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1. Generative imaging and image processing via generative encoder

   https://doi.org/10.3934/ipi.2021060  

   Ong, Yong Zheng ; Yang, Haizhao ( January 2022 , Inverse Problems & Imaging)

   This paper introduces a novel generative encoder (GE) framework for generative imaging and image processing tasks like image reconstruction, compression, denoising, inpainting, deblurring, and super-resolution. GE unifies the generative capacity of GANs and the stability of AEs in an optimization framework instead of stacking GANs and AEs into a single network or combining their loss functions as in existing literature. GE provides a novel approach to visualizing relationships between latent spaces and the data space. The GE framework is made up of a pre-training phase and a solving phase. In the former, a GAN with generator \begin{document}$ G $\end{document} capturing the data distribution of a given image set, and an AE network with encoder \begin{document}$ E $\end{document} that compresses images following the estimated distribution by \begin{document}$ G $\end{document} are trained separately, resulting in two latent representations of the data, denoted as the generative and encoding latent space respectively. In the solving phase, given noisy image \begin{document}$ x = \mathcal{P}(x^*) $\end{document}, where \begin{document}$ x^* $\end{document} is the target unknown image, \begin{document}$ \mathcal{P} $\end{document} is an operator adding an addictive, or multiplicative, or convolutional noise, or equivalently given such an image \begin{document}$ x $\end{document}more »in the compressed domain, i.e., given \begin{document}$ m = E(x) $\end{document}, the two latent spaces are unified via solving the optimization problem

   \begin{document}$ z^* = \underset{z}{\mathrm{argmin}} \|E(G(z))-m\|_2^2+\lambda\|z\|_2^2 $\end{document}

   and the image \begin{document}$ x^* $\end{document} is recovered in a generative way via \begin{document}$ \hat{x}: = G(z^*)\approx x^* $\end{document}, where \begin{document}$ \lambda>0 $\end{document} is a hyperparameter. The unification of the two spaces allows improved performance against corresponding GAN and AE networks while visualizing interesting properties in each latent space.

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2. Undulated bilayer interfaces in the planar functionalized Cahn-Hilliard equation

   https://doi.org/10.3934/dcdss.2022035  

   Promislow, Keith ; Wu, Qiliang ( January 2022 , Discrete and Continuous Dynamical Systems - S)

   Experiments with diblock co-polymer melts display undulated bilayers that emanate from defects such as triple junctions and endcaps, [8]. Undulated bilayers are characterized by oscillatory perturbations of the bilayer width, which decay on a spatial length scale that is long compared to the bilayer width. We mimic defects within the functionalized Cahn-Hillard free energy by introducing spatially localized inhomogeneities within its parameters. For length parameter \begin{document}$ \varepsilon\ll1 $\end{document}, we show that this induces undulated bilayer solutions whose width perturbations decay on an \begin{document}$ O\!\left( \varepsilon^{-1/2}\right) $\end{document} inner length scale that is long in comparison to the \begin{document}$ O(1) $\end{document} scale that characterizes the bilayer width.
3. Highly accurate operator factorization methods for the integral fractional Laplacian and its generalization

   https://doi.org/10.3934/dcdss.2022016  

   Wu, Yixuan ; Zhang, Yanzhi ( January 2022 , Discrete & Continuous Dynamical Systems - S)

   In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$ (- \Delta)^\frac{{ \alpha}}{{2}} $\end{document} for \begin{document}$ \alpha \in (0, 2) $\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}$ {\mathcal O}(h^2) $\end{document}, while \begin{document}$ {\mathcal O}(h^4) $\end{document} for quadratic basis functions with \begin{document}$ h $\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$ \alpha \in (0, 2) $\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$ u \in C^{m, l}(\bar{ \Omega}) $\end{document} for \begin{document}$ m \in {\mathbb N} $\end{document} and \begin{document}$ 0 < l < 1 $\end{document}, our method has an accuracy of \begin{document}$more »{\mathcal O}(h^{\min\{m+l, \, 2\}}) $\end{document} for constant and linear basis functions, while \begin{document}$ {\mathcal O}(h^{\min\{m+l, \, 4\}}) $\end{document} for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

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4. Instantaneous smoothing and exponential decay of solutions for a degenerate evolution equation with application to Boltzmann's equation

   https://doi.org/10.3934/krm.2022012  

   Nazarov, Fedor ; Zumbrun, Kevin ( January 2022 , Kinetic and Related Models)

   We establish an instantaneous smoothing property for decaying solutions on the half-line \begin{document}$ (0, +\infty) $\end{document} of certain degenerate Hilbert space-valued evolution equations arising in kinetic theory, including in particular the steady Boltzmann equation. Our results answer the two main open problems posed by Pogan and Zumbrun in their treatment of \begin{document}$ H^1 $\end{document} stable manifolds of such equations, showing that \begin{document}$ L^2_{loc} $\end{document} solutions that remain sufficiently small in \begin{document}$ L^\infty $\end{document} (i) decay exponentially, and (ii) are \begin{document}$ C^\infty $\end{document} for \begin{document}$ t>0 $\end{document}, hence lie eventually in the \begin{document}$ H^1 $\end{document} stable manifold constructed by Pogan and Zumbrun.
5. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

   https://doi.org/10.3934/eect.2022007  

   Triggiani, Roberto ; Wan, Xiang ( January 2022 , Evolution Equations and Control Theory)

   We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case,more »by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].

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