reduction numericalschemesbasedon introduction voronoi ... · monge-ampere equation seismicwave...

Voronoi’sreduction

Jean-MarieMirebeau

IntroductionAnisotropy inPDEs

Lattice geometry

Voronoi’s firstreduction

RiemannianoperatorsAnisotropicdiffusion

Eikonal equation

HJBoperatorsMonge-Ampereequation

Seismic wavetravel times

Conclusion

Numerical schemes based onVoronoi’s first reduction

Jean-Marie Mirebeau

University Paris Sud, CNRS, University Paris-Saclay

November 29, 2019

Modélisation, Analyse, SimulationConférence à l’occasion des 50 ans du LJLL

In collaboration with F. Bonnans, L. Metivier,PhD students G. Bonnet, F. Desquilbet.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction

Riemannian operatorsAnisotropic diffusionEikonal equation

Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times

Conclusion and perspectives

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Anisotropy in Partial Differential Equations (PDEs)

Anisotropy is the existence of preferred directions, locally, in adomain. The phenomenon is generic and ubiquitous, and mayhave a variety of causes, such as:I Micro-structure, either biological, geologic, synthetic.I Different nature of the domain dimensions, e.g. R2 × S1.I Proximity of the domain boundary, or of discontinuities.

In the numerical analysis of PDEs, (strong) anisotropy is asource of difficulties.I Increased numerical cost, accuracy loss, instabilities or

failure of the numerical methods.Several approaches can be envisioned to address these.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Anisotropy in Partial Differential Equations (PDEs)

Anisotropy is the existence of preferred directions, locally, in adomain. The phenomenon is generic and ubiquitous, and mayhave a variety of causes, such as:I Micro-structure, either biological, geologic, synthetic.I Different nature of the domain dimensions, e.g. R2 × S1.I Proximity of the domain boundary, or of discontinuities.

In the numerical analysis of PDEs, (strong) anisotropy is asource of difficulties.I Increased numerical cost, accuracy loss, instabilities or

failure of the numerical methods.Several approaches can be envisioned to address these.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

First approach: adapt the domain representation

I Encode the problem anisotropy in a Riemannian metric.I Create an anisotropic mesh of the domain.

Figure: Adaptive interpolation of a function with a sharp transition.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

First approach: adapt the domain representationI Encode the problem anisotropy in a Riemannian metric.

I Create an anisotropic mesh of the domain.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

First approach: adapt the domain representationI Encode the problem anisotropy in a Riemannian metric.I Create an anisotropic mesh of the domain.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Second approach: adapt the numerical scheme

I A basic cartesian grid is used throughout this work.

I Local adaptive stencils are created independently at eachpoint, without any consistency constraint.

I Easily address asymmetric or three dimensional anisotropy.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I A basic cartesian grid is used throughout this work.

I Local adaptive stencils are created independently at eachpoint, without any consistency constraint.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I A basic cartesian grid is used throughout this work.I Local adaptive stencils are created independently at each

point, without any consistency constraint.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

First Last

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Second approach: adapt the numerical schemeI A basic cartesian grid is used throughout this work.I Local adaptive stencils are created independently at each

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Comparative advantages of the two approachesAdaptive meshes

I Locally adjust the sampling density.I Domains of arbitrary shape, and topology.

I Conforming meshes are a pre-requisite for someapplications (e.g. finite volume or finite element schemes).

Adaptive stencils on cartesian grids

I Simplicity of implementation.I Numerical cost.I Tools of lattice geometry.

I Cartesian grids are a pre-requisite for some applications(e.g image processing).

Different tools for different applications.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Locally adjust the sampling density.I Domains of arbitrary shape, and topology.

I Conforming meshes are a pre-requisite for someapplications (e.g. finite volume or finite element schemes).

I Simplicity of implementation.I Numerical cost.I Tools of lattice geometry.

