Level-set volume-preserving diffusions Yann Brenier CNRS, Centre de mathématiques Laurent Schwartz Ecole Polytechnique, FR 91128 Palaiseau Fluid Mechanics, Hamiltonian Dynamics, and Numerical Aspects of Optimal Transportation, MSRI, October 14-18, 2013 Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 1 / 18 EQUIVALENCE OF TWO DIFFERENT FUNCTIONS WITH LEVEL SETS OF EQUAL VOLUME ' ∼ '0 1.5 volume and topology preservation 1 0.5 0 -0.5 -1 -1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 2 / 18 MOTIVATION: MINIMIZATION PROBLEMS WITH VOLUME CONSTRAINTS ON LEVEL SETS 1.5 volume and topology preservation 1 0.5 0 -0.5 -1 -1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 This goes back to Kelvin. See Th. B. Benjamin, G. Burton etc.... Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 3 / 18 This can be rephrased as Z Z 1 2 inf sup jr'(x)j dx + [F('(x)) − F('0(x))]dx ': ! D R F:R!R 2 D D Optimal solutions are formally solutions to − 4' + F0(') = 0 for some function F : R ! R , and, in 2d, are just stationary solutions to the Euler equations of incompressible fluids. An example in fluid mechanics d d Here D = T = (R=Z) is the flat torus and '0 is a given function on D. We want to minimize the Dirichlet integral among all ' ∼ '0. Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 4 / 18 Optimal solutions are formally solutions to − 4' + F0(') = 0 for some function F : R ! R , and, in 2d, are just stationary solutions to the Euler equations of incompressible fluids. An example in fluid mechanics d d Here D = T = (R=Z) is the flat torus and '0 is a given function on D. We want to minimize the Dirichlet integral among all ' ∼ '0. This can be rephrased as Z Z 1 2 inf sup jr'(x)j dx + [F('(x)) − F('0(x))]dx ': ! D R F:R!R 2 D D Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 4 / 18 , and, in 2d, are just stationary solutions to the Euler equations of incompressible fluids. An example in fluid mechanics d d Here D = T = (R=Z) is the flat torus and '0 is a given function on D. We want to minimize the Dirichlet integral among all ' ∼ '0. This can be rephrased as Z Z 1 2 inf sup jr'(x)j dx + [F('(x)) − F('0(x))]dx ': ! D R F:R!R 2 D D Optimal solutions are formally solutions to − 4' + F0(') = 0 for some function F : R ! R Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 4 / 18 An example in fluid mechanics d d Here D = T = (R=Z) is the flat torus and '0 is a given function on D. We want to minimize the Dirichlet integral among all ' ∼ '0. This can be rephrased as Z Z 1 2 inf sup jr'(x)j dx + [F('(x)) − F('0(x))]dx ': ! D R F:R!R 2 D D Optimal solutions are formally solutions to − 4' + F0(') = 0 for some function F : R ! R , and, in 2d, are just stationary solutions to the Euler equations of incompressible fluids. Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 4 / 18 This is a so-called "quadratic assignment problem", a well known NP problem of combinatorial optimization (also related to the works of F. Memoli and K.T. Sturm, recently presented at MSRI). Discrete version: a NP problem in combinatorics After discretizing the Dirichlet integral on a lattice with N grid points Ai, we have to find a permutation σ that achieves N X inf cσ σ λij σ i j i;j=1 where λ is a matrix that depends on the lattice and 2 cij = j'0(Ai) − '0(Aj)j Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 5 / 18 Discrete version: a NP problem in combinatorics After discretizing the Dirichlet integral on a lattice with N grid points Ai, we have to find a permutation σ that achieves N X inf cσ σ λij σ i j i;j=1 where λ is a matrix that depends on the lattice and 2 cij = j'0(Ai) − '0(Aj)j This is a so-called "quadratic assignment problem", a well known NP problem of combinatorial optimization (also related to the works of F. Memoli and K.T. Sturm, recently presented at MSRI). Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 5 / 18 THIS SUGGESTS THE CONSTRUCTION OF "GRADIENT FLOWS" OF THE DIRICHLET INTEGRAL (i.e. DIFFUSION EQUATIONS) THAT ARE "LSVP": THEY PRESERVE THE VOLUME OF 1.5 ALL LEVEL SETS volume and topology preservation 1 0.5 0 -0.5 -1 -1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 6 / 18 If v is smooth enough, this just means '(t; ξ(t; x)) = '0(x) where ξ is the time-dependent family of volume and orientation-preserving diffeomorphisms defined by @tξ(t; x) = v(t; ξ(t; x)); ξ(0; x) = x By Ambrosio-DiPerna-Lions (resp. Cauchy-Lipschitz) theory on ODEs, bounded variation (resp. Lipschitz) regularity of v in x is enough to preserve the volume (resp. the topology) of level sets. Transport of functions by divergence-free vector fields A canonical way to preserve the volume of each level set of a time-dependent scalar function (t; x) ! '(t; x) 2 R is the transport by a time-dependent divergence-free velocity field v(t; x) 2 Rd: @t' + r · (v') = 0; r · v = 0 Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 7 / 18 where ξ is the time-dependent family of volume and orientation-preserving diffeomorphisms defined by @tξ(t; x) = v(t; ξ(t; x)); ξ(0; x) = x By Ambrosio-DiPerna-Lions (resp. Cauchy-Lipschitz) theory on ODEs, bounded variation (resp. Lipschitz) regularity of v in x is enough to preserve the volume (resp. the topology) of level sets. Transport of functions by divergence-free vector fields A canonical way to preserve the volume of each level set of a time-dependent scalar function (t; x) ! '(t; x) 2 R is the transport by a time-dependent divergence-free velocity field v(t; x) 2 Rd: @t' + r · (v') = 0; r · v = 0 If v is smooth enough, this just means '(t; ξ(t; x)) = '0(x) Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 7 / 18 By Ambrosio-DiPerna-Lions (resp. Cauchy-Lipschitz) theory on ODEs, bounded variation (resp. Lipschitz) regularity of v in x is enough to preserve the volume (resp. the topology) of level sets. Transport of functions by divergence-free vector fields A canonical way to preserve the volume of each level set of a time-dependent scalar function (t; x) ! '(t; x) 2 R is the transport by a time-dependent divergence-free velocity field v(t; x) 2 Rd: @t' + r · (v') = 0; r · v = 0 If v is smooth enough, this just means '(t; ξ(t; x)) = '0(x) where ξ is the time-dependent family of volume and orientation-preserving diffeomorphisms defined by @tξ(t; x) = v(t; ξ(t; x)); ξ(0; x) = x Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 7 / 18 Transport of functions by divergence-free vector fields A canonical way to preserve the volume of each level set of a time-dependent scalar function (t; x) ! '(t; x) 2 R is the transport by a time-dependent divergence-free velocity field v(t; x) 2 Rd: @t' + r · (v') = 0; r · v = 0 If v is smooth enough, this just means '(t; ξ(t; x)) = '0(x) where ξ is the time-dependent family of volume and orientation-preserving diffeomorphisms defined by @tξ(t; x) = v(t; ξ(t; x)); ξ(0; x) = x By Ambrosio-DiPerna-Lions (resp. Cauchy-Lipschitz) theory on ODEs, bounded variation (resp. Lipschitz) regularity of v in x is enough to preserve the volume (resp. the topology) of level sets. Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 7 / 18 The usual diffusion equation does not preserve the volume of level sets The usual linear diffusion equation @t' = 4' cannot be written in form @t' + r · (v') = 0; r · v = 0 and, therefore, cannot preserve the volume of the level sets of '. Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 8 / 18 and v is a divergence-free vector-field (with zero mean in x). When v has Sobolev regularity Ws;2 in x, s ≥ 1 is enough to enforce the volume preservation of all level sets (DiPerna-Lions’ theory), but not their topology unless s > d=2 + 1. In the case s = 0, we will focus on for notational simplicity, we will speak only of "formal preservation". A concept of "admissible" solutions DEFINITION We say that a pair ('; v) on [0; T ] × Td is admissible if ' is transported by v @t' + r · (v') = 0 Yann Brenier (CNRS) Level-set volume-preserving diffusions October 2013 9 / 18 When v has Sobolev regularity Ws;2 in x, s ≥ 1 is enough to enforce the volume preservation of all level sets (DiPerna-Lions’ theory), but not their topology unless s > d=2 + 1.