computing the convex envelope using a nonlinear...

October 18, 2007 19:18 WSPC/INSTRUCTION FILEConvexEnvelopeNumericsM3AS

COMPUTING THE CONVEX ENVELOPE USING A NONLINEARPARTIAL DIFFERENTIAL EQUATION

ADAM M. OBERMAN

Department of Mathematics, Simon Fraser University

Burnaby, British Columbia, Canada

[email protected]

Received (Day Month Year)

Revised (Day Month Year)Communicated by (xxxxxxxxxx)

A fully nonlinear partial differential equation for the convex envelope was recently intro-duced by the author. In this article, the equation is discretized using a finite difference

method. The resulting scheme yields an explicit local method to compute the convex

envelope. The scheme is shown to converge. Computational results are presented forsmooth and non-smooth data. Extensions to higher dimensions and unstructured grids

are discussed.

Keywords: convex envelope; partial differential equation; finite difference method

AMS Subject Classification: 35J70, 26B25, 52A41, 65N06

1. Introduction

The convex envelope of a given function g(x) is a natural mathematical object whichhas been the subject of study for many years. In the recent past, the possibilitythat the convex envelope satisfies a differential equation had been suggested12, anda computational method for computing the convex envelope had been introducedwhich used a related equation28. However, it was only very recently observed25 thatthe convex envelope is the solution of a Partial Differential Equation (PDE).

The convex envelope, u(x), of the function g(x) : Rn → R, is defined as thesupremum of all convex functions which are majorized by g,

u(x) = supv(x) | v convex, v(y) ≤ g(y) for all y ∈ Rn. (CE)

In a recent work25, it was shown that the convex envelope is given by the solutionof the fully nonlinear, degenerate elliptic PDE

max u(x)− g(x),−λ1[u](x) = 0. (PDE)

Here λ1[u](x) is the smallest eigenvalue of the Hessian D2u(x). See Figure 1. Theequation (PDE) is an obstacle problem involving the function g(x), and the fullynonlinear second order PDE, λ1[u](x). In one dimension, this equation reduces tothe classical obstacle problem. But the differential operator λ1[u] is fully nonlinear

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2

u - g = 0-λ1[u] < 0

u - g < 0-λ1[u] = 0

Fig. 1. Illustration of the equation

in dimensions two and higher, and the problem of computing the convex envelopeis correspondingly more challenging.

Despite the fact that the convex envelope is well-understood, the equation (PDE)offers some advantages not apparent in other descriptions of the envelope. It pro-vides a local characterization of the convex envelope, in contrast to the global natureof its definition. This allows for numerical methods to be built which involve onlylocal conditions, eliminating costly global constraints. The equation can also beused to provide an explicit certificate that a candidate function is the convex en-velope: computing partial derivatives and verifying (PDE) is easier than disputingthe existence of a larger convex function majorized by g(x). Finally, the equationmakes possible a notion of approximate solutions. This makes more tractable theproblem of computing projections onto the set of convex functions, a project whichwill be investigates in a separate work.

1.1. Related work

Our objective is to work with continuous functions, which have non-trivial curva-ture. The challenge here is to discretize the problem in a way that is robust enoughto handle functions which are non-differentiable (for example convex polyhedralfunctions) but also delicate enough to match derivatives at contact points whendealing with smooth functions.

In the special case where the given function g(x) is polyhedral, computing theconvex envelope reduces to a fully discrete problem, and methods from Computa-tional Geometry10 may be applied. There is an large literature on this aspect ofthe subject, and a search for software yields many results. For example, Matlab hasa built in function for computing the convex hull of a curve. The computation ofconvex hulls of point sets or triangulated sets is a typical problem in computationalgeometry. However, these algorithms don’t work if we wish to consider curved ob-jects. Our main algorithm works for polyhedral functions: in this special case thediscretization error is zero.


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We next turn our attention to methods for computing convex envelopes of non-polyhedral functions. This is a challenging problem, for the reasons given above.Some previous efforts6,28,8 have used numerical methods which resemble methodsfor solving partial differential equations, notwithstanding the lack of availability ofa governing PDE at the time. Vese28 uses a PDE-based method to evolve in timethe original function g(x) to the convex envelope, by moving the level sets. Thescheme is the closest to ours, in that it involves the minimum eigenvalue of theHessian. However, it stops short of identifying a PDE for the steady solution. Afinite element type approach has also been used.6 The one-dimensional case hasalso been studied.14

A motivating problem for this work is variational problems with convexity con-straints. The objective is to minimize a convex functional over the set of convexfunctions. The problem is well-posed, although there is no Euler-Lagrange equationavailable, and computing solutions is a challenge. This type of problem occurs inthe classical Newton’s problem7,16 of shape optimization. See also Ref. [17].