I Cartesian grids are a pre-requisite for some applications(e.g image processing).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Locally adjust the sampling density.I Domains of arbitrary shape, and topology.I Conforming meshes are a pre-requisite for some

applications (e.g. finite volume or finite element schemes).

I Simplicity of implementation.I Numerical cost.I Tools of lattice geometry.I Cartesian grids are a pre-requisite for some applications

(e.g image processing).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Locally adjust the sampling density.I Domains of arbitrary shape, and topology.I Conforming meshes are a pre-requisite for some

applications (e.g. finite volume or finite element schemes).

I Simplicity of implementation.I Numerical cost.I Tools of lattice geometry.I Cartesian grids are a pre-requisite for some applications

(e.g image processing).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Lattice geometryIs the simultaneous study of a positive quadratic form, and of adiscrete subgroup of a vector space.

Sample applications

I What is the densest periodic packing of spheres ?

I Which integers are sums of three squares ?I Message coding: error correction, cryptography.

What are the prime factors of RSA-768 ?1230186684530117755130494958384962720772853569595334792197322452151726400507263657518745202199786469389956474942774063845925192557326303453731548268507917026122142913461670429214311602221240479274737794080665351419597459856902143413

Conway, Sloane, Sphere packings, lattices and groups, 1998, has100+ pages of references,

and has been cited 6000+ times.

I Few applications to PDE discretization. Bonnans et al, 04.M. et al, 14, ...

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

I What is the densest periodic packing of spheres ?I Which integers are sums of three squares ?

I Message coding: error correction, cryptography.What are the prime factors of RSA-768 ?

1230186684530117755130494958384962720772853569595334792197322452151726400507263657518745202199786469389956474942774063845925192557326303453731548268507917026122142913461670429214311602221240479274737794080665351419597459856902143413

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

I What is the densest periodic packing of spheres ?I Which integers are sums of three squares ?

Legendre’s theorem:

N = i2 + j2 + k2; (i , j , k) ∈ Z3 t 4a(8b + 7); a, b ∈ N.

I Message coding: error correction, cryptography.What are the prime factors of RSA-768 ?

1230186684530117755130494958384962720772853569595334792197322452151726400507263657518745202199786469389956474942774063845925192557326303453731548268507917026122142913461670429214311602221240479274737794080665351419597459856902143413

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

I What is the densest periodic packing of spheres ?I Which integers are sums of three squares ?I Message coding: error correction, cryptography.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

Conway, Sloane, Sphere packings, lattices and groups, 1998, has100+ pages of references, and has been cited 6000+ times.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Sample applications

Conway, Sloane, Sphere packings, lattices and groups, 1998, has100+ pages of references, and has been cited 6000+ times.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Voronoi’s reductions associate a discrete object to each positivedefinite quadratic form on Rd , and are covariant with GL(Zd).I ρ : S++

d → X , ρ(ATDA) = A · ρ(D) for A ∈ GL(Zd).

Reduction to a fundamental domainI If x1, · · · , xn is a system of representatives of X modulo

A, then ρ−1(x1, · · · , xn) is (under assumptions) afundamental domain in D modulo A.

I Classify positive quadratic forms modulo isomorphisms ofthe lattice Zd .

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Intended use: following closely related ideas, Lagrange providesa table of the positive quadratic forms with integer entries andsmall determinant.

Figure: Lagrange, Recherches d’arithmétique, 1774

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1. Unit ball of ‖ · ‖D .

I Voronoi’s second reduction of D =germ of Delaunay’s triangulation of Z2 w.r.t. ‖ · ‖D .

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

I Dimension 1 2 3 4 5# Inequivalent triangulations 1 1 1 3 222

Shurmann, Computational geometry of positive quadratic forms,2009

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Voronoi’s first reductionAssociates to a given D ∈ S++

d the solution M ∈ S++d to

Tr(DM) subject to ‖e‖2M ≥ 1, ∀e ∈ Zd \ 0.

where ‖e‖M :=√〈e,Me〉.

