ,hℓ − δ− xk,hkδ+ xℓ,hℓ + δ− xk,hkδ− xℓ,hℓ

• Uα

= hkhℓδ2

xk,hkδ2 xℓ,hℓUα

= hkhℓδ2

xℓ,hℓδ2 xk,hkUα,

an O

• h2

k + h2 ℓ

• approximation of uxkxkxℓxℓ(xα) scaled by hkhℓ.

1d×d :

• D++

h

Uα − D+−

h

Uα − D−+

h

Uα + D−−

h

Uα

• ≈ h2∆2u(xα),

Numerical Moment II

Remark:

◮ Here the numerical moment plays a similar role for the 2rd

• rder fully nonlinear PDEs to what the numerical viscosity

does for the 1st order fully nonlinear PDEs.

◮ The introduction of numerical moments is very much

consistent with the vanishing moment method (at the PDE level) proposed and analyzed by F . and Neilan (’07-’12): ε∆2uε + F(D2uε, ∇uε, uε, x) = 0, with the special choice of the parameter ε = αh2.

◮ Is there an analogue of E. Tadmor’s result for 2rd order

PDEs?

Convergence I

Assumption: F is uniformly elliptic and Lipschitz continuous in the Hessian argument, and A and b are uniformly bounded for the family of linear operators defining the HJB problem with Dirichlet boundary data. Assume c ≡ 0.

Theorem (F-Lewis, ’16)

The Lax-Friedrichs-like scheme is admissible, consistent, stable, and generalized-monotone. Main ingredients of proof:

◮ Consistency is trivial. ◮ Generalized monotonicity is also “easy" by choosing the

numerical moment large enough (related to the Lipschitz constant of F).

Convergence II

◮ Admissibility and stability are the most difficult to verify.

The idea is to show that the mapping Mρ : Uα → Uα defined by

• ∆h

Uα = ∆hUα − ρ F[Uα] is monotone and a contraction for small enough ρ > 0. ∆h is an enhanced discrete Laplacian.

◮ Stability follows from applying Crandall-Tartar lemma to

Mρ which commutes with additive constants (this is the reason to assume c ≡ 0).

Convergence III

Theorem (F-Lewis, ’16)

In addition, suppose F[u] = 0 satisfies the comparison

• principle. Let U be the solution to the Lax-Friedrichs-like

scheme and uh be its piecewise constant extension. Then uh converges to u locally uniformly as h → 0+. Main ingredients of proof

◮ Follow and modify Barles-Souganidis’ proof to show that

u(x) := lim inf uh(x) and u(x) := lim sup uh(x) are respectively viscosity super- and sub-solution.

◮ First work with quadratic test functions, then with general

test functions.

◮ Use the numerical moment to control the off-diagonal

entries in the discrete Hessians (this is the most difficult and technical step).

◮ Use the consistency and comparison principle to get u = u.

The Local Stencil (2D)

◮ The extra nodes in the Cartesian directions smooth the

approximation through the diagonal components of the vanishing moment.

◮ Directional resolution is no longer necessary avoiding the

need for a wide-stencil.

Overcoming a Lack of Monotonicity

xk v

x0 δ2

xk ,hk v(x0 ± hk ek ) < δ2 xk ,hk v(x0) < 0

δ2

xk ,hk v(x0) − δ2 xk ,hk v(x0) < 0

xk v

x0 δ2

xk ,hk v(x0 ± hk ek ) > 0 and δ2 xk ,hk v(x0) < 0

δ2

xk ,hk v(x0) − δ2 xk ,hk v(x0) > 0

Outline

• Motivation and Background
• A Narrow-Stencil Finite Difference Method
• High Order Extensions
• Numerical Experiments
• Conclusion

Extensions

Goals: To develop high order methods and to use unstructured meshes

◮ “Bad" news: It is not possible to construct higher than 2nd

• rder monotone FD schemes (we have yet given up since

g-monotonicity is a weaker requirement)

Extensions

Goals: To develop high order methods and to use unstructured meshes

◮ “Bad" news: It is not possible to construct higher than 2nd

• rder monotone FD schemes (we have yet given up since

g-monotonicity is a weaker requirement)

◮ “Good" news: Inspired by a work of Yan-Osher (JCP

, ’11) for fully nonlinear 1st order Hamilton-Jacobi equations, we are able to construct high order MDG (mixed DG) and LDG (local DG) methods.

◮ Ideas of MDG: write F(uxx, x) = 0 as

• F(

↑

pR,

↓

pM,

↑

pR, x) = 0 pL = u−

xx

pM = ua

xx

pR = u+

xx

and use different numerical fluxes on the linear equations.

