Stochastic variational inequalities and applications to the total - PowerPoint PPT Presentation

Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative noise Michael R¨ ockner (University of Bielefeld) joint work with Viorel Barbu (Romanian Academy, Iasi) R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 1 / 39

Contents Introduction and framework 1 Definition of (solutions to) SVI and the main existence and uniqueness result 2 The equivalent random PDE 3 Method of proof 4 Extinction in finite time 5 R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 2 / 39

1. Introduction and framework Consider the nonlinear diffusion equation � � dX ( t ) = div sign ( ∇ X ( t )) dt + X ( t ) dW ( t ) on (0 , T ) × O , X = 0 on (0 , T ) × ∂ O , (SPDE) X (0) = x ∈ L 2 ( O ) , where T > 0 is arbitrary and O := bounded, convex, open set in R N , ∂ O smooth; ∞ � W ( t , ξ ) := µ k e k ( ξ ) β k ( t ) , ( t , ξ ) ∈ (0 , ∞ ) × O with µ k ∈ R , β k , k ∈ N , independent k =1 BM’s on (Ω , F , ( F t ) , P ) and e k , k ∈ N , eigenbasis of Dirichlet Laplacian ∆ D on O . Furthermore, sign : R N → 2 R N (multi-valued!) � u if u ∈ R N \{ 0 } , | u | , sign u := { u ∈ R N : | u | ≤ 1 } , if u = 0 ∈ R N . R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 3 / 39

1. Introduction and framework Standing assumption: � ∞ � ∞ (H1) C 2 µ 2 k | e k | 2 ∞ := ∞ < ∞ and µ k |∇ e k | ∞ < ∞ . k =1 k =1 Set � ∞ µ 2 k e 2 µ ( ξ ) := k ( ξ ) , k =1 i.e. � W ( · , ξ ) � t = µ ( ξ ) · t , t ≥ 0 , ξ ∈ O . R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 4 / 39

1. Introduction and framework Modelling: (i) In nonlinear diffusion theory, (SPDE) is derived from the continuity equation perturbed by a Gaussian process proportional to the density X ( t ) of the material, that is, dX ( t ) = div J ( ∇ X ( t )) dt + X ( t ) dW ( t ) , where J = sgn is the flux of the diffusing material. (See [Y. Giga, R. Kobayashi 2003], [M.H. Giga, Y. Giga 2001], [Y. Giga, R.V. Kohn 2011].) (ii) (SPDE) is also relevant as a mathematical model for faceted crystal growth under a stochastic perturbation as well as in material sciences (see [R. Kobayashi, Y. Giga 1999] for the deterministic model and complete references on the subject). As a matter of fact, these models are based on differential gradient systems corresponding to a convex and nondifferentiable potential (energy). R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 5 / 39

1. Introduction and framework (iii) Other recent applications refer to the PDE approach to image recovery (see, e.g., [A. Chamballe, P.L. Lions 1997] and also [T. Barbu, V. Barbu, V. Biga, D. Coca 2009], [T. Chan, S. Esedogly, F. Park, A. Yip 2006]). In fact, if x ∈ L 2 ( O ) is the blurred image, one might find the restored image via the total variation flow X = X ( t ) generated by the stochastic equation � ∇ X ( t ) � dX ( t ) = div dt + X ( t ) dW ( t ) in (0 , T ) × O , |∇ X ( t ) | (SPDE’) X (0) = x in O . In its deterministic form, this is the so-called total variation based image restoration model and its stochastic version (SPDE’) arises naturally in this context as a perturbation of the total variation flow by a Gaussian (Wiener) noise (which explains the title of the talk). R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 6 / 39

1. Introduction and framework In [V. Barbu, G. Da Prato, M. R., SIAM 2009], a complete existence and uniqueness result was proved for variational solutions to (SPDE) in the case of additive noise, that is, dX ( t ) − div [ sgn ( ∇ X ( t ))] dt = dW ( t ) in (0 , T ) × O , X (0) = x in O , X ( t ) = 0 on (0 , T ) × ∂ O , if 1 ≤ N ≤ 2. For the multiplicative noise X ( t ) dW ( t ), only the existence of a variational solution was proved and uniqueness remained open. (See, however, the work olle, arXiv 2011] for recent results on this line, if x ∈ H 1 [B. Gess, J.M. T¨ 0 ( O ).) R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 7 / 39

1. Introduction and framework In [V. Barbu, M. R., ArXiv 2012], we prove the existence and uniqueness of variational solutions to (SPDE) in all dimensions N ≥ 1 and all initial conditions x ∈ L 2 ( O ). We would like to stress that one main difficulty is when x ∈ L 2 ( O ) \ H 1 0 ( O ), while the case x ∈ H 1 0 ( O ) is more standard. Furthermore, we prove the finite-time extinction of solutions with positive probability, if N ≤ 3, generalizing corresponding results from [F. Andreu, V. Caselles, J. D´ ıaz, J. Maz´ on, JFA 2002] and [F. Andreu-Vaillo, V. Caselles, J.M. Maz´ on, Birkh¨ auser 2004] obtained in the deterministic case. R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 8 / 39

