Xiliang Lu

• Professor (Associate) at Wuhan University

In recent years, stochastic algorithms have been introduced to solve ill-posed inverse problems. These algorithms select a random subset of equations during each iteration, displaying excellent scalability and competitive performance in large-scale inverse problems. However, given the inherent ill-posed nature of the underlying problems and the pre...

Background One of the main challenges of utilizing spot‐scanning proton arc therapy (SPArc) in routine clinics is treatment delivery efficiency. Spot reduction, which relies on spot sparsity optimization (SSO), is crucial for achieving high delivery efficiency in SPArc. Purpose This study aims to develop a novel SSO approach based on the alternati...

Background Intensity‐modulated proton therapy (IMPT) optimizes spot intensities and position, providing better conformability. However, the successful application of IMPT is dependent upon addressing the challenges posed by range and setup uncertainties. In order to address the uncertainties in IMPT, robust optimization is essential. Purpose This...

Efficient quantum compiling is essential for complex quantum algorithms realization. The Solovay–Kitaev (S–K) theorem offers a theoretical lower bound on the required operations for approaching any unitary operator. However, it is still an open question that this lower bound can be actually reached in practice. Here, we present an efficient quantum...

In this article, we present a novel sampling method to improve the training accuracy of Physics-Informed Neural Networks (PINNs). Inspired by the idea of incremental learning in artificial intelligence, we propose a risk min–max framework to do the adaptive sampling. Within this framework, we develop a simple yet effective strategy known as Gaussia...

To effectively implement quantum algorithms on noisy intermediate-scale quantum (NISQ) processors is a central task in modern quantum technology. NISQ processors feature tens to a few hundreds of noisy qubits with limited coherence times and gate operations with errors, so NISQ algorithms naturally require employing circuits of short lengths via qu...

In this work, we investigate a numerical procedure for recovering a space-dependent diffusion coefficient in a (sub)diffusion model from the given terminal data, and provide a rigorous numerical analysis of the procedure. By exploiting decay behavior of the observation in time, we establish a novel H{\"o}lder type stability estimate for a large ter...

In this paper, we investigate solving the elliptical multiple eigenvalue (EME) problems using a Feedforward Neural Network. Firstly, we propose a general formulation for computing EME based on penalized variational forms of elliptical eigenvalue problems. Next, we solve the penalized variational form using the Deep Ritz Method. We establish an uppe...

In this paper, we introduce CDII-PINNs, a computationally efficient method for solving CDII using PINNs in the framework of Tikhonov regularization. This method constructs a physics-informed loss function by merging the regularized least-squares output functional with an underlying differential equation, which describes the relationship between the...

Consider linear ill-posed problems governed by the system Aix=yi for i=1,…,p , where each A i is a bounded linear operator from a Banach space X to a Hilbert space Y i . In case p is huge, solving the problem by an iterative regularization method using the whole information at each iteration step can be very expensive, due to the huge amount of mem...

In this paper, we focus on approximating a natural class of functions that are compositions of smooth functions. Unlike the low-dimensional support assumption on the covariate, we demonstrate that composition functions have an intrinsic sparse structure if we assume each layer in the composition has a small degree of freedom. This fact can alleviat...

In this paper, we consider recovering n-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-$$\end{document}dimensional signals from m binary measurements corrupted by no...

The membership matrix is a key element in fuzzy clustering, enabling novel data representation in multiple clusters. The row vectors of the membership matrix represent each sample's degree of membership to different clusters. Notably, researchers have confirmed the presence of the local affinity among these row vectors, effectively preserving the l...

In this work we analyze the inverse problem of recovering a space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based o...

We propose an inverse problem to derive the stress distributions that drive the tissue flows during gastrulation in the epiblast of the chick embryo, from measurements of the tissue velocity fields at different stages of development. We assume that the embryonic tissue can be described as a highly viscous fluid, characterised by the Stokes equation...