A more recent application is to mathematical economics26, where the convexfunctions represent utility functions. This variational problem has been solvednumerically8, and convex envelopes have been computed in the same work.

We also mention Ref. [18], which gives a characterization of the dual cone ofconvex functions, and Ref. [13], which uses a penalty method to deal with convexityconstraints.

It is a well-known fact from Convex Analysis27 that the convex envelope ofg(x), is the result of applying the Legendre Transform twice successively to thegiven function g(x). This leads to the notation g∗∗, for the convex envelope, whichwe avoid here. The Legendre Transform has been implemented numerically5,19,20,but the method has difficulties with the convex envelope.20

Several related notions (quasi-convexity, rank-one convexity, directional con-vexity, . . . ) of convexity appear in Material Science, for vector valued functionsas well as scalars. The notion of directionally convex functions21 is related to anapproximate notion of convexity used herein. In fact, we show below that direc-tionally convex functions converge to convex functions as the directional resolutionincreases. Further references to notions of convexity appearing in material sciencecan be found in Ref. [11] or Ref. [1].

1.2. The convex envelope of scattered data points

In this section we give a numerical formulation of the convex envelope on scattereddata points. It can be regarded as the exact solution given scattered point data.

A natural approach is to use the characterization of a convex functions in termsof supporting hyperplanes.

Definition 1.1 (supporting hyperplane). A supporting hyperplane to the func-tion u : Rn → R at the point x is a plane P (y) = u(x) + λ · (y − x), which touches


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the graph of u from below, i.e. which satisfies

u(y) ≥ u(x) + λ · (y − x),

for all y ∈ Rn. The vector λ is a subgradient of u at x.

The function u(x) is convex if and only if there exists a supporting hyperplaneto u at each point x.

A discrete version of the subgradient condition on scattered data points can beused to build the convex envelope of discrete data, see Section 6.5.5 of Ref. 4.

Given data gi at the points xi, in order for u(x) to be the discrete convexenvelope of g, we require the following to hold. First u(x) is majorized by g(x),

u(xi) ≤ gi, i = 1, . . . m. (1.1)

Second, u(x) is convex, which is verified using the supporting hyperplane condition.I.e. there exists vectors λ1, . . . , λm such that

u(xj) ≥ u(xi) + λi · (xj − xi), i, j = 1, . . . ,m. (1.2)

The convex envelope is given by maximizing each of the values u(xi) over functionswhich satisfy (1.1), (1.2). The conditions result in a convex program4, which canbe solved by standard methods.

Definition 1.2. We call the largest solution of the convex program (1.1), (1.2) theconvex envelope of discrete data.

Remark 1.1. While the program above solves the discrete convex envelope prob-lem, it requires the introduction of new variables (for the subgradients), and thesatisfaction of a large number of constraints. In fact, the number of constraints isquadratic in the number of points (one constraint for every pair of points). Theresult can be a quite costly convex program. It should be clear that the vast ma-jority of constraints will be inactive: the inequality (1.2) is unlikely to be activewhen |xi − xj | is large. We can thus reduce the cost with a trade-off: localize theconstraints to obtain an approximate convexity, which we quantify below.

Remark 1.2 (Interpolation and extrapolation of convex functions). Thecharacterization above gives a natural method for extrapolating values of a givenconvex function with values prescribed on scattered data point. The interpolant,u(z), is then given as the maximum of the supporting hyperplanes,

u(z) = maxi=1,...,m

(u(xi) + λi · (z − xi)).

This resolves the problem posed by Example 6.4.


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1.3. A nonlinear characterization

Before introducing the main scheme, we discuss another approach to solving (PDE).In two dimensions, solve the characteristic equation for the eigenvalues of theHessian to obtain λ = (Tr ±

√Tr2 − 4 det)/2, where Tr = uxx + uyy, and

det = uxxuyy − u2xy. This simplifies to

λ1[u] =uxx + uyy −

√(uxx − uyy)2 + (2uxy)2

2. (1.3)

The scheme obtained using finite differences for this last equation yields a reason-able method for solving (PDE). However, the scheme does not generalize easilyto higher dimensions, and is not monotone. Consequently, there is no convergenceproof available.

The finite difference scheme above results in a method which was not invariantfor convex polyhedral functions. In fact, for g(x, y) = |x − y|, simply applying thefinite difference operator results in large positive values for λ1 on the line x = y,but large negative values for λ1 at the adjacent grid points.

This is related to the fact that the numerical Hessian of a convex function neednot be positive definite.

1.4. Wide stencil finite difference schemes

The schemes we introduce below fit into the framework of wide stencil finite dif-ference schemes, which have been used to build monotone, convergent schemes forfunctions of the eigenvalues22, motion of level sets by mean curvature23 and theInfinity Laplace equation24. The schemes converge to the unique viscosity solutionof the equation as the spatial resolution, h, and the directional resolution, dθ, goto zero. In practice, we work with reasonably small stencils (up to four times thegrid spacing). The computational time for the wide stencil schemes scales no worsethan linearly with the stencil size, depending on the particular choice of stencil.