Geometric interpretation

I M defines a disjoint ellipsoid packing, with centers in Zd .

(z + 12BM)z∈Zd where BM := x ∈ Rd ; ‖x‖M < 1.

I Minimize∑d

i=1 r−2i , where ri is radius along the i-th

principal axis, in the geometry defined by D.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Voronoi’s first reductionAssociates to a given D ∈ S++

d the solution M ∈ S++d to

Tr(DM) subject to ‖e‖2M ≥ 1, ∀e ∈ Zd \ 0.

where ‖e‖M :=√〈e,Me〉.

Geometric interpretation

I M defines a disjoint ellipsoid packing, with centers in Zd .

(z + 12BM)z∈Zd where BM := x ∈ Rd ; ‖x‖M < 1.

I Minimize∑d

i=1 r−2i , where ri is radius along the i-th

principal axis, in the geometry defined by D.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Unit ball defined by D =

(1 + a bb 1− a

), a2 + b2 < 1.

I Optimal tiling of Z2 w.r.t. ‖ · ‖D .

Optimal M (×2).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

I Optimal tiling of Z2 w.r.t. ‖ · ‖D .

Optimal M (×2).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

I Optimal tiling of Z2 w.r.t. ‖ · ‖D . Optimal M (×2).

2 -1-1 2

2 11 2

2 -3-3 6

6 -3-3 2

6 33 2

2 33 6

2 -5-5 14

6 -9-9 14

14 -9-9 6

14 -5-5 2

14 55 2

14 99 6

6 99 14

2 55 14

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Voronoi’s first reduction is a linear minimization problemon Ryskov’s polyhedron

P = M ∈ Sd ;∀e ∈ Zd \ 0, 〈e,Me〉 ≥ 1.

I Locally finite polyhedron, vertices are called perfect forms.

Theorem (Voronoi, 1905)In any dimension d , Voronoi’s first reduction is a feasible linearprogram. In addition, there is only a finite number ofinequivalent perfect forms.

Dimension 1 2 3 4 5 6 7 8 9# Ineq perf forms 1 1 1 2 3 7 33 10916 >500 000

Shurmann, Computational geometry of positive quadratic forms, 09

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

P = M ∈ Sd ;∀e ∈ Zd \ 0, 〈e,Me〉 ≥ 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

P = M ∈ Sd ;∀e ∈ Zd \ 0, 〈e,Me〉 ≥ 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Voronoi’s first reduction is a linear program, in Sd , subjectto infinitely many constraints. (Locally: finitely many)

I Dual program is an optimal tensor decomposition problem.

Voronoi’s first reduction (dual formulation)Associates to D ∈ S++

d a decomposition

D =∑

1≤i≤Iλieie

Ti , with λi ≥ 0, ei ∈ Zd \ 0,

which maximizes the criterion

(w.l.o.g. I ≤ d(d + 1)/2)

∑1≤i≤I

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

d a decomposition

D =∑

1≤i≤Iλieie

(w.l.o.g. I ≤ d(d + 1)/2)

∑1≤i≤I

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

d a decomposition

D =∑

1≤i≤Iλieie

(w.l.o.g. I ≤ d(d + 1)/2)

∑1≤i≤I

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

d a decomposition

D =∑

1≤i≤Iλieie

which maximizes the criterion (w.l.o.g. I ≤ d(d + 1)/2)∑1≤i≤I

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

I Support of the optimal decomposition.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Practical decomposition in dimension d ∈ 2, 3I A superbase of Zd is a tuple (e0, · · · , ed) ∈ (Zd)d+1 s.t.

e0 + · · ·+ ed = 0, | det(e1, · · · , ed)| = 1.

I Selling’s algorithm, 1874.Input: matrix D ∈ S++

d , init superbase b = (e0, · · · , ed).While there exists i , j , such that 〈ei ,Dej〉 > 0

(d = 2) b ← (ei ,−ej , ej − ei ),

(d = 3) b ← (ei ,−ej , ek + ej , el + ej).