Consider linear ill-posed problems governed by the system $A_i x = y_i$ for $i =1, \cdots, p$, where each $A_i$ is a bounded linear operator from a Banach space $X$ to a Hilbert space $Y_i$. In case $p$ is huge, solving the problem by an iterative regularization method using the whole information at each iteration step can be very expensive, due to...

Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating it...

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) [47] for second order elliptic equations with Neumann boundary conditions. We establish the fi...

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the...

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discret...

For ℓ0 penalized (nonlinear) regression problems, most existing algorithms carried out theoretical analysis and numerical calculation with a fixed step size. However, the selection of an appropriate step size and the guarantee of good performance depend heavily on the parameters of the restricted strong convexity and smoothness of the loss function...

Efficient quantum compiling tactics greatly enhance the capability of quantum computers to execute complicated quantum algorithms. Due to its fundamental importance, a plethora of quantum compilers has been designed in past years. However, there are several caveats to current protocols, which are low optimality, high inference time, limited scalabi...

Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating it...

In this work we analyze the inverse problem of recovering the space-dependent potential coefficient in an elliptic / parabolic problem from distributed observation. We establish novel (weighted) conditional stability estimates under very mild conditions on the problem data. Then we provide an error analysis of a standard reconstruction scheme based...

In this paper, we propose and analyze a reconstruction algorithm for imaging an anisotropic conductivity tensor in a second-order elliptic PDE with a nonzero Dirichlet boundary condition from internal current densities. It is based on a regularized output least-squares formulation with the standard $L^2(\Omega)^{d,d}$ penalty, which is then discret...

In this work, we propose a nonconvex stochastic alternating minimizing (SAM) method for sparse phase retrieval, where an $s$ -sparse vector of length $n$ is recovered from $m$ phaseless linear measurements. In each iteration of SAM, a batch of measurements is chosen randomly to form a sparse constrained least square subproblem, and then we em...

In this work we propose a nonconvex two-stage \underline{s}tochastic \underline{a}lternating \underline{m}inimizing (SAM) method for sparse phase retrieval. The proposed algorithm is guaranteed to have an exact recovery from $O(s\log n)$ samples if provided the initial guess is in a local neighbour of the ground truth. Thus, the proposed algorithm...

We propose a pathwise semismooth Newton algorithm (PSNA) for sparse recovery in high-dimensional linear models. PSNA is derived from a formulation of the KKT conditions for Lasso and Enet based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regression coefficients al...

In this work, we consider the algorithm to the (nonlinear) regression problems with $\ell_0$ penalty. The existing algorithms for $\ell_0$ based optimization problem are often carried out with a fixed step size, and the selection of an appropriate step size depends on the restricted strong convexity and smoothness for the loss function, hence it is...

Deep Ritz methods (DRM) have been proven numerically to be efficient in solving partial differential equations. In this paper, we present a convergence rate in $H^{1}$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary condition, where the error depends on the depth and width in the deep neural networks and the number of sampl...

The Normalized Difference Vegetation Index (NDVI) can reflect the plant life cycle of growth and senescence, and has become a widely used tool for many applications related to phenology, ecology, and the environment. However, unwanted disturbance from cloud, snow, and other atmospheric effects greatly lowers the NDVI quality and hinders its further...

We study the problem of find a global minimizers of $V(x):\mathbb{R}^d\rightarrow\mathbb{R}^1$ approximately via sampling from a probability distribution $\mu_{\sigma}$ with density $p_{\sigma}=\frac{\exp(-V(x)/\sigma)}{\int_{\mathbb R^d} \exp(-V(y)/\sigma) dy }$ with respect to the Lebesgue measure for $\sigma \in (0,1)$ small enough. We analyze a...

In this paper, we consider the sparse phase retrieval problem, recovering an s-sparse signal x♮∈Rn from m phaseless samples yi=|〈x♮,ai〉| for i=1,…,m. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard thresholding pursuit (HTP) algorithm in compressed sensing, we propose an e...