Extensions (continued)

◮ Ideas of LDG: write F(uxx, ux, u, x) = 0 as

• F(

↑

p1,

↓

p2,

↓

p3,

↑

p4,

↑

q1,

↓

q2, u, x) = 0, q1 = ux(x−), q2 = ux(x+), p1 = q1x(x−), p2 = q1x(x+), p3 = q2x(x−), p4 = q2x(x+),

where ux(x−) (resp. qjx(x−)) and ux(x+) (resp. qjx(x+)) denote the left and right limits of ux (resp. qjx) at x. They dictate how numerical fluxes should be chosen in the

• discretization. ↑ and ↓ stands for monotone increasing

and decreasing, respectively. r = 0 is allowed!

Outline

• Motivation and Background
• A Narrow-Stencil Finite Difference Method
• High Order Extensions
• Numerical Experiments
• Conclusion

1-D simulations Test 1: Consider the problem −u2

xx + 1 = 0,

0 < x < 1 u(0) = 0, u(1) = 0.5 This problem has the pointwise solutions u+(x) = 0.5x2 (convex), u−(x) = −0.5x2 + x (concave) Computed using Lax-Friedrichs-like scheme F1 with α = 1 (left) and α = −1 (right)

Test 2: Consider the problem min

θ∈{1,2} −Aθuxx − S(x) = 0,

−1 < x < 1 u(−1) = −1, u(1) = 1 A1 = 1, A2 = 2, S(x) =

• 12x2,

if x < 0 −24x2, if x ≥ 0 This problem has the exact solution u(x) = x|x|3

Test 3: Consider the problem −u3

xx + 8 sign(x) = 0,

−1 < x < 1 u(−1) = −1, u(1) = 1. This problem has the exact solution u(x) = x|x| ∈ C1([−1, 1]) which is not classical

2-D simulations Test 4: Let Ω = (0, 1)2. Consider Monge-Ampére equation det(D2u) = 1 in Ω u = 0

• n ∂Ω

Remark: No explicit solution formula and solution is not classical.

( α < 0: concave solution) ( α > 0: convex solution)

Test 5: Let Ω = (−1, 1)2. Consider Monge-Ampére equation det(D2u(x, y)) = 0 in Ω u = g

• n ∂Ω

Choose g such that the exact solution is u(x, y) = |x|.

hx L∞ norm

• rder

L2 norm

• rder

2.50E-01 3.86E-02 3.42E-02 1.25E-01 2.08E-02 0.89 1.85E-02 0.88 8.33E-02 1.38E-02 1.02 1.24E-02 0.99

Top: α = I, r = 1, and hy = 1/3 fixed. Bottom: α = I, r = 1, and

• dd number of intervals in the x-direction.

3-D Simulations Test 6: Let Ω = (0, 1)3. Consider Monge-Ampére equation det(D2u) = f in Ω u = g

• n ∂Ω

f(x, y, z) = (1 + x2 + y2 + z2)e

3(x2+y2+z2) 2

g(x, y, z) =                          e

y2+z2 2

if x = 0 e

x2+z2 2

if y = 0 e

x2+y2 2

if z = 0 e

y2+z2+1 2

if x = 1 e

x2+z2+1 2

if y = 1 e

x2+y2+1 2

if z = 1 The exact solution is u0(x, y, z) = e

x2+y2+z2 2

(computed solution) (error function)

Test 7: Let Ω = (0, 1)3. Consider Monge-Ampére equation det(D2u) = 1 in Ω u = 0

• n ∂Ω

Remark: No explicit solution formula and solution is not classical.

computed solution (α > 0)

Outline

• Motivation and Background
• A Narrow-Stencil Finite Difference Method
• High Order Extensions
• Numerical Experiments
• Conclusion

Concluding Remarks

◮ The generalized monotone structure allows us to deal with low

regularity functions by considering multiple gradient and Hessian approximations.

◮ The key new concept is that of a numerical moment which can

be used to design generalized monotone methods such as the Lax-Friedrichs-like scheme.

◮ By using narrow stencil schemes that can be expressed using

• nly forward and backward difference quotients, the methods

can be easily extended to higher-order and non-Cartesian domains/grids using the DG techniques (or DG Finite Element Calculus).

◮ Formulation easily extends to Monge-Ampère type equations

either directly or based on its HJB reformulation, although the analysis is not yet.

◮ Many open problems/issues: such as extension to degenerate

PDES, rate of convergence in C0-norm, boundary layers, parabolic PDEs, nonlinear solvers, etc.

References

◮ XF, R. Glowinski and M. Neilan, “Recent Developments in

Numerical Methods for Fully Nonlinear 2nd Order PDEs", SIAM Review, 55:1-64, 2013

◮ M. Neilan, A. Salgado and W. Zhang, “Numerical Analysis of

Strongly Nonlinear PDEs", arxiv.org/abs/1610.07992, to appear in Acta Numerica.

◮ Check arxiv.org for more (and new) references.

Thanks for Your Attention!