Finite Difference Methods for Fully Nonlinear Second Order ... · method using C1 ﬁnite elements....

Finite Difference Methods for Fully NonlinearSecond Order PDEs

Xiaobing Feng

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

Crete, September 22, 2011

Collaborators:Tom Lewis, University of TennesseeChiu-Yen Kao, Ohio State UniversityMichael Neilan, University of Pittsburgh

Supported in part by NSF

Outline

Introduction and Background

A Finite Difference Framework: 1-D Case

High Dimension and High Order Extensions

Conclusion

Terminologies:

• Semilinear PDEs: Nonlinear in unknown functions but linearin all their derivatives. e.g.

−∆u = eu, ut −∆u + (u2 − 1)u = 0

• Quasilinear PDEs: Nonlinear in lower order derivatives ofunknown functions but linear in highest order derivatives. e.g.

−div (|∇u|p−2∇u) = f , ut − div( ∇u√|∇u|2 + 1

)= 0

• Fully nonlinear PDEs: Nonlinear in highest order derivativesof unknown functions

First order fully nonlinear PDEs

F (Du,u, x) = 0

Examples:• Eikonal equation: |Du| = f• Hamilton-Jacobi equations: ut + H(Du,u, x , t) = 0

Second order fully nonlinear PDEs

F (D2u,Du,u, x) = 0

Examples:• Monge-Ampére equation: det(D2u) = f• HJ-Bellman equation: infν∈V (Lνu − fν) = 0

Fully nonlinear PDEs arise in

• Differential Geometry• Mass Transportation• Optimal Control• Semigeostrophic Flow• Meteorology• Antenna Design• Image Processing and Computer Vision• Gas Dynamics• Astrophysics• Grid generation

Example 1: (Minkowski Problem) Let K > 0 be a constant, findu : Ω ⊂ Rn → R such that the Gauss curvature of the graph of uis equal to K at every point x in Ω

Ω = [−0.57,0.57]2; BC: u = x2 + y2 − 1; K = 0.1,1,2,2.1

In the PDE language, the problem is described by

det(D2u)

(1 + |Du|2)n+2

2

= K or det(D2u) = K (1 + |Du|2)n+2

2

Example 2: (Monge-Kantorovich Optimal Transport Problem)Let ρ+, ρ− be two nonnegative (density) functions on Rn withequal mass. Let A denote the set of all mass preserving mapswith respect to ρ+ and ρ−, that is, Φ : Rn → Rn satisfies

ρ+(x) = det(∇Φ(x))ρ−(Φ(x)) ∀x ∈ supp(ρ+). (MPc)

Given c : Rn × Rn → [0,∞) (cost density function), the MKOptimal Transport Problem seeks Ψ ∈ A such that

I(Ψ) = infΦ∈A

I(Φ) :=

∫Rd

c(x ,Φ(x))ρ+(x)dx

If Ψ = ∇u (i.e., u is a potential of Ψ), then (MPc) implies

ρ+(x) = det(D2u(x))ρ−(∇u(x)) ∀x ∈ supp(ρ+)

Example 3: (Stochastic Optimal Control) Suppose a stochasticprocess x(τ) is governed by the stochastic differential equation

dx(τ) = f (τ,x(τ),u(τ)) + 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

tL (τ,x(τ),u(τ)) dτ + g (x(T ))

].

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

Bellman Principle

Suppose u∗ ∈ U such that

u∗ ∈ argminu∈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 initialvalue x(t) = x , and u∗ is the optimal control that attains theminimum.