Recovering sparse signals from observed data is an important topic in signal/imaging processing, statistics and machine learning. Nonconvex penalized least squares have been attracted a lot of attentions since they enjoy nice statistical properties. Computationally, coordinate descent (CD) is a workhorse for minimizing the nonconvex penalized least...

In recent years, physical informed neural networks (PINNs) have been shown to be a powerful tool for solving PDEs empirically. However, numerical analysis of PINNs is still missing. In this paper, we prove the convergence rate to PINNs for the second order elliptic equations with Dirichlet boundary condition, by establishing the upper bounds on the...

In this paper, we consider the problem of recovering a sparse signal based on penalized least squares formulations. We develop a novel algorithm of primal-dual active set type for a class of nonconvex sparsity-promoting penalties, including ℓ 0 , bridge, smoothly clipped absolute deviation, capped ℓ 1 and minimax concavity penalty. First, we establ...

We propose a fast Newton algorithm for \(\ell _0\) regularized high-dimensional generalized linear models based on support detection and root finding. We refer to the proposed method as GSDAR. GSDAR is developed based on the KKT conditions for \(\ell _0\)-penalized maximum likelihood estimators and generates a sequence of solutions of the KKT syste...

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establis...

In this paper, we construct neural networks with ReLU, sine and $2^x$ as activation functions. For general continuous $f$ defined on $[0,1]^d$ with continuity modulus $\omega_f(\cdot)$, we construct ReLU-sine-$2^x$ networks that enjoy an approximation rate $\mathcal{O}(\omega_f(\sqrt{d})\cdot2^{-M}+\omega_{f}\left(\frac{\sqrt{d}}{N}\right))$, where...

In this paper, we study the optimization algorithm to compute the smallest eigenvalue of the Schrödinger operator with volume constraint. A finite element discretization of this problem is established. We provide the error estimate for the numerical solution. The optimal solution can be approximated by a fixed point iteration scheme. Then a monoton...

We propose an Euler particle transport (EPT) approach for generative learning. The proposed approach is motivated by the problem of finding an optimal transport map from a reference distribution to a target distribution characterized by the Monge-Ampere equation. Interpreting the infinitesimal linearization of the Monge-Ampere equation from the per...

Accurate extracted ion chromatograms (XIC or EIC) is of great importance for Liquid chromatography - mass spectrometry (LC-MS) based quantitative proteomics. However, current preprocessing methods for XIC mainly focus on removing the peaks of low quality. Such operations are helpful for quantitation, but also contain two potential disadvantages: on...

We consider the variational regularization for inverse problems in a general form. Based on the discrepancy principle, we propose a heuristic parameter choice rule for choosing the regularization parameter which does not require the information on the noise level and is therefore purely data driven. Under variational source conditions, we obtain a...

The main goal of 1-bit compressive sampling is to decode $n$ dimensional signals with sparsity level $s$ from $m$ binary measurements. This is a challenging task due to the presence of nonlinearity, noises and sign flips. In this paper, the cardinality constraint least square is proposed as a desired decoder. We prove that, up to a constant $c$, wi...

In this paper, we consider the sparse phase retrival problem, recovering an $s$-sparse signal $\bm{x}^{\natural}\in\mathbb{R}^n$ from $m$ phaseless samples $y_i=|\langle\bm{x}^{\natural},\bm{a}_i\rangle|$ for $i=1,\ldots,m$. Existing sparse phase retrieval algorithms are usually first-order and hence converge at most linearly. Inspired by the hard...

Screening and working set techniques are important approaches to reducing the size of an optimization problem. They have been widely used in accelerating first-order methods for solving large-scale sparse learning problems. In this paper, we develop a new screening method called Newton screening (NS) which is a generalized Newton method with a buil...

