Trapped dynamics widely appears in nature, e.g., the motion of particles in viscous cytoplasm. The famous continuous time random walk (CTRW) model with power law waiting time distribution (having diverging first moment) describes this phenomenon. Because of the finite lifetime of biological particles, sometimes it is necessary to temper the power law measure such that the waiting time measure has convergent first moment. Then the time operator of the Fokker-Planck equation corresponding to the CTRW model with tempered waiting time measure is the so-called tempered fractional derivative. This paper focus on discussing the properties of the time tempered fractional derivative, and studying the wellposedness and the Jacobi-predictor-corrector algorithm for the tempered fractional ordinary differential equation. By adjusting the parameter of the proposed algorithm, any desired convergence order can be obtained and the computational cost linearly increases with time. And the effectiveness of the algorithm is numerically confirmed.
Based on the superconvergent approximation at some point (depending on the fractional order 伪, but not belonging to the mesh points) for Gr眉nwald discretization to fractional derivative, we develop a series of high order quasi-compact schemes for space fractional diffusion equations. Because of the quasi-compactness of the derived schemes, no points beyond the domain are used for all the high order schemes including second order, third order, fourth order, and even higher order schemes; moreover, the algebraic equations for all the high order schemes have the completely same matrix structure. The stability and convergence analysis for some typical schemes are made; the techniques of treating the nonhomogeneous boundary conditions are introduced; and extensive numerical experiments are performed to confirm the theoretical analysis or verify the convergence orders.
The tempered evolution equation describes the trapped dynamics, widely appearing in nature, e.g., the motion of living particles in viscous liquid. This paper proposes the fast predictor-corrector approach for the tempered fractional ordinary differential equations by digging out the potential 'very' short memory principle. The algorithms basing on the idea of equidistributing are detailedly described; their effectiveness and low computation cost, being linearly increasing with time t, are numerically demonstrated.
Figure 1. We propose a deep Dual-Encoder network for denoising Monte Carlo rendering to produce high quality images. We train our network to learn the complicated relationship between noisy images with low sampling rate and corresponding reference with high sampling rate (a). The learned model is then applied to denoise other rendering result with low sampling rate to predict noise-free results (b).
AbstractIn this paper, we present DEMC, a deep Dual-Encoder network to remove Monte Carlo noise efficiently while preserving details. Denoising Monte Carlo rendering is different from natural image denoising since inexpensive by-products (feature buffers) can be extracted in the rendering stage. Most of them are noisefree and can provide sufficient details for image reconstruction. However, these feature buffers also contain redundant information, which makes Monte Carlo denoising different from natural image denoising. Hence, the main challenge of this topic is how to extract useful information and reconstruct clean images. To address this problem, we propose a novel network structure, Dual-Encoder network with a feature fusion subnetwork, to fuse feature buffers firstly, then encode the fused feature buffers and a noisy image simultaneously, and finally reconstruct a clean image by a decoder network. Compared with the state-of-the-art methods, our 1 arXiv:1905.03908v1 [cs.MM] 10 May 2019 model is more robust on a wide range of scenes, and is able to generate satisfactory results in a significantly faster way.
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