Bellman Principle (Continued)

Let Ω ⊂ Rn, T > 0, and U ⊂ Rm. The Bellman Principle says vis the solution of

vt = F (D2v ,∇v , v , x , t) in (0,T ]× Ω, (1)

for

F (D2v ,∇v , v , x , t) = infu∈U

(Luv − hu) ,

Luv =n∑

i=1

n∑j=1

aui,j(t , x)vxi xj +

n∑i=1

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

with

Au :=12σσT bu := f (t , x ,u)

cu := 0 hu := L (t , x ,u)

Bellman Principle (Continued)

After v is found, to find the control u∗ and the correspondingstochastic process x,

I Solve u∗ = argminu∈U

[Luv − fu]

I Plug u∗ into the SDE and solve for x

Thus, solving the Stochastic Optimal Control Problem can berecast as solving a fully nonlinear 2nd order (parabolic)Bellman equation

Fully Nonlinear 2rd Order Elliptic PDE Theories I

Definition: F [u] := F (D2u,∇u,u, x) is said to be elliptic if

F (A,p, r , x) ≤ F (B,p, r , x) ∀A,B ∈ SL(n), A− B > 0

Remark:When F (A,p, r , x) is differentiable in A, then F is elliptic at u if∂F [u]

∂A> 0, OR if the linearization of F at u is elliptic as a linear

operator

Fully Nonlinear 2rd Order Elliptic PDE Theories II

(1) Classical solution theory (before 1985): [Chapter 17, Gilbarg& Trudinger], [Chapter 16, Lieberman]

(2) Viscosity solution theory (after 1985). Solution conceptwas due to Crandall and Lions (1983). Two breakthroughs:

I R. Jensen (1986): Uniqueness by maximum principleI H. Ishii (1987): Existence by Perron’s method

(3) A complete regularity theory was established in ’80s and’90s, some key milestones are

I Krylov-Safonov Cα estimates for linear elliptic PDEs ofnon-divergence form (’79) (which replaces De Giorgi-NashCα estimates for linear elliptic PDEs of divergence form)

I Evans-Krylov C2,α estimates (’82)I Carffarelli’s W 2,p estimates for the Monge-Ampère equation

and general elliptic PDEs

Fully Nonlinear 2rd Order Elliptic PDE Theories III

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 asub- and supersolution of F [u] = 0

where u∗(x) := lim supx ′→x

u(x ′) and u∗(x) := lim infx ′→x

u(x ′) are the

upper and lower semi-continuous envelops of u

Geometrically,

Figure: A geometric interpretation of viscosity solutions

Remark: The concept of viscosity solution is non-variational. Itis based on a “differentiation by parts" approach, instead of theusual integration by parts approach used to define weaksolutions for linear, semilinear, and quasilinear PDEs

$64K Question:

How to compute viscosity solutions?

Available Numerical Methodologies

I Finite Difference Methods based on directlyapproximating derivatives by difference quotients

I Galerkin-type Methods based on variational principlesand approximating infinite dimension spaces by finitedimension spaces (finite element, finite volume, spectralGalerkin, discontinuous Galerkin )

I Everything else such as Collocation Method, MeshlessMethod, Lattice Boltzmann method, radial basis functionmethods, etc.

Numerical Challenges

In contrast with the success of PDE analysis, little progress ondeveloping numerical methods was made until very recently

I None of above methods work (directly) for general fullynonlinear 2rd order PDEs

I Situation is even worse for Galerkin-type methods becausethere is no variational principle to start with! Recall that theconcept of viscosity solutions is non-variational!!

I Conditional uniqueness is hard to deal with numericallyI Tons of other numerical issues such convergence,

efficiency, fast solvers, implementations, etc.

Remark: On the other hand, to certain degree, all abovemethods work for linear, semilinear, and quasilinear PDEs. Atleast they can be formulated without too much difficulties

Consider Monge-Ampère problem

det(D2u) = f in Ω = (0,1)2

u = g on ∂Ω

f and g are chosen such that u = e(x21 +x2

2 )/2 ∈ C∞(Ω) is theexact solution. We discretize the Monge-Ampère equationusing the standard nine-point finite difference method(

D2xxuij

)(D2

yyuij)−(D2

xyuij)

= fi,j ,

Figure: 16 computed solutions of the Monge-Ampère equation usinga nine-point stencil on a 4× 4 grid. 2(N−2)2

solutions on an N ×N grid

Nevertheless, there are some recent attempts for constructingGalerkin-type methods for Monge-Ampère type equations.

I Dean and Glowinski (’03-’06) proposed some least squaresand augmented Lagrange methods for Monge-Ampèretype equations.