Feature selection is important for modeling high-dimensional data, where the number of variables can be much larger than the sample size. In this paper, we develop a support detection and root finding procedure to learn the high dimensional sparse generalized linear models and denote this method by GSDAR. Based on the KKT condition for $\ell_0$-pen...

Sparse phase retrieval plays an important role in many fields of applied science and thus attracts lots of attention. In this paper, we propose a \underline{sto}chastic alte\underline{r}nating \underline{m}inimizing method for \underline{sp}arse ph\underline{a}se \underline{r}etrieval (\textit{StormSpar}) algorithm which {emprically} is able to rec...

The code is provided as is. Please run demo_simulation.m in Matlab. Please cite: L. Guo, L. Chen, X. Lu and C. L. P. Chen, "Membership Affinity Lasso for Fuzzy Clustering," in IEEE Transactions on Fuzzy Systems. doi: 10.1109/TFUZZ.2019.2905114

Fuzzy clustering generates a membership vector for each data point in the data set to indicate its belongingness to different clusters. This procedure can be regarded as an encoding process and the obtained vectors of memberships are the new representations of original data. Naturally, the affinities between new representations or the vectors of me...

A natural image u is often sparse under a given transformation W, one can use \(L_0\) norm of Wu as a regularisation term in image reconstructions. Since minimizing the \(L_0\) norm is a NP hard problem, the \(L_1\) norm is widely used as an replacement. However, recent studies show that nonconvex penalties, e.g., MCP, enjoy better performance for...

Stochastic gradient descent (SGD) and its variants are among the most successful approaches for solving large-scale optimization problems. At each iteration, SGD employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for handling truly...

We propose a semismooth Newton algorithm for pathwise optimization (SNAP) for the LASSO and Enet in sparse, high-dimensional linear regression. SNAP is derived from a suitable formulation of the KKT conditions based on Newton derivatives. It solves the semismooth KKT equations efficiently by actively and continuously seeking the support of the regr...

Destriping is a classical problem in remote sensing image processing. Although considerable effort has been made to remove stripes, few of the existing methods can eliminate stripe noise with arbitrary orientations. This situation makes the removal of oblique stripes in the higher-level remote sensing products become an unfinished and urgent issue....

We propose a constructive approach to estimating sparse, high-dimensional linear regression models. The approach is a computational algorithm motivated from the KKT conditions for the `0-penalized least squares solutions. It generates a sequence of solutions iteratively, based on support detection using primal and dual information and root finding....

Stochastic gradient descent is one of the most successful approaches for solving large-scale problems, especially in machine learning and statistics. At each iteration, it employs an unbiased estimator of the full gradient computed from one single randomly selected data point. Hence, it scales well with problem size and is very attractive for truly...

In 1-bit compressive sensing (1-bit CS) where a target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads y = η sign(Ψx∗ + ), where x∗ ∈ Rⁿ, y ∈ Rm, Ψ ∈ Rm×ⁿ, and is the random error before quantization and η ∈ Rⁿ is a random vector modeling the sig...

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the pre...

In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: $y = \eta \odot\textrm{sign} (\Psi x^* + \epsilon)$, where $x^{*}\in \mathcal{R}^{n}, y\in \mathcal{R}^{m}$, $\Psi \in \mathcal{R}^{m\times n}$,...

Striping effects are a common phenomenon in remote-sensing imaging systems, and they can exhibit considerable differences between different sensors. Such artifacts can greatly degrade the quality of the measured data and further limit the subsequent applications in higher level remote-sensing products. Although a lot of destriping methods have been...

In this paper, a novel numerical algorithm is presented to compute the optimal time of a time optimal control problem where the governing system is a linear ordinary differential equation. By the equivalence between time optimal control problem and norm optimal control problem, computation of the optimal time can be obtained by solving a sequence o...

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the pre...

In this note, we analyze an iterative soft / hard thresholding algorithm with homotopy continuation for recovering a sparse signal $x^\dag$ from noisy data of a noise level $\epsilon$. Under suitable regularity and sparsity conditions, we design a path along which the algorithm can find a solution $x^*$ which admits a sharp reconstruction error $\|...

