Dissipation enhancement for a degenerated parabolic equation

In this paper, we quantitatively consider the enhanced-dissipation effect of the advection term to the parabolic $p$-Laplacian equations. More precisely, we show the mixing property of flow for the passive scalar enhances the dissipation process of the $p$-Laplacian in the sense of $L^2$ decay, that is, the $L^2$ decay can be arbitrarily fast. The main ingredient of our argument is to understand the underlying iteration structure inherited from the parabolic $p$-Laplacian equations. This extends the dissipation enhancement result of the advection diffusion equation by Yuanyuan Feng and Gautam Iyer into a non-linear setting.Comment: 22 page

Suppression of epitaxial thin film growth by mixing

We consider following fourth-order parabolic equation with gradient nonlinearity on the two-dimensional torus with and without advection of an incompressible vector field in the case $2<p<3$: \begin{equation*} \partial_t u + (-\Delta)^2 u = -\nabla\cdot(|\nabla u|^{p-2}\nabla u). \end{equation*} The study of this form of equations arises from mathematical models that simulate the epitaxial growth of the thin film. We prove the local existence of mild solutions for any initial data lies in $L^2$ in both cases. Our main result is: in the advective case, if the imposed advection is sufficiently mixing, then the global existence of solution can be proved, and the solution will converge exponentially to a homogeneous mixed state. While in the absence of advection, there exist initial data in $H^2\cap W^{1,\infty}$ such that the solution will blow up in finite time.Comment: 33 page

Tumor boundary instability induced by nutrient consumption and supply

We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model derived from taking the incompressible limit of a cell density model. We analyze the boundary stability/instability in two scenarios: 1) the front of the traveling wave; 2) the radially symmetric boundary. In each scenario, we investigate the boundary behaviors under two different nutrient supply regimes, in vitro, and in vivo. Our main conclusion is that for either scenario, the in vitro regime always stabilizes the tumor's boundary regardless of the nutrient consumption rate. However, boundary instability may occur when the tumor cells aggressively consume nutrients, and the nutrient supply is governed by the in vivo regime.Comment: 33 pages, 10 figure

The uncertainty estimation of feature-based forecast combinations

Forecasting is an indispensable element of operational research (OR) and an important aid to planning. The accurate estimation of the forecast uncertainty facilitates several operations management activities, predominantly in supporting decisions in inventory and supply chain management and effectively setting safety stocks. In this paper, we introduce a feature-based framework, which links the relationship between time series features and the interval forecasting performance into providing reliable interval forecasts. We propose an optimal threshold ratio searching algorithm and a new weight determination mechanism for selecting an appropriate subset of models and assigning combination weights for each time series tailored to the observed features. We evaluate our approach using a large set of time series from the M4 competition. Our experiments show that our approach significantly outperforms a wide range of benchmark models, both in terms of point forecasts as well as prediction intervals

The uncertainty estimation of feature-based forecast combinations

Forecasting is an indispensable element of operational research (OR) and an important aid to planning. The accurate estimation of the forecast uncertainty facilitates several operations management activities, predominantly in supporting decisions in inventory and supply chain management and effectively setting safety stocks. In this paper, we introduce a feature-based framework, which links the relationship between time series features and the interval forecasting performance into providing reliable interval forecasts. We propose an optimal threshold ratio searching algorithm and a new weight determination mechanism for selecting an appropriate subset of models and assigning combination weights for each time series tailored to the observed features. We evaluate our approach using a large set of time series from the M4 competition. Our experiments show that our approach significantly outperforms a wide range of benchmark models, both in terms of point forecasts as well as prediction intervals