I Böhmer (’08) proposed and analyzed an L2 projectionmethod using C1 finite elements.

I Brenner-Gudi-Neilan-Sung (’09-’10) constructed and analyzedsome C0 discontinuous Galerkin L2-projection methods forMonge-Ampère type equations.

I F.-Neilan (’06-’10) has developed an indirect approachbased on combining the newly developed vanishingmoment method(VMM) and Galerkin type methods.

I F.-Lewis (’10-’11) extended the vanishing moment method(VMM) to Hamilton-Jacobi-Bellman type equations

Outline




Conclusion

Goals:

I To construct some (2rd order) finite difference methods(FDMs) which guarantee to converge to viscosity solutionsof the underlying fully nonlinear 2rd order PDE problems

I To develop a blueprint/receipt for designing FDMs and anew FD convergence framework/theory which is simplerthan Barles-Souganidis’ (’91) framework and more suitablefor FDMs and local discontinuous Galerkin (LDG) methods

Remark:To simplify the presentation, I shall describe the ideas andmethods using 1-D equations. High dimension (and high order)generalizations will be given at the end

Notation:

Let Ω = [a,b] and Ωh := xjJj=1 be a (uniform) mesh on Ω withmesh size h = hx . Define the forward and backward differenceoperators by

δ+x v(x) :=

v(x + h)− v(x)

h, δ−x v(x) :=

v(x)− v(x − h)

h

for a continuous function v defined in Ω and

δ+x Vj :=

Vj+1 − Vj

h, δ−x Vj :=

Vj − Vj−1

h,

for a grid function V on Ωh

Remark:

δ+x and δ−x will be our building blocks in the sense that we shall

approximate 1st and 2rd order derivatives using combinationsand compositions of them

Existing Finite Difference Methods

I Barles-Souganidis (’91) proposed an abstract framework forapproximating F [u] = 0 by S[uρ] := S(uρ, x , ρ) = 0 (FD orotherwise). They proved that uρ converges to u locallyuniformly if S is monotone, stable, and consistent. AnO(hα) convergence rate in C0-norm was establishedrecently by Caffarelli-Souganidis (’08).

I Oberman (’08) constructed a wide-stencil FD scheme forthe Monge-Ampére equation which is the only FD schemeknown to satisfy Barles-Souganidis’ criterion so far.

I Krylov (’02-05) proposed several monotone FD schemes forBellman type equations F [u] = 0.

I Kao-Trudinger (’90-’93) developed several monotone/positivefinite difference methods for F [u] = 0.

I Barles-Souganidis (’91) proposed an abstract framework forapproximating F [u] = 0 by S[uρ] := S(uρ, x , ρ) = 0 (FD orotherwise). They proved that uρ converges to u locallyuniformly if S is monotone, stable, and consistent.

Remark:

I Barles-Souganidis’ framework does not tell how toconstruct a convergent scheme, no hints are given either.

I Their monotonicity and consistency are difficult to verify fora given scheme.

I We feel that imposing monotonicity on S[uρ] (as a functionof uρ) is too restrictive because differential operator F [u]does not have that property.

Ideas used for 1st order Hamilton-Jacobi equationsConsider fully nonlinear 1st order HJ equation

H(ux , x) = 0

A general FD scheme on the mesh Th := xjJj=1 has the form

H(δ−x Uj , δ+x Uj , xj) = 0

H is called a numerical Hamiltonian

Consistency: H(p,p, xj) = H(p, xj)

Monotonicity: H(↑, ↓, xj)

Remark/Observation:I H is a function of both δ−x Uj and δ+

x Uj . This is crucialbecause ux may be discontinuous at xj .