1.5. The stencils: spatial and directional resolution

The indices of neighboring grid points for successively wider stencil schemes areindicated in Table 1. The first two levels are displayed in Figure 2(a).

Table 1. Neighbors of reference point (i,j) in the first quadrant. The neighbors in

the other quadrants are given by rotation. Values of dθ and tan2 dθ.

Level Neighbors of (i,j) in First Quadrant dθ tan2(dθ)

1 (i+1, j) (i+1, j+1) .39 .172 (i+1, j) (i+1, j+1) (i+2, j+1) (i+1, j+2) .23 .056

3 (i+1, j) (i+1, j+1) (i+2, j+1) (i+1, j+2) .16 .026(i+1, j+3) (i+3, j+1) (i+2, j+3) (i+3, j+2)


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Fig. 2. Computational stencils for the 9 and 17 point schemes. Computational stencil for the 17

point scheme near the boundary.

On a regular cartesian grid, let the stencil at the reference point x0 consistof the neighbors x1, . . . , xN (illustrated by open and closed circles respectively inFigure 2). Define the direction vectors,

vi ≡xi − x0

|xi − x0|, i = 1, . . . , N.

To compute directional second derivatives, we assume

if vi is a direction vector, then so is −vi. (1.4)

Define the spatial and directional resolution

h ≡ maxi|xi − x0| (1.5)

dθ ≡ max‖w‖=1

mini

cos−1(wT vi). (1.6)

Note dθ measures the maximum angle between an arbitrary vector, and the directionvectors for the stencil. For the schemes in two dimensions, dθ is half the maximumangle between direction vectors. Sample values for dθ can be found in Table 1.

2. Analysis of semi-discrete schemes

To illustrate ideas, we begin with a semi-discrete scheme. Using the classicalRayleigh-Ritz characterization of the eigenvalues, write

λ1[u](x) = min|v|=1

vT D2u v. (2.1)

Substitute the standard centered difference approximation to the second derivativein the direction of v, to arrive at the semi-discrete scheme

λh1 [u](x) ≡ min

|v|=1

u(x + hv)− 2u(x) + u(x− hv)h2

.

Inserting the last equation into (PDE) and solving for u(x) gives

u(x) = min|v|=1

(g(x),

u(x + hv) + u(x− hv)2

). (2.2)


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2.1. Properties of the semi-discrete scheme

While the scheme above is only semi-discrete, we set aside that consideration fornow and study its properties. Many of these properties will also hold for the fullydiscrete schemes which follow.

This scheme is second order accurate, by which we mean,

λ1[φ](x)− λh1 [φ](x) = O(h2), for every twice differentiable function φ(x).

The scheme is defined locally, but by extending this definition to all points in thedomain (with possible adjustments near the boundary) we can arrive at a schemefor the equation in a domain. While we make local arguments, the properties belowapply globally. More precisely, the properties below apply to the solution operatorof the scheme.

The scheme is monotone: increasing the value at any of the neighboring valuesu(x + hv) cannot decrease the value at u(x).

The scheme is degenerate: changing the value at some of the neighbors mighthave no effect on the value at u(x).

The scheme is stable, in fact it is non-expansive in the uniform norm. Considera perturbation δ(x), write u(x+hv) = u(x+hv)+ δ(x+hv), and call the resultingvalue u(x). It follows that

|u(x)− u(x)| ≤ max|y−x|=h

|u(y)− u(y)| = max|y−x|=h

|δ(y)| .

The scheme is nonlinear, but the nonlinearity is polyhedral: it consists of a minimumof affine terms.

2.2. Approximate convexity

Convexity is a rigid notion. The characterization of convexity via a partial differen-tial equation allows for a notion of approximate convexity. This is done by relaxingthe notion of directional convexity: instead of testing an inequality in all directions,test only a finite number of directions. The definition of the directional resolution ofa collection of direction vectors is provided in (1.6). Using this measure, which wedenote by dθ, we can quantify the degree of non-convexity allowed by the schemewith directional resolution dθ.

Proposition 2.1. Let u(x) be a twice-continuously differentiable function definedon Rn. Let vik

i=1 be a set of direction vectors, with directional resolution dθ.Assume dθ ≤ π/4. If

d2u

dv2i

≥ 0, i = 1, . . . , k, (2.3)

thenλ1

λn≥ − tan2(dθ). (2.4)


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where λ1 ≤ · · · ≤ λn are the eigenvalues of the Hessian of u. If in addition

d2u

dv2i

≥ sin2(dθ)λn, i = 1, . . . , k, (2.5)

then u is convex.