Output: b = (e0, · · · , ed), now an obtuse superbase.I Voronoi’s first reduction M = 1

0≤i≤d eieTi .

I Tensor decomposition, with 〈vij , ek〉 = δij − δjk

D = −∑

0≤i<j≤d〈ei ,Dej〉vijvTij .

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

e0 + · · ·+ ed = 0, | det(e1, · · · , ed)| = 1.

(d = 2) b ← (ei ,−ej , ej − ei ),

(d = 3) b ← (ei ,−ej , ek + ej , el + ej).

Output: b = (e0, · · · , ed), now an obtuse superbase.

I Voronoi’s first reduction M = 12∑

0≤i≤d eieTi .

D = −∑

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

e0 + · · ·+ ed = 0, | det(e1, · · · , ed)| = 1.

(d = 2) b ← (ei ,−ej , ej − ei ),

(d = 3) b ← (ei ,−ej , ek + ej , el + ej).

0≤i≤d eieTi .

D = −∑

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

e0 + · · ·+ ed = 0, | det(e1, · · · , ed)| = 1.

(d = 2) b ← (ei ,−ej , ej − ei ),

(d = 3) b ← (ei ,−ej , ek + ej , el + ej).

0≤i≤d eieTi .

D = −∑

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Divergence form anisotropic diffusion

I Consider the anisotropic elliptic energy of u : Ω→ R

E(u) =

∫Ω‖∇u(x)‖2D(x)dx .

I Taking the L2 gradient flow yields

∂tu = div(D(x)∇u(x))I Existing approaches fail the maximum principle for large

anisotropies, or use very wide stencils (Weickert, 96).

Proposed discretization (Ferenbach, M., 2014)I Given x ∈ Ω, assume D(x) =

∑Ii=1 λieie

Ti . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2

I Insert the second order accurate finite difference

〈∇u(x), ei 〉2 ≈12

[(u(x)− u(x + hei )

)2+(u(x)− u(x − hei )

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

E(u) =

∫Ω‖∇u(x)‖2D(x)dx .

∑Ii=1 λieie

Ti . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2

〈∇u(x), ei 〉2 ≈12

[(u(x)− u(x + hei )

)2+(u(x)− u(x − hei )

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

E(u) =

∫Ω‖∇u(x)‖2D(x)dx .

∑Ii=1 λieie

Ti . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2

〈∇u(x), ei 〉2 ≈12

[(u(x)− u(x + hei )

)2+(u(x)− u(x − hei )

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

I Application to image processing, with J. Fehrenbach.Weickert’s coherence enhancing diffusion D = D(u).

Theorem (Equivalence with anisotropic Delaunay, in 2D)Assume Ω = [0, 1]2 equipped with periodic bc, and discretizedon 0, h, · · · , 1− h2 where h = 1/n. Assume D : Ω→ S++

d isLipschitz. Denote Eh the AD-LBR energy, and Eh the finiteelement energy on Shewchuk’s anisotropic Delaunay. Then

(1− ch)Eh(u) ≤ Eh(u) ≤ (1 + ch)Eh(u), ∀u,∀0 < h ≤ h0

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Computing minimal Riemannian geodesics

I Solve the Riemannian eikonal equation, D(x) := M(x)−1,

‖∇u(x)‖2D(x) = 1 (+b.c .)

I Backrack the minimal paths with gradient descent

γ(t) = V (γ(t)), where V (x) = D(x)∇u(x).

I Existing approaches use either wide stencils (Kimmel,Sethian, 01) or multi-pass methods (Bornemann, 06).

Proposed discretization (M, 2019)

I Given x ∈ Ω, assume D(x) =∑I

i=1 λieieTi . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2.

