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Ji Li

Publications and source records attributed to Ji Li.

At least 235 records · Page 13Linked to original sources
arXiv2016

Numerical Optimization Algorithm of Wavefront Phase Retrieval from Multiple Measurements

Wavefront phase retrieval from a set of intensity measurements can be formulated as an optimization problem. Two nonconvex objective models (MLP and its variants LS) based on maximum likelihood estimation are investigated. We develop numerical optimization algorithms for real-valued function of complex variables and apply them to solve the wavefront phase retrieval problem efficiently. Numerical simulation is given with application to three wavefront phase retrieval problems. LS model shows better numerical performances than MLP model. An explanation for this is that the distribution of the eigenvalues of Hessian matrix of LS model is more clustered than MLP model. LBFGS shows more robust performance and takes fewer calculations than other line search methods.

math.OC↗
arXiv2016

On Relaxed Averaged Alternating Reflections (RAAR) Algorithm for Phase Retrieval from Structured Illuminations

In this paper, as opposed to the random phase masks, the structured illuminations with a pixel-dependent deterministic phase shift are considered to derandomize the model setup. The RAAR algorithm is modified to adapt to two or more diffraction patterns, and the modified RAAR algorithm operates in Fourier domain rather than space domain. The local convergence of the RAAR algorithm is proved by some eigenvalue analysis. Numerical simulations is presented to demonstrate the effectiveness and stability of the algorithm compared to the HIO (Hybrid Input-Output) method. The numerical performances show the global convergence of the RAAR in our tests.

math.NA↗
arXiv2016

On Gradient Descent Algorithm for Generalized Phase Retrieval Problem

In this paper, we study the generalized phase retrieval problem: to recover a signal $\bm{x}\in\mathbb{C}^n$ from the measurements $y_r=\lvert \langle\bm{a}_r,\bm{x}\rangle\rvert^2$, $r=1,2,\ldots,m$. The problem can be reformulated as a least-squares minimization problem. Although the cost function is nonconvex, the global convergence of gradient descent algorithm from a random initialization is studied, when $m$ is large enough. We improve the known result of the local convergence from a spectral initialization. When the signal $\bm{x}$ is real-valued, we prove that the cost function is local convex near the solution $\{\pm\bm{x}\}$. To accelerate the gradient descent, we review and apply several efficient line search methods. We also perform a comparative numerical study of the line search methods and the alternative projection method. Numerical simulations demonstrate the superior ability of LBFGS algorithm than other algorithms.

math.OC↗
arXiv2016

The odd-isotope fractions of Barium in CEMP-r/s star HE 0338-3945 and r-II star CS 31082-001

We report the first measurement of the odd-isotope fractions for barium, \fodd\, in two extremely metal-poor stars: a CEMP-r/s star \he\ (\feh\,$=-2.42\pm0.11$) and an r-II star \cs\ (\feh\,$=-2.90\pm0.13$). The measured \fodd\ values are $0.23\pm0.12$ corresponding to $34.3\pm34.3$\% of the r-process contributions for \he\ and $0.43\pm0.09$ corresponding to $91.4\pm25.7$\% of the r-process contribution to Ba production for \cs. The high r-process signature of barium in \cs\ ($91.4\pm25.7\%$) suggests that the majority of the heavy elements in this star were synthesised via an r-process path, while the lower r-process value ($34.3\pm34.3\%$) found in \he\ indicates that the heavy elements in this star formed through a mix of s-process and r-process synthesis. These conclusions are consistent with studies based on AGB model calculations to fit their abundance distributions.

astro-ph.SR↗
arXiv2016

Localized spatially nonlinear matter waves in atomic-molecular Bose-Einstein condensates with space-modulated nonlinearity

The intrinsic nonlinearity is the most remarkable characteristic of the Bose-Einstein condensates (BECs) systems. Many studies have been done on atomic BECs with time- and space- modulated nonlinearities, while there is few work considering the atomic-molecular BECs with space-modulated nonlinearities. Here, we obtain two kinds of Jacobi elliptic solutions and a family of rational solutions of the atomic-molecular BECs with trapping potential and space-modulated nonlinearity and consider the effect of three-body interaction on the localized matter wave solutions. The topological properties of the localized nonlinear matter wave for no coupling are analysed: the parity of nonlinear matter wave functions depends only on the principal quantum number $n$, and the numbers of the density packets for each quantum state depend on both the principal quantum number $n$ and the secondary quantum number $l$. When the coupling is not zero,the localized nonlinear matter waves given by the rational function, their topological properties are independent of the principal quantum number $n$, only depend on the secondary quantum number $l$. The Raman detuning and the chemical potential can change the number and the shape of the density packets. The stability of the Jacobi elliptic solutions depends on the principal quantum number $n$, while the stability of the rational solutions depends on the chemical potential and Raman detuning.