Proof. Suppose the minimum of λ1[u](x) occurs at x. Let w1 be the eigenvectorcorresponding to λ1. Let θ be the angle between w1 and the nearest grid direction,vi. By (1.6), θ ≤ dθ. Decompose vi = cos θw1 + sin θw, where w is a unit vectororthogonal to w1. Then compute

d2u

dv2i

= (cos θw1 + sin θw)T D2u (cos θw1 + sin θw)

= cos2 θwT1 D2u w1 + sin2 θwT D2u w + 2 sin θ cos θwT

1 D2u w

= cos2 θλ1 + sin2 θwT D2u w ≤ cos2 θλ1 + sin2 θλn.

In the computation above, we have used the fact that w1 is an eigenvalue and w

is orthogonal to w1 to eliminate the cross term, and we have used the estimatewT D2u w ≤ λn.

Apply the previous calculation to (2.3), to obtain

λ1

λn≥ − tan2(dθ),

since θ ≤ dθ. Likewise, apply the previous calculation to (2.5), to obtain

sin2(dθ)λn ≤d2u

dv2i

= cos2 θλ1 + sin2 θwT D2u w ≤ cos2 θλ1 + sin2 θλn

which yields λ1 ≥ 0.

2.3. The uniformly convex envelope

Motivated by the results of the previous section, we discuss solutions of

max u(x)− g(x),−λ1[u](x) + ε = 0. (2.6)

for ε > 0 and small. Solutions of (2.6) are called the uniformly convex envelope ofg(x).

In the definition of the convex envelope, (CE), it is sufficient to restrict the classof test functions v(x) to affine functions. (This assertion follows from the supportinghyperplane characterization of convexity). The solution of (2.6) can be describedas

u(x) = supv(x) | v ∈ P ε, v(y) ≤ g(y) for all y ∈ Rn. (SCE)

where P ε is the set of quadratic functions q(x), with λ1[q] ≥ ε.


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3. The schemes

Equation (2.2) gives a semi-discrete approximation to the equation (PDE). In orderto fully discretize the scheme, we need to take second derivatives only in directionsθ which are available on the computational grid. The limited directional resolutionof a finite grid requires the introduction of a second parameter dθ, in addition to thespatial resolution, h. While the spatial resolution is improved by using more gridpoints, the directional resolution is improved by increasing the size of the stencil.

3.1. Definition of the monotone scheme

We proceed to define the fully discrete monotone scheme. Discretize the character-ization of the eigenvalues (2.1) using the directions vi available on the grid,

λdx,dθ1 [u](x0) = min

i

u(x0 + vi)− 2u(x0) + u(x0 − vi)|vi|2

(3.1)

Inserting the last equation into (PDE) and solving for u(x) gives

uM (x0) ≡ mini

(g(x0),

u(x0 + vi) + u(x0 − vi)2

)(3.2)

For any fixed stencil, because of the limited directional resolution, the numericalsolution is going to be below the function g(x), but possibly above the convexenvelope.

3.2. A modified scheme which ensures convexity

Since the base scheme (3.2) allows for mildly non-convex solutions, with approxi-mate convexity quantified by (2.4), we introduce a modification to ensure convexityof solutions.

Solving directly λ1 = sin2(dθ)λn, as in Proposition 2.1, ensures convexity, butthe resulting scheme is non-monotone, (in fact it is unstable). Instead, replace λn

with an upper bound, M , for λn[u](x). The result is the following scheme.

uS(x0) ≡ mini

(g(x0),

u(x0 + vi) + u(x0 − vi)2

− sin2(dθ)h2M

)(3.3)

The solution of this scheme is a subsolution of the discrete convex envelope. ByProposition 2.1, the solution will be convex, provided M is an upper bound forλn[u].

Remark 3.1. Even if the solution is not twice-differentiable, for discrete data on agiven grid, an upper bound M can be computed which ensures (discrete) convexity.In fact, for a given function, the discrete version of λn can by computed using theformula (3.1), with the minimum replaced by a maximum. In most cases, M can becomputed using the function g(x). For a better estimate, M can be computed froman intermediate solution.


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Remark 3.2. Because of the factor h in the term sin2(dθ)h2M , the correctionneeded for uniform convexity goes to zero, even for a fixed stencil. This means thatfor this scheme a narrow stencil can be used.

3.3. Boundary grid points

Near the boundary of the domain, the stencil width is limited by the availabilityof grid points. Non-centered second derivatives in a given direction are computedusing function values at intermediate points on the boundary. To give a concreteexample, in a typical calculation, we might compute

d2u

dv2=

15h2

(u(i + 2, j + i)− 2u(i, j) + u(i− 2, j − 1)) + O(h2), v =(2, 1)√

5.