I Insert the first order accurate upwind finite difference

|〈∇u(x), ei 〉| ≈ max0, u(x)− u(x + hei ), u(x)− u(x − hei )

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

‖∇u(x)‖2D(x) = 1 (+b.c .)

i=1 λieieTi . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

‖∇u(x)‖2D(x) = 1 (+b.c .)

i=1 λieieTi . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

‖∇u(x)‖2D(x) = 1 (+b.c .)

i=1 λieieTi . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

‖∇u(x)‖2D(x) = 1 (+b.c .)

i=1 λieieTi . Then

‖∇u(x)‖2D(x) =∑

1≤i≤Iλi 〈∇u(x), ei 〉2.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Application to tubular structure segmentationI Segmenting the retinal vascular tree using Reeds-Shepp

model, with data driven c(x , n). With R. Duits et al.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Application to tubular structure segmentationI Segmenting the retinal vascular tree using Reeds-Shepp

model, with data driven c(x , n). With R. Duits et al.

Figure: Density plot of the cost function c(x , y , θ).Related: (radius lift) Li and Yezzi 07, (θ lift) Péchaud et al 09.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Application to tubular structure segmentation

I Segmenting the retinal vascular tree using Reeds-Sheppmodel, with data driven c(x , n). With R. Duits et al.

Figure: Subriemannian control sets

Related: (radius lift) Li and Yezzi 07, (θ lift) Péchaud et al 09.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion Figure: Optimal paths for the Reeds-Shepp car in R3 × S2.Application to (simulated) dMRI data, with Duits et al (2018)

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Anisotropy and non-linearity

I In HJB equations, non-linearity can (often) be seen assimultaneous anisotropy in multiple incompatible directions:

Λu(x) = maxD∈D(x)

Tr(D∇2u(x)), where D(x) ⊆ S++d .

I Finite difference discretization of Tr(D∇2u(x))

∆hDu(x) :=

∑1≤i≤I

λiu(x − hei )− 2u(x) + u(x + hei )

where D =∑

1≤i≤I λieieTi .

Proposed discretization (Bonnet, Bonnans, M. 2019)

I Partition S++d based on the support of Voronoi’s optimal

decompositionS++d =

⊔b∈BSb.

I Write Λ = maxb∈B Λb where Λbu(x) = maxD∈D(x)∩Sb

∆hDu(x).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Anisotropy and non-linearity

I In HJB equations, non-linearity can (often) be seen assimultaneous anisotropy in multiple incompatible directions:

Λu(x) = maxD∈D(x)

Tr(D∇2u(x)), where D(x) ⊆ S++d .

I Finite difference discretization of Tr(D∇2u(x))

∆hDu(x) :=

∑1≤i≤I

λiu(x − hei )− 2u(x) + u(x + hei )

Proposed discretization (Bonnet, Bonnans, M. 2019)

I Partition S++d based on the support of Voronoi’s optimal

decompositionS++d =

⊔b∈BSb.

I Write Λ = maxb∈B Λb where Λbu(x) = maxD∈D(x)∩Sb

∆hDu(x).

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

(1 + a bb 1− a

), a2 + b2 < 1.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

The Monge-Ampere equation

I A fully non-linear second order equation

det(∇2u) = f , u convex.

I Variants, with appropriate boundary conditions, appear inoptimal transport, optical design, etc.

I Existing computational approaches for OT : entropicrelaxation (Cuturi 2013), semi-discrete (Aurenhammer1998, Merigot 2011), ...

Proposed discretization

I Express the non-linear operator as an extremum of linearoperators: for any M 0

d(detM)1d = inf

detD=1D0

Tr(DM)

I The minimization associated with each supportb = (e0, e1, e2) is solvable in analytical form

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

d(detM)1d = inf

detD=1D0

Tr(DM)

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

d(detM)1d = inf

detD=1D0

Tr(DM)

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

d(detM)1d = inf

detD=1D0

Tr(DM)

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Λhu(x) = min(e0,e1,e2) superbase

H(∆he0u(x),∆h

e1u(x),∆he2u(x))

where ∆heu(x) = (u(x + h)− 2u(x) + u(x − h))/h2, and

H(a, b, c) =

ab if a + b ≤ c12(ab + bc + ca)− 1

4(a2 + b2 + c2) otherwise.

