Gi Ren Liu

Scaling limits for time-fractional diffusion-wave systems with random initial data

Let w (x, t) := (u, v)(x, t), x ∈ ℝ3, t > 0, be the ℝ2-valued spatial-temporal random field w = (u, v) arising from a certain two-equation system of time-fractional linear partial differential equations of reaction-diffusion-wave type, with given random initial data u(x,0), ut(x,0), and v(x,0), vt(x,0). We discuss the scaling limit, under proper homogenization and renormalization, of w(x,t), subject to suitable assumptions on the random initial conditions. Since the component fields u,v depend on the interactions present within the system, we employ a certain stochastic decoupling method to tackle this component dependence. The work shows, in particular, the various non-Gaussian scenarios proposed in [4, 13, 17] and the references therein, for the single diffusion type equations, in classical or in fractional time/space derivatives, can be studied for the two-equation system, in a significant way.

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Alternating projection (AP) of various forms, including the parallel AP (PAP), real-constrained AP (RAP) and the serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed to produce good-quality initial guess. Numerical experiments show that the null initialization is more accurate than the spectral initialization and that AP converges faster to the true object than other iterative schemes such as the Wirtinger flow (WF). In numerical experiments AP with the null initialization converges globally to the true object.

Phase retrieval by linear algebra

The null vector method, based on a simple linear algebraic concept, is proposed as an initialization method for nonconvex approaches to the phase retrieval problem. For the stylized measurement with random complex Gaussian matrices, a nonasymptotic error bound is derived, stronger than that of the spectral vector method. Numerical experiments show that the null vector method also has a superior performance for the realistic measurement of coded diffraction patterns in coherent diffractive imaging.

Scaling Limits for Some P.D.E. Systems with Random Initial Conditions

Let X = {X(x, t), x ∈ R n , t ∈ R +} be the R 2-valued spatial-temporal random field X = (u, v) arising from a certain two-equation system of parabolic linear partial differential equations with a given random initial condition X 0 = (u 0, v 0). We discuss the scaling limit of X under suitable conditions on X 0. Since the component fields u, v are dependent, even when the initial data u 0, v 0 are independent, the scaling limit is not readily reduced to the known single equation case. The correlated structure of random vector (u(x, t), v(x′, t′)) and the Hermite expansion associated with (u 0, v 0) play the essential roles in our study. The work shows, in particular, the non-Gaussian scenario proposed by Anh and Leonenko [2] for the single heat equation can be discussed for the two-equation system, in a significant way.

Multi-scaling limits for relativistic diffusion equations with random initial data

Let

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u(t,\mathbf{x}),\ t>0,\ \mathbf{x}\in \mathbb{R}^{n},

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be the spatial-temporal random field arising from the solution of a relativistic diffusion equation with the spatial-fractional parameter

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections of the Null Vector

\alpha\in (0,2)

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and the mass parameter

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\mathfrak{m}> 0