Near the boundary, if the i− 2 value is not available, we compute instead

d2u

dv2=

415h2

(u(i + 2, j + i)− 3u(i, j) + 2u(i− 1, j − 1/2)) + O(h).

Mathematically, we can assume the u(i− 1, j − 1/2) value to be given, since it lieson the boundary. In practice, it may be simpler to compute the value, using inter-polation. Quadratic interpolation is non-monotone, and in practice can introducesignificant errors. Linear interpolation is monotone, so it is preferred.

3.4. The centered finite-difference scheme (in two dimensions)

Starting from (1.3), applying finite differences, and solving for the reference variablegives the nonlinear scheme

uQ(x0) ≡14

(u(1,0) + u(0,1) +

√(u(1,0) − u(0,1))2 + (u(1,1) − u(1.−1))2

)(3.4)

where, using finite difference notation, ui+k,j+l ≡ u(x + kh, y + lh),

u(1,0) = ui+1,j + ui−1,j u(0,1) = ui,j+1 + ui,j−1

u(1,1) =ui+1,j+1 + ui−1,j−1

2u(1,−1) =

ui+1,j−1 + ui−1,j+1

2.

This scheme is accurate to O(dx2). It is non-monotone, and therefore we can’t makeclaims about convergence.

3.5. A hybrid scheme

The next scheme is a hybrid scheme. It achieves the higher accuracy of the finite dif-ference scheme in smooth regions of the solution, while maintaining the robustnessof the monotone scheme in non-smooth regions.

This is accomplished using a thresholding function which selects a convex com-bination of the two schemes based on a regularity measure, as determined by thetest ‖D2u‖ < O(1/h). The hybrid scheme is described by the following algorithm.


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(1) Compute numerically ‖D2u‖ =√

u2xx + 2u2

xy + u2yy.

(2) Pass the result, multiplied by the spatial resolution, h, to a threshold func-tion, s(t), to obtain T (x) = s(h‖D2u(x)‖). The threshold function should beincreasing with values in [0, 1]. The following function was used.

s(t) =

0 t ≤ t1

log(t/t1)/ log(t2/t1) t1 ≤ t ≤ t2

1 t2 ≤ t

with t1 = .1, t2 = .5.(3) Propagate the threshold values to adjacent neighbors, so that

T (x) = max|x−y|≤h

T (y)

This means that T (x) is a slightly non-local function of x.(4) Update the values using a convex combination of the monotone scheme and the

finite difference scheme, with weight T (x)

uH(x) = T (x)uM (x) + (1− T (x))uQ(x). (3.5)

4. Viscosity Solutions

The advantage of using viscosity solutions9 is that the theory applies to whole classesof equations; the machinery for proving existence, uniqueness and stability resultsis the same for each equation. In particular the convergence theory of numericalschemes 2, applies to a broad class of equations.

4.1. Definition of Viscosity Solutions

For the reader’s convenience we recall the main result from Ref. [25].

Definition 4.1. The upper semicontinuous function u is a viscosity solution of−λ1[u] ≤ 0 if for every twice-differentiable function φ(x),

−λ1[φ](x) ≤ 0, whenever x is a local maximum of u− φ. (4.1)

Next we define viscosity solutions of (PDE) in terms of sub and supersolutions:a function is a viscosity solution of (PDE) if it is both a subsolution and a super-solution.

Definition 4.2. The upper semicontinuous function u is a viscosity subsolutionof (PDE) if for every twice-differentiable function φ(x),

φ(x)−g(x) ≤ 0 and −λ1[φ(x)] ≤ 0, whenever x is a local maximum of u− φ.

The lower semicontinuous function u is a viscosity supersolution of (PDE) if forevery twice-differentiable function φ(x),

φ(x)−g(x) ≥ 0 or −λ1[φ(x)] ≥ 0, whenever x is a local minimum of u− φ.


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It is well known that twice differentiable convex functions are characterized bythe property that the Hessian is everywhere positive semidefinite.3 This result isgeneralized to continuous functions by the following result. The continuous functionu : Rn → R is convex if and only if it is a viscosity solution of −λ1[u] ≤ 0. Finally,the following theorem was proved in Ref. [25].

Theorem 4.1. The convex envelope of the function g is a viscosity solutionof (PDE).

4.2. The comparison principle and monotonicity

It is clear from the definition of the convex envelope (CE) that increasing g(x)cannot decrease the convex envelope.