Figure: Numerical optimal Transport, Benamou, Duval (2017), basedon Benamou, Collino, M, 2016.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

ConclusionFigure: Top : various stencil choices. Above: reconstruction ofsynthetic solution |x +

√10| by a Monge-Ampere equation.

Uniformly elliptic variant of the scheme: Bonnet, Bonnans, M, Monotone and consistent schemes forthe Pucci and Monge-Ampere equation, preprint, 2019

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Minimal travel-time of seismic wavesI The high frequency asymptotic yields a Finslerian eikonal

equation1 = max

D∈D(x)‖∇u(x)‖D

where D(x) ⊆ S++d depends on the Hooke elasticity tensor.

I In the TTI (Tilted Transversely Isotropic) case

1 = maxs∈[0,1]

‖∇u(x)‖D(x ,s)

I Existing numerical methods are multi-pass

Proposed discretization (Desquilbet, Metivier, M)

I For a given x ∈ Ω, find 0 = s0 < · · · < sn = 1 such thatthe optimal decomposition support is constant over each[si , si+1], 0 ≤ i < n.

I Use the Riemannian eikonal discretization, along with 1Doptimization, on each [si , si+1].

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

equation1 = max

1 = maxs∈[0,1]

‖∇u(x)‖D(x ,s)

Proposed discretization (Desquilbet, Metivier, M)

I For a given x ∈ Ω, find 0 = s0 < · · · < sn = 1 such thatthe optimal decomposition support is constant over each[si , si+1], 0 ≤ i < n.

I Use the Riemannian eikonal discretization, along with 1Doptimization, on each [si , si+1].

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

equation1 = max

1 = maxs∈[0,1]

‖∇u(x)‖D(x ,s)

Figure: Example of 1D problem involved in the numerical scheme

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Figure: Work in progress with L. Metivier and PhD student F.Desquilbet. (Top) Constant 2D medium. (Bottom) Varying tilt 2Dmedium. (Pics based on closely related approach)

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Perspective: dimension >3

Structure of Voronoi’s decomposition

I Voronoi’s first reduction is nice in dimension d ≤ 3:Unique decomposition depending continuously on thematrix. Unifying concept of superbase. The decompositionsupport spans Zd additively.

I These good properties are lost in dimension d > 3.

Potential applications with d > 3

I (d = 4) Shortest paths with a trailer, in R2 × S× S.I (d = 5) MRI data processing, in R3 × S2.I (d = 6) Three dimensional elasticity.

Hooke tensor in S++(S(R3)), and dim(S(R3)) = 6.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Perspective: dimension >3

Structure of Voronoi’s decomposition

I Voronoi’s first reduction is nice in dimension d ≤ 3:Unique decomposition depending continuously on thematrix. Unifying concept of superbase. The decompositionsupport spans Zd additively.

I These good properties are lost in dimension d > 3.

Potential applications with d > 3

I (d = 4) Shortest paths with a trailer, in R2 × S× S.I (d = 5) MRI data processing, in R3 × S2.I (d = 6) Three dimensional elasticity.

Hooke tensor in S++(S(R3)), and dim(S(R3)) = 6.

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Thanks for your attention.

I Cartesian grids, which are natural for numerousapplications, are not incompatible with anisotropic PDEs.

I The tools of lattice geometry yield a new class of adaptivenumerical schemes, which can be fast, robust, accurate.

I Handling (strongly) anisotropic PDEs allows to addressnew models and applications.

Numerical codes, demo notebooks, available atgithub.com/mirebeau/

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

Jean-MarieMirebeau

Lattice geometry

Eikonal equation

Conclusion

reduction numericalschemesbasedon introduction voronoi ... · monge-ampere equation seismicwave...

Documents