I Both monotonicity and consistency are not hard to verify inpractice. The uniform convergence for such schemes wasproved by Crandall-Lions (’84)

Finite difference approximations of uxx in F (uxx , x) = 0

I Since uxx may be discontinuous at xj , it should beapproximated from both sides of xj

I There are 3 possible FD approximations of uxx (xj):

δ−x δ−x u(xj), δ+

x δ−x u(xj), δ+

x δ+x u(xj)

I It is trivial to check

δ−x δ−x u(x) = δ2

x u(xj − h) = δ2x u(xj−1)

δ+x δ−x u(x) = δ2

x u(xj)

δ+x δ

+x u(x) = δ2

x u(xj + h) = δ2x u(xj+1)

where

δ2x u(xj ) :=

u(xj − h)− 2u(xj ) + u(xj + h)

h2 =u(xj−1)− 2u(xj ) + u(xj+1)

h2

A general finite difference framework for F [u] = 0

Inspired by the above observations, we propose the followingform of FD methods for F [u] = 0:

Fh[U, xj ] := F (δ2x Uj−1, δ

2x Uj , δ

2x Uj+1, xj) = 0

Definition: F is called a numerical operator

Questions:I What is F?I Is there a guideline for constructing F? (Recall that such a

guideline is missing in Barles-Souganidis’ framework)

Criterions for “good" numerical operator F

I Consistency: F (p,p,p, xj) = F (p, xj)

I Monotonicity: F (↑, ↓, ↑, xj)

I Solvability and Stability: ∃h0 > 0 and C0 > 0, which isindependent h, such that Fh[U, xj ] = 0 has a (unique)solution U and ‖U‖`∞(Th) < C0 for h < h0

Remark:I If F is not continuous, F may not be continuous either. In

this case the consistency definition should be changed to

lim infpk →p,k=1,2,3

ξ→x

F (p1,p2,p3, ξ) ≥ F∗(p, x) and

lim suppk →p,k=1,2,3

ξ→x

F (p1,p2,p3, x) ≤ F ∗(p, x)

I Above consistency and monotonicity are different fromBarles-Souganidis’ definitions, but the solvability andstability are same

For consistent, monotone and stable FD schemes, we have thefollowing theorem, which may be regarded as an analogue ofCrandall-Lions’ convergence theorem for fully nonlinear 1storder equations

Theorem (F., Kao, Lewis, ’11)

Let uh denote the piecewise constant extension of U on[xj− 1

2, xj+ 1

2]. Suppose the numerical operator F is consistent,

monotone and stable, then uh converges to the unique viscositysolution of the Dirichlet problem for F (uxx , x) = 0.

Idea of Proof: Follow the PDE analysis and modify Barles-Souganidis’ proof.

Guidelines for constructing F

Once again, we look for hints/ideas from FD methods for fullynonlinear 1st order Hamilton-Jacobi equations (and hyperbolicconservation laws)!

Theorem (E. Tadmor (’97))

Every “convergent" monotone finite difference scheme for HJequations (and hyperbolic conservation laws) must contain anumerical diffusion (or viscosity) term.

In other words, every “convergent" monotone finite differencescheme for HJ equations implicitly approximates the differentialequation

−αh“∆u” + H(∇u, x) = 0

for sufficiently large and nonlnear α > 0,

Lax-Friedrichs scheme:

HLF (p1,p2) := H(p1 + p2

2

)−αLF (p2 − p1)

Godunov scheme:

HG(p1,p2) := extp∈I(p1,p2)

H(p) = HG(p1,p2)−αG(p2 − p1)

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

extp∈I(p1,p2)

:=

minp∈I(p1,p2) if p1 ≤ p2

maxp∈I(p1,p2) if p1 > p2

Question: What “quantity" plays the role of “numerical diffusion(viscosity) term" as above for fully nonlinear 2rd order PDEs?

Answer/Conjecture: Numerical moment! We conjecture that a“good" scheme for fully nonlinear 2rd order PDEs shouldcontain some kind numerical moment term. In other words,such a “good" scheme implicitly approximates the regularizedPDE

αh2“∆2u” + F (D2u,∇u,u, x) = 0

for sufficiently large (and nonlinear) α > 0. This is perfectlyconsistent with the vanishing moment method (at the PDElevel) introduced and studied by F. and Neilan (’07-’10):

ε∆2uε + F (D2uε,∇uε,uε, x) = 0

with ε = αh2

Examples of “good" numerical operators/schemes I

Lax-Friedrichs-like schemes (FKL ’11):