The comparison principle is a nonlinear generalization of the maximum princi-ple. A consequence of the comparison principle is the monotonicity of the solutionoperator. If u1, u2 are solutions of (PDE) for functions g1(x), g2(x), respectively,then

g1 ≤ g2 implies u1 ≤ u2 in Ω. (4.2)

Write S : C(Ω) → C(Ω) for the solution operator, which takes data g, to thesolution u. Then the monotonicity property (4.2) becomes

g1 ≤ g2 implies S(g1) ≤ S(g2). (4.3)

In this form, monotonicity of the solution operator takes the same form for thesolution operator of the numerical scheme. Embracing a duplication of notation, wewrite S for the solution operator of the corresponding numerical scheme, and take(4.3) as the definition of monotonicity.

4.3. Convergence of approximation schemes

One advantage of the viscosity solutions formulation is that we need only checkconsistency on twice continuously differentiable functions, even though we allow fornon-differentiable solutions. The fundamental result for the convergence of numeri-cal schemes for fully nonlinear, degenerate equations is due to Barles and Souganidis.

Theorem 4.2 (Convergence of Approximation Schemes2). Consider a de-generate elliptic equation for which there exist unique viscosity solutions. A consis-tent, stable approximation scheme converges uniformly on compact subsets to theviscosity solution, provided it is monotone.

Note the theorem gives only a uniform convergence rate. As mentioned above,the lack of regularity of solutions in the general setting prevents asserting conver-gence rates like O(h2), despite second order consistency for schemes.

Monotonicity is an essential property which we discuss below. The definition ofconsistency is in terms of twice differentiable test functions: it amounts to checking


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Taylor series computations, see Definition 5.1. The theorem requires only a mildform of stability: our schemes always satisfy a stronger notion of stability: they arenonexpansive in the uniform norm.

5. Convergence of the schemes

Next we prove convergence of the schemes. By Theorem 4.2, we need only to ver-ify that the schemes are consistent and monotone. The first property involves aTaylor series calculation. The added complication of the directional resolution pa-rameter, dθ, necessitates an additional argument beyond the standard Taylor seriesarguments used for consistency in h.

5.1. Monotonicity

Lemma 5.1. The schemes (3.2), (3.3) are monotone.

Proof. The values uM (x0) or uS(x0) are non-decreasing functions of the valuesg(x0) and the neighbors u(xj).

5.2. Consistency

Definition 5.1 (Consistency). We say the scheme Fh,dθ is consistent with theequation (PDE) at x0 if for every twice continuously differentiable function φ(x)defined in a neighborhood of x0,

Fh,dθ[φ](x0) → F [φ](x0) as h, dθ → 0.

Lemma 5.2 (Consistency). Let x0 be a reference point with neighbors x1, . . . , xN ,and direction vectors vi = xi − x0, i = 1, . . . , N , arranged symmetrically so that(1.4) holds. The scheme for λ1 is given by (3.1). If φ(x) is a twice continuouslydifferentiable function defined in a neighborhood of the grid, then

λ1[φ](x0)− λdx,dθ1 [φ](x0) = O(h2 + dθ2). (5.1)

In addition, the explicit schemes (3.2), (3.3) are consistent.

Proof. Given the Hessian D2u(x), let w1 be the eigenvector corresponding to λ1.For an arbitrary unit vector v, decompose v = cos θw1 + sin θw, where w is a unitvector orthogonal to w1. Compute (as in the proof of Proposition 2.1),

d2u

dv2= (cos θw1 + sin θw)T D2u (cos θw1 + sin θw)

= cos2 θλ1 + sin2 θvT2 D2u v2

= λ1 + sin2 θ(vT2 D2u v2 − λ1)

This last equation recovers the Rayleigh-Ritz characterization of the eigenvalues:minimizing over θ yields λ1.


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Next minimize over a discrete set of direction vectors, vi, i = 1, . . . , N, withresolution dθ, defined by (1.6). Calculate

mini=1,...,N

d2u

dv2i

= λ1 + sin2 θ((v′i)T D2u v′i − λ1),

where v′i is the projection of vi onto the space orthogonal to the eigenspace of λ1.By the definition of dθ, we have

mini=1,...,N

d2u

dv2i

≤ λ1 + sin2(dθ)((v′i)T D2u v′i − λ1),

≤ λ1 + sin2(dθ)(λn − λ1).

The standard Taylor series computation gives

u(x0 + vi)− 2u(x0) + u(x0 − vi)|vi|2

=d2u

dv2i

+ O(h2i ).

Combining the h error with the dθ error gives (5.1).Finally, inserting (5.1) into the schemes (3.2) (3.3) and solving for the reference

variable shows consistency of the explicit schemes.

5.3. Convergence

Theorem 5.1 (Convergence). The explicit finite difference schemes givenby (3.2),(3.3) converge uniformly on compacts subsets of Ω to the unique viscos-ity solution of (PDE).

Proof. By virtue of Theorem 4.2, we need only show that the schemes are consis-tent and monotone. Consistency follows from Lemma 5.2 and monotonicity followsfrom Lemma 5.1.

