Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction

Stability of Discrete Stokes Operators in Fractional Sobolev Spaces

Using a general approximation setting having the generic properties of finite-elements, we prove uniform boundedness and stability estimates on the discrete Stokes operator in Sobolev spaces with fractional exponents. As an application, we construct approximations for the timedependent Stokes equations with a source term in L(0, T ;L(Ω)) and prove uniform estimates on the time derivative and di...

Sobolev Spaces

Lifting in Sobolev Spaces

for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this...

Structural inpainting

Scene-agnostic visual inpainting remains very challenging despite progress in patch-based methods. Recently, Pathak et al. [25] have introduced convolutional “context encoders” (CEs) for unsupervised feature learning through image completion tasks. With the additional help of adversarial training, CEs turned out to be a promising tool to complete complex structures in real inpainting problems. ...

Composition in Fractional Sobolev Spaces

1. Introduction. A classical result about composition in Sobolev spaces asserts that if u ∈ W k,p (Ω)∩L ∞ (Ω) and Φ ∈ C k (R), then Φ • u ∈ W k,p (Ω). Here Ω denotes a smooth bounded domain in R N , k ≥ 1 is an integer and 1 ≤ p < ∞. This result was first proved in [13] with the help of the Gagliardo-Nirenberg inequality [14]. In particular if u ∈ W k,p (Ω) with kp > N and Φ ∈ C k (R) then Φ • ...