F1(p1,p2,p3) := F(p1 + p2 + p3

3

)+ α(p1 − 2p2 + p3)

F2(p1,p2,p3, xj) := F (p2) + α(p1 − 2p2 + p3)

F3(p1,p2,p3) := F(p1 + p3

2

)+ α(p1 − 2p2 + p3)

Examples of “good" numerical operators/schemes II

Godunov-like schemes (FKL ’11):

F4(p1,p2,p3) := extp∈I(p1,p2,p3)

F (p)

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

extp∈I(p1,p2,p3)

:=

minp∈I(p1,p2,p3)

if p2 ≥ maxp1,p3,

maxp∈I(p1,p2,p3)

if p2 ≤ minp1,p3,

minp1≤p≤p2

if p1 < p2 < p3,

minp3≤p≤p2

if p3 < p2 < p1.

Examples of “good" numerical operators/schemes III

Godunov-like schemes (FKL ’11) (continued):

F5(p1,p2,p3) := extrp∈I(p1,p2,p3)

F (p)

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

extrp∈I(p1,p2,p3)

:=

minp∈I(p1,p2,p3)

if p2 ≥ maxp1,p3,

maxp∈I(p1,p2,p3)

if p2 ≤ minp1,p3,

maxp2≤p≤p3

if p1 < p2 < p3,

maxp2≤p≤p1

if p3 < p2 < p1.

Outline




Conclusion

High dimension and high order extensions

High dimension extensionsI The main issue is how to discretize uxk xù(x). There are 4

possible ways:

δ−xkδ−xù(x), δ+

xkδ+

xù(x), δ−xkδ+

xù(x), δ+xkδ−xù(x)

we are saved by the following identity:

δ−xkδ−xÙij + δ+

xkδ+

xÙij − δ−xkδ+

xÙij − δ+xkδ−xÙij

= h2xk

h2x` δ

2xkδ2

xÙij = h2xk

h2x` δ

2x`δ

2xk

Uij

because it is an O(h2xk

+ h2x`) approximation to uxk xk x`x` , which is

a prototypical 4th order mixed derivative term in ∆2u

I To ensure consistency, monotonicity and stability, hence,convergence, the multi-D numerical operator F must depend onall four approximate derivatives δ−xk δ

−xÙij , δ+

xkδ+

xÙij ,δ−xk δ

+xÙij , δ+

xkδ−xÙij , and F is decreasing (↓) in first two

arguments and increasing (↑) in last two arguments


High order extensionsI “Bad" news: We tried but failed (as expected) to construct

higher than 2rd order monotone FD schemes.

I “Good" news: Inspired by a recent work by Yan-Osher(JCP, ’11) for fully nonlinear 1st order Hamilton-Jacobiequations, we are able to construct high order LDG (localdiscontinuous Galerkin) methods.


High order extensionsI “Bad" news: We tried but failed (as expected) to construct

higher than 2rd order monotone FD schemes.I “Good" news: Inspired by a recent work by Yan-Osher

(JCP, ’11) for fully nonlinear 1st order Hamilton-Jacobiequations, we are able to construct high order LDG (localdiscontinuous Galerkin) methods.

Outline




Conclusion

Concluding Remarks

I We propose a new and “easy to verify" framework fordeveloping “good" FD (and LDG) methods for fullynonlinear 2rd elliptic PDEs

I We also give a guideline for designing consistent,monotone and stable FD schemes. Key concept isnumerical moment, which may be regarded as ananalogue of numerical viscosity from 1st order PDEs

I We see a strong interplay between PDE analysis andnumerical analysis in this work

I Many open problems and challenging theoretical andpractical issues need to be addressed: rate ofconvergence in C0-norm, boundary layers, parabolicPDEs, nonlinear solvers, etc.

References

I arxiv.org/abs/1109.1183 (small book on VMM)I Review paper by XF, R. Glowinski and M. Neilan (’10): “Recent

Developments in Numerical Methods for Fully Nonlinear 2rdOrder PDEs (submitted to SIAM Review)

Thanks for Your Attention !

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