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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
First approach: adapt the domain representationI 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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
First approach: adapt the domain representationI 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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 each
point, without any consistency constraint.
I Easily address asymmetric or three dimensional anisotropy.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 each
point, without any consistency constraint.
First
First
Last
Last
I Easily address asymmetric or three dimensional anisotropy.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 each
point, without any consistency constraint.
First Last
First
Last
I Easily address asymmetric or three dimensional anisotropy.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
point, without any consistency constraint.
I Easily address asymmetric or three dimensional anisotropy.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 some
applications (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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 some
applications (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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 ?
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
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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, ...
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I D =
(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 .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Voronoi’s second reduction of D =germ of Delaunay’s triangulation of Z2 w.r.t. ‖ · ‖D .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Voronoi’s second reduction of D =germ of Delaunay’s triangulation of Z2 w.r.t. ‖ · ‖D .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Voronoi’s second reduction of D =germ of Delaunay’s triangulation of Z2 w.r.t. ‖ · ‖D .
I Dimension 1 2 3 4 5# Inequivalent triangulations 1 1 1 3 222
Shurmann, Computational geometry of positive quadratic forms,2009
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
Voronoi’s first reductionAssociates to a given D ∈ S++
d the solution M ∈ S++d to
minM
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
Voronoi’s first reductionAssociates to a given D ∈ S++
d the solution M ∈ S++d to
minM
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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).
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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).
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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).
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
λi .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
λi .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
λi .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
λi .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I Unit ball defined by D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Support of the optimal decomposition.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I Unit ball defined by D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Support of the optimal decomposition.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
2∑
0≤i≤d eieTi .
I Tensor decomposition, with 〈vij , ek〉 = δij − δjk
D = −∑
0≤i<j≤d〈ei ,Dej〉vijvTij .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 = 12∑
0≤i≤d eieTi .
I Tensor decomposition, with 〈vij , ek〉 = δij − δjk
D = −∑
0≤i<j≤d〈ei ,Dej〉vijvTij .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
2∑
0≤i≤d eieTi .
I Tensor decomposition, with 〈vij , ek〉 = δij − δjk
D = −∑
0≤i<j≤d〈ei ,Dej〉vijvTij .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
2∑
0≤i≤d eieTi .
I Tensor decomposition, with 〈vij , ek〉 = δij − δjk
D = −∑
0≤i<j≤d〈ei ,Dej〉vijvTij .
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
h
)2+(u(x)− u(x − hei )
h
)2]
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
h
)2+(u(x)− u(x − hei )
h
)2]
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
h
)2+(u(x)− u(x − hei )
h
)2]
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
/h
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
/h
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
/h
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
/h
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
/h
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion Figure: Optimal paths for the Reeds-Shepp car in R3 × S2.Application to (simulated) dMRI data, with Duits et al (2018)
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
h2
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).
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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 )
h2
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).
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I Unit ball defined by D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Support of the optimal decomposition.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
I Unit ball defined by D =
(1 + a bb 1− a
), a2 + b2 < 1.
I Support of the optimal decomposition.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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].
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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].
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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
Figure: Example of 1D problem involved in the numerical scheme
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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)
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
Conclusion
IntroductionAnisotropy in PDEsLattice geometryVoronoi’s first reduction
Riemannian operatorsAnisotropic diffusionEikonal equation
Hamilton-Jacobi-Bellman operatorsMonge-Ampere equationSeismic wave travel times
Conclusion and perspectives
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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.
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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/
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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/
Voronoi’sreduction
Jean-MarieMirebeau
IntroductionAnisotropy inPDEs
Lattice geometry
Voronoi’s firstreduction
RiemannianoperatorsAnisotropicdiffusion
Eikonal equation
HJBoperatorsMonge-Ampereequation
Seismic wavetravel times
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/