nlin.PS↗
arXiv2016

Characterizations of product Hardy spaces in Bessel setting

In this paper, we work in the setting of Bessel operators and Bessel Laplace equations studied by Weinstein, Huber, and the harmonic function theory in this setting introduced by Muckenhoupt--Stein, especially the generalised Cauchy--Riemann equations and the conjugate harmonic functions. We provide the equivalent characterizations of product Hardy spaces associated with Bessel operators in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions defined via Poisson and heat semigroups, based on the atomic decomposition, the extension of Merryfield's result which connects the product non-tangential maximal function and area function, and on the grand maximal function technique which connects the product non-tangential and radial maximal function. We then obtain directly the decomposition of the product BMO space associated with Bessel operators. These results are a first extension for product Hardy and BMO associated to a differential operator other than the Laplacian and are a major step beyond the Chang--Fefferman setting.

math.AP↗
arXiv2016

Compactness of Riesz transform commutator associated with Bessel operators

Let $λ>0$ and $\triangle_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$ be the Bessel operator on $\mathbb R_+:=(0,\infty)$. We first introduce and obtain an equivalent characterization of ${\rm CMO}(\mathbb R_+,\, x^{2λ}dx)$. By this equivalent characterization and establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function $b\in {\rm BMO}(\mathbb R_+,\, x^{2λ}dx)$ is in ${\rm CMO}(\mathbb R_+,\, x^{2λ}dx)$ if and only if the Riesz transform commutator $[b, R_{Δ_λ}]$ is compact on $L^p(\mathbb R_+, x^{2λ}dx)$ for any $p\in(1, \infty)$.

math.CA↗
arXiv2015

Criterion of the boundedness of singular integrals on spaces of homogeneous type

It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric $d$ and the measure $μ.$ To achieve our goal, we prove that the atomic Hardy spaces introduced by Coifman and Weiss coincide with the Hardy spaces defined in terms of wavelet coefficients and develop the molecule theory for this general setting. The main tools used in this paper are atomic decomposition, the orthonormal wavelet basis constructed recently by Auscher and Hytönen, the discrete Calderón-type reproducing formula, the almost orthogonal estimates, implement various stopping time arguments and the duality of the Hardy spaces with the Carleson measure spaces.

math.CA↗
arXiv2015

Geometric characterizations of embedding theorems

The embedding theorem arises in several problems from analysis and geometry. The purpose of this paper is to provide a deeper understanding of analysis and geometry with a particular focus on embedding theorems on spaces of homogeneous type in the sense of Coifman and Weiss. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov and Triebel-Lizorkin spaces.

math.CA↗
arXiv2015

Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type

Let $X_1$ and $X_2$ be metric spaces equipped with doubling measures and let $L_1$ and $L_2$ be nonnegative self-adjoint second-order operators acting on $L^2(X_1)$ and $L^2(X_2)$ respectively. We study multivariable spectral multipliers $F(L_1, L_2)$ acting on the Cartesian product of $X_1$ and $X_2$. Under the assumptions of the finite propagation speed property and Plancherel or Stein--Tomas restriction type estimates on the operators $L_1$ and~$L_2$, we show that if a function~$F$ satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator $F(L_1, L_2)$ is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space $X_1\times X_2$. We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner--Riesz means.

math.CA↗
arXiv2015

$T1$ criterions for generalised Calderón--Zygmund type operators on Hardy and BMO spaces associated to Schrödinger operators and applications

Suppose $L=-Δ+V$ is a Schrödinger operator on $\mathbb{R}^n$ with a potential $V$ belonging to certain reverse Hölder class $RH_σ$ with $σ\geq n/2$. The main aim of this paper is to provide necessary and sufficient conditions in terms of $T1$ criteria for a generalised Calderón--Zygmund type operator with respect to $L$ to be bounded on Hardy spaces $H^p_L(\mathbb{R}^n)$ and on BMO type spaces BMO$_L^α(\mathbb{R}^n)$ associated with $L$. As applications, we prove the boundedness for several singular integral operators associated to $L$. Our approach is flexible enough to prove the boundedness of the Riesz transforms related to $L$ with $n/2 \leq σ<n$ which were investigated in \cite{MSTZ} under the stronger condition $σ\geq n$. Thus our results not only recover existing results in \cite{MSTZ} but also contains new results in literature.

math.AP↗
arXiv2015

Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type

The aim of this article is to develop the theory of product Hardy spaces associated with operators which possess the weak assumption of Davies--Gaffney heat kernel estimates, in the setting of spaces of homogeneous type. We also establish a Calderón--Zygmund decomposition on product spaces, which is of independent interest, and use it to study the interpolation of these product Hardy spaces. We then show that under the assumption of generalized Gaussian estimates, the product Hardy spaces coincide with the Lebesgue spaces, for an appropriate range of~$p$.

math.CA↗
arXiv2015

End-point estimates for singular integrals with non-smooth kernels on product spaces

The main aim of this article is to establish boundedness of singular integrals with non-smooth kernels on product spaces. Let $L_1$ and $L_2$ be non-negative self-adjoint operators on $L^2(\mathbb{R}^{n_1})$ and $L^2(\mathbb{R}^{n_2})$, respectively, whose heat kernels satisfy Gaussian upper bounds. First, we obtain an atomic decomposition for functions in $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ where the Hardy space $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ associated with $L_1$ and $L_2$ is defined by square function norms, then prove an interpolation property for this space. Next, we establish sufficient conditions for certain singular integral operators to be bounded on the Hardy space $H^1_{L_1,L_2}(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2})$ when the associated kernels of these singular integrals only satisfy regularity conditions significantly weaker than those of the standard Calderón--Zygmund kernels. As applications, we obtain endpoint estimates of the double Riesz transforms associated to Schrdingier operators and a Marcinkiewicz-type spectral multiplier theorem for non-negative self-adjoint operators on product spaces.