1. Introduction and framework Notation L p ( O ) := standard L p -spaces with norm | · | p , p ∈ [1 , ∞ ] W 1 , p (0) O := standard (Dirichlet) Sobolev spaces in L p ( O ) , p ∈ [1 , ∞ ) with norm �� � 1 / p |∇ u | p d ξ � u � 1 , p := ( d ξ = Lebesgue measure on O ) O H 1 0 ( O ) := W 1 , 2 H 2 ( O ) := W 2 , 2 ( O ) . ( O ) , 0 BV ( O ) := space of functions u : O → R with bounded variation �� � u div ϕ d ξ : ϕ ∈ C ∞ 0 ( O ; R N ) , | ϕ | ∞ ≤ 1 � Du � := sup O � |∇ u | d ξ, if u ∈ W 1 , 1 ( O )) . (= O BV 0 ( O ) := all u ∈ BV ( O ) vanishing on ∂ O . R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 9 / 39

1. Introduction and framework − div ∇ u |∇ u | as subdifferential of � Du � : Consider φ 0 : L 1 ( O ) → R = ( −∞ , + ∞ ] � if u ∈ BV 0 ( O ) , � Du � φ 0 ( u ) := + ∞ otherwise , and let cl φ 0 denote the lower semicontinuous closure of φ 0 in L 1 ( O ), that is, � � lim inf φ 0 ( u n ); u n → u ∈ L 1 ( O ) cl φ 0 ( u ) := inf . R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 10 / 39

1. Introduction and framework Hence, by [H. Attouch, G. Buttazzo, M. Gerard 2006], for u ∈ L 1 ( O ), �   | γ 0 ( u ) | d H N − 1 � Du � + if u ∈ BV ( O ) , cl φ 0 ( u ) = ∂ O  + ∞ otherwise, where γ 0 ( u ) is the trace of u on the boundary and d H N − 1 is the Hausdorff measure. Let φ denote the restriction of cl φ 0 ( u ) to L 2 ( O ), i.e., �   | γ 0 ( u ) | d H N − 1 , if u ∈ BV ( O ) ∩ L 2 ( O ) , � Du � + φ ( u ) := ∂ O  if u ∈ L 2 ( O ) \ BV ( O ) . + ∞ , Note that (as in the deterministic case) the initial Dirichlet boundary condition is lost during this procedure as a price for getting φ to be lower semicontinuous on L 1 ( O ). Hence in (SPDE) the boundary condition only holds in this generalized sense. R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 11 / 39

1. Introduction and framework By ∂φ : D ( ∂φ ) ⊂ L 2 ( O ) → L 2 ( O ), we denote the subdifferential of φ , that is, ∂φ ( u ) := { η ∈ L 2 ( O ); φ ( u ) − φ ( v ) ≤ � η, u − v � , ∀ v ∈ D ( φ ) } , where D ( φ ) := { u ∈ L 2 ( O ); φ ( u ) < ∞} = BV ( O ) ∩ L 2 ( O ) . Then it turns out that ∂φ ( u ) := {− div z | z ∈ L ∞ ( O ; R N ) , | z | ∞ ≤ 1 , � z , ∇ u � = φ ( u ) } ���� measure! (where div and pairing � , � in sense of Schwartz distributions). Heuristically, � ∇ u � � � �� int. by parts “ |∇ u | d ξ ” = “ − div u d ξ ” . |∇ u | O O Rigorously, for − div z ∈ ∂φ ( u ) � Def. of div φ ( u ) = � z , ∇ u � = ( − div ���� ) u d ξ. z ���� O measure! ξ �→ z ( ξ ) is section of ξ �→ ∇ u |∇ u | = sign ( ∇ u ) (multi-valued!) R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 12 / 39

1. Introduction and framework Hence (again) we can rewrite (SPDE) as dX ( t ) + ∂φ ( X ( t )) dt ∋ X ( t ) dW ( t ) , t ∈ [0 , T ] , (SPDE”) X (0) = x ∈ L 2 ( O ) . However, since the multi-valued mapping ∂φ : L 2 ( O ) → L 2 ( O ) is highly singular, at present for arbitrary initial conditions x ∈ L 2 ( O ) no general existence result for stochastic infinite dimensional equations of subgradient type is applicable to the present situation. Our approach is to rewrite (SPDE”) (hence (SPDE), (SPDE’)) as a stochastic variational inequality (SVI). R¨ ockner (Bielefeld) Stochastic variational inequalities and applications to the total variation flow perturbed by linear multiplicative 13 / 39