6. Examples and counterexamples

Here we discuss several examples, some related to regularity, and some which illus-trate numerical difficulties in working with convex functions.

Example 6.1 (Lack of differentiability). Convex functions (and thus the con-vex envelope) need not be differentiable: take u(x) = g(x) = |x|.

Example 6.2 (Lack of continuity to boundary). Consider g(x, y) = −x2, on[−1, 1]2, then the convex envelope is a constant.

On unbounded domains, the regularity result of Ref. [15] shows that if g(x) isdifferentiable, then so is u(x). However, on a bounded domain, we can have g(x) isdifferentiable, u(x) continuous up to the boundary, but not differentiable.

Example 6.3 (The convex envelope does not inherit differentiability ona bounded domain). Consider

g(x, y) =α

4(x + y)2 − β

4(x− y)2


15

on [−1, 1]2. Then the convex envelope is given by averaging g along the lines y+x =C. In particular, let (x′, y′) be the projection of (x, y) onto the line y = sign(C),then u(x, y) = g(x′, y′). Set C = x + y, then

u(x, y) =α

4C2 − β

4(C − 2 sign(C))2, C = x + y.

The following example illustrates problems with triangulated domains, and sug-gests why wider stencils are needed.

Example 6.4 (Linear interpolation does not preserve convexity). Con-vexity is not preserved by may standard operations in numerical approximation: inparticular, the piecewise linear interpolant of a convex function may not be convex.8

For example, consider the function g(x, y) = (x + y)2. Sample values of g on thecorners of the unit square: (0, 0), (1, 0), (0, 1), (1, 1). There are two ways of doinglinear interpolation on this square: one results in a convex function; the other in aconcave function.

7. Numerical solution of the equation

It is desirable to design a scheme which behaves in a robust and consistent fashionover the wide range of regularity which is supported by the equation. In particular,convex functions include both (i) polyhedral (piecewise linear) functions, and (ii)smooth, or locally quadratic functions.

As the estimates in a previous section show, for a given stencil, the monotonescheme (3.2) allows for mild nonconvexity. The modification (3.3) forces solutionsto be convex, but introduces some strict convexity to convex polyhedral data.

The hybrid scheme (3.5) uses higher accuracy computations when the solutionis smooth, and falls back on the monotone scheme when the solution is non-smooth.Although the scheme is not provably convergent, it combines some of the advantagesof both schemes: leaving convex polyhedral functions invariant, and detecting andcorrecting the mild non-convexity which the main scheme (3.2) misses.

7.1. Preliminary numerical results

Using the basic scheme (3.2), we computed sample solutions. The first example used

g(x, y) = max(.5x, y − .5, 2x + y − 1,−5x + y − 4, 2− 10x2 + 10y2),

which has for exact solution, u(x, y) = max(.5x, y − .5, 2x + y − 1,−5x + y − 4).The scheme computed the exact solution, to within numerical error. The secondexample used

g(x, y) = sin(π(x− y))/(exp(2(x2 + y2))) + x2 + y2.

To illustrate robustness to discontinuous solutions, we took g(x, y) = y/2−y2, whichhas a linear exact solution. Next we added spatial noise (with amplitude 1/2) toa quadratic function, which resulted in a highly irregular function. The results aredisplayed in Figure 3.


16

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Fig. 3. The function g(x) and its numerically computed convex envelope (inverted in some plots).


17

7.2. Results with a nonconvex quadratic

Using the function from Example 6.3, the exact solution was captured, up to nu-merical accuracy. Since the direction of non-convexity lines up with the grid, the dθ

error is zero, for this example.The next example uses the more general nonconvex quadratic function,

gα,θ(x, y) = (cos2 θ +α sin2 θ)x2

2+(1−α) cos θ sin θxy +(cos2 θα+sin2 θ)

y2

2(7.1)

with α < 1, so that λ1 = α, λn = 1, and the eigenvectors are at an angle of θ fromthe coordinate axes.

Numerical results confirmed the estimates in Proposition 2.1. Choosing the worstpossible directions, which is midway between the grid directions for the Level 2scheme, we get θ = arctan(1/2)/2. Then the threshold for detection of non-convexitylies near α = −0.055. The worst angle for the Level 3 point scheme is an angle ofθ = arctan(1/3)/2. In this case, the threshold for detection of non-convexity is nearα = −.0264.

Next we solved with g(x, y) given by (7.1) with α = −.5, θ = arctan(1/3)/2, theworst possible angle, on a 30×30 grid. Solving using (3.2) gave a mildly nonconvexsolution, with λ1 approximately constant away from the boundary, with value nearthe critical value −.055.

Using uS , the solution was convex, and in some places strictly convex. Using uH

we obtained λ1(x) close to a constant away from the boundary, with values near−.01. Using uQ gave a nonconstant λ1(x), with values as negative as −.004. Theresults are displayed in Figure 4.