math.CA↗
arXiv2015

Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type

We give an explicit construction of Haar functions associated to a system of dyadic cubes in a geometrically doubling quasi-metric space equipped with a positive Borel measure, and show that these Haar functions form a basis for $L^p$. Next we focus on spaces $X$ of homogeneous type in the sense of Coifman and Weiss, where we use these Haar functions to define a discrete square function, and hence to define dyadic versions of the function spaces $H^1(X)$ and ${\rm BMO}(X)$. In the setting of product spaces $\widetilde{X} = X_1 \times \cdots \times X_n$ of homogeneous type, we show that the space ${\rm BMO}(\widetilde{X})$ of functions of bounded mean oscillation on $\widetilde{X}$ can be written as the intersection of finitely many dyadic ${\rm BMO}$ spaces on $\widetilde{X}$, and similarly for $A_p(\widetilde{X})$, reverse-Hölder weights on $\widetilde{X}$, and doubling weights on $\widetilde{X}$. We also establish that the Hardy space $H^1(\widetilde{X})$ is a sum of finitely many dyadic Hardy spaces on $\widetilde{X}$, and that the strong maximal function on $\widetilde{X}$ is pointwise comparable to the sum of finitely many dyadic strong maximal functions. These dyadic structure theorems generalize, to product spaces of homogeneous type, the earlier Euclidean analogues for ${\rm BMO}$ and $H^1$ due to Mei and to Li, Pipher and Ward.

math.CA↗
arXiv2015

Factorization for Hardy spaces and characterization for BMO spaces via commutators in the Bessel setting

Fix $λ>0$. Consider the Hardy space $H^1(\mathbb{R}_+,dm_λ)$ in the sense of Coifman and Weiss, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. Also consider the Bessel operators $Δ_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$, and $S_λ:=-\frac{d^2}{dx^2}+\frac{λ^2-λ}{x^2}$ on $\mathbb{R_+}$. The Hardy spaces $H^1_{Δ_λ}$ and $H^1_{S_λ}$ associated with $Δ_λ$ and $S_λ$ are defined via the Riesz transforms $R_{Δ_λ}:=\partial_x (Δ_λ)^{-1/2}$ and $R_{S_λ}:= x^λ\partial_x x^{-λ} (S_λ)^{-1/2}$, respectively. It is known that $H^1_{Δ_λ}$ and $H^1(\mathbb{R}_+,dm_λ)$ coincide but they are different from $H^1_{S_λ}$. In this article, we prove the following: (a) a weak factorization of $H^1(\mathbb{R}_+,dm_λ)$ by using a bilinear form of the Riesz transform $R_{Δ_λ}$, which implies the characterization of the BMO space associated to $Δ_λ$ via the commutators related to $R_{Δ_λ}$; (b) the BMO space associated to $S_λ$ can not be characterized by commutators related to $R_{S_λ}$, which implies that $H^1_{S_λ}$ does not have a weak factorization via a bilinear form of the Riesz transform $R_{S_λ}$.

math.CA↗
arXiv2015

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this paper includes product~$H^p$, the dual $\cmo^p$ of $H^p$ with the special case $\bmo = \cmo^1$, and the predual $\vmo$ of $H^1$. We also use the wavelet expansion to establish the Calderón--Zygmund decomposition for product $H^p$, and deduce an interpolation theorem. We make no additional assumptions on the quasi-metric or the doubling measure for each factor space, and thus we extend to the full generality of product spaces of homogeneous type the aspects of both one-parameter and multiparameter theory involving the Littlewood--Paley theory and function spaces. Moreover, our methods would be expected to be a powerful tool for developing wavelet analysis on spaces of homogeneous type.

math.CA↗
arXiv2013

Optimal Transportation for Generalized Lagrangian

In this paper, we study the optimal transportation for generalized Lagrangian $L=L(x, u,t)$, and consider the cost function as following: $$c(x, y)=\inf_{\substack{x(0)=x\\x(1)=y\\u\in\mathcal{U}}}\int_0^1L(x(s), u(x(s),s), s)ds.$$ Where $\mathcal{U}$ is a control set, and $x$ satisfies the following ordinary equation: $$\dot{x}(s)=f(x(s),u(x(s),s)).$$ We prove that under the condition that the initial measure $μ_0$ is absolutely continuous w.r.t. the Lebesgue measure, the Monge problem has a solution, and the optimal transport map just walks along the characteristic curves of the corresponding Hamilton-Jacobi equation: \begin{equation*} \begin{cases} V_t(t, x)+\sup_{\substack{u\in\mathcal{U}}} =0.\\ V(0,x)=ϕ_0(x) \end{cases} \end{equation*}

math.DS↗