7.3. Results with a convex affine function

The next computational experiments were performed on various convex affine func-tions. Since these functions are solutions of (PDE), the question was whether theschemes introduced an error. While all the schemes appeared to converge as h → 0,it was considered a significant advantage when the schemes left the solution in-variant. The results for uM and uS were as expected: the former scheme left affinefunctions invariant, the latter introduced a small degree of strict convexity. (In factthese properties can be proven and quantified for the schemes).

The hybrid scheme uH succeeded in leaving convex affine functions invariant.

7.4. Accelerating Iterations

The iteration (3.2) is a simple, explicit, convergent method to find the solution of thedifference equation. On the largest grids used, the number of iterations numberedin the thousands, and the time to solve was at most five minutes using MATLABon a laptop.

A simple method to accelerate convergence was to use as initial data the inter-polated solution from a coarser grid. Given a grid size n×n, we solved the equation


18

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Fig. 4. (a) (i) g(x) quadratic, with θ lined up on the grid. (ii) The corresponding solution uM . (b)

(iii) g(x) quadratic, with θ not lined up on the grid. (iv) The corresponding solution uM . (v) Thecorresponding solution uS . (vi) The difference uM − uS .

on a coarser grid of n/2 × n/2, and interpolated the solution onto the finer grid.This proved to be faster than solving with a less accurate initial guess.

We compared the number of iterations to get a solution using the two meth-ods. The results are presented in Table 2. The results suggest that the number of


19

iterations using the coarse interpolant appears to be O(n), compared to superlineargrowth using less accurate initial data. Each iteration consists of an application ofthe scheme at each grid point, using data from the previous iteration. (The perfor-mance could be doubled by simply using a Gauss-Seidel iteration).

Table 2. Number of iterations required for convergence

comparing initial data given by a rough guess with theinterpolated solution from a coarser grid.

n 16 32 64 128 256

guess 240 770 2700 9620 47400

interpolated 170 430 920 1810 3490ratio 1.4 1.8 2.9 5.3 13.6

8. Conclusions and Future Work

8.1. Conclusions

We introduced several schemes for solving the nonlinear PDE for the convex enve-lope. The main scheme, (3.2), is an explicit scheme which involves the minimum ofdirectional second derivatives. This scheme leaves convex affine functions invariantand it is also robust under non-smooth data.

This scheme used a wide stencil: on a narrow stencil, solutions could be mildlynon-convex. The degree of convexity was quantified using the dθ parameter, whichmeasured the directional resolution of the scheme. The solution converges to theconvex envelope as dθ → 0.

Based on the convexity estimate, a modification of the scheme was introducedwhich ensured solutions were convex. This was done at the cost of introducing a milduniform convexity of solutions. (So that linear functions became mildly quadratic).This scheme computed the strictly convex envelope.

A centered finite difference scheme was also introduced using a formula (availablein two dimensions) for the smallest eigenvalue. This scheme was not monotone. Infact, applying the scheme to a convex affine function results in negative values forλ1. Formally it is more accurate, but the scheme introduces additional errors whenthe data is not smooth.

Finally a hybrid scheme was introduced: the hybrid scheme combines the mainscheme with the more accurate scheme, using a smoothness monitor to determinewhich scheme is active. The hybrid scheme appears to combine the advantages ofboth schemes: greater sensitivity to negative eigenvalues, as well as robustness tonon-smooth data.

Methods for accelerating convergence were discussed.


20

8.2. Future work

Higher dimensions

The main schemes (3.2) and (3.3) generalize directly to higher dimensions. The sim-plest scheme would involve all the nearest neighbors, including the diagonals, givinga stencil with 27 points in three dimensions. If some degree of uniform convexityis acceptable, using the scheme (3.3) will ensure the solution is convex; no largerstencil is needed.

Unstructured or triangulated grids

The main assumption made about the data points was the symmetry assump-tion (1.4). The scheme can be adapted to allow for unstructured grids. First orderapproximations to second derivatives are available on non-structured grids. Usingnotation from §1.5, consider a local stencil with direction vectors vi. For each linearlydependent subset of direction vectors, we have constants αj such that

∑j αjvj = 0.

Then ∑j

αj(u(x0 + vj)− u(x0)) =∑

j

αjd2u

dv2j

+ O(h).

Enforcing non-negativity of the left hand side of the previous equation for all linearlyindependent subsets of the direction vectors gives and approximation to λ1 ≥ 0.

Other solution methods

The numerical solution is currently found using an explicit iterative method. Futurework will involve investigating whether fast implicit solvers can be implemented.

An alternative characterization of (PDE) was given in Ref. [25] as the valuefunction of a stochastic optimal control problem. It might be possible to build anumerical method based on this interpretation.

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