In this note, we analyze an iterative soft / hard thresholding algorithm with homotopy continuation for recovering a sparse signal $x^\dag$ from noisy data of a noise level $\epsilon$. Under suitable regularity and sparsity conditions, we design a path along which the algorithm can find a solution $x^*$ which admits a sharp reconstruction error $\|...

We develop a constructive approach to estimating sparse, high-dimensional linear regression models. The approach is a computational algorithm motivated from the KKT conditions for the $\ell_0$-penalized least squares solutions. It generates a sequence of solutions iteratively, based on support detection using primal and dual information and root fi...

We propose a preconditioned alternating direction method of multipliers (ADMM) to solve linear inverse problems in Hilbert spaces with constraints, where the feature of the sought solution under a linear transformation is captured by a possibly non-smooth convex function. During each iteration step, our method avoids solving large linear systems by...

In this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic...

Remote sensing images are often contaminated by varying degrees of stripes, which severely affects the visual quality and subsequent application of the data. Unlike with conventional methods, we achieve the destriping by separating the stripe component based on a full analysis of the various stripe properties. Under an optimization framework, an l0...

In this work we propose and analyze a novel approach for recovering group sparse signals, which arise naturally in a number of practical applications. It is based on regularized least squares with an $\ell^0(\ell^2)$ penalty. One distinct feature of the new approach is that it has the built-in decorrelation mechanism within each group, and thus can...

In this paper, a stabilized finite element method for optimal control problems governed by a convection dominated diffusion equation is investigated. The state and the adjoint variables are approximated by piecewise linear continuous functions with bubble functions. The control variable either is approximated by piecewise linear functions (called t...

In this paper, an extremal eigenvalue problem corresponding to an inhomogeneous membrane which is composed of two different materials with different densities is investigated. The convergence of the finite element discretization and the error order for the smallest eigenvalue are obtained. A monotonic decreasing algorithm is presented to solve the...

The inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov re...

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with general convex penalty term. When the data is given exactly, we give a convergence analysis of our ADMM algorithm without assuming the existence of Lagrange multiplier. In case the da...

A two-grid variational multiscale (VMS) method based on two local Gauss integrations for the convection dominated nonlinear convection diffusion equation is investigated. This method combines the two-grid strategy with the variational multiscale method which chooses polynomial bubble functions as subgrid scale. Two local Gauss integrations are appl...

In this paper we consider a nonconvex model of recovering low-rank matrices from the noisy measurement. The problem is formulated as a nonconvex regularized least square optimization problem, in which the rank function is replaced by a matrix minimax concave penalty function. An alternating direction method with a continuation (ADMc) technique (on...

We consider a boundary value problem involving a Riemann-Liouville fractional derivative of order $\alpha\in (3/2,2)$ on the unit interval $(0,1)$. The standard Galerkin finite element approximation converges slowly due to the presence of singularity term $x^{\alpha-1}$ in the solution representation. In this work, we develop a simple technique, by...

In this article, the P 2−P 2-stabilized finite element method based on two local Gaussian quadratures is applied to discretize the Stokes eigenvalue problem, and the corresponding convergence analysis is given. Furthermore, a two-level scheme, which solves the Stokes eigenvalue problem on a coarser grid and a Stokes problem on the fine grid, is emp...

The success of compressed sensing relies essentially on the ability to efficiently find an approximately sparse solution to an under-determined linear system. In this paper, we developed an efficient algorithm for the sparsity promoting -regularized least squares problem by coupling the primal dual active set strategy with a continuation technique...

The success of compressed sensing relies essentially on the ability to efficiently find an approximately sparse solution to an under-determined linear system. In this paper, we developed an efficient algorithm for the sparsity promoting l1-regularized least squares problem by coupling the primal dual active set strategy with a continuation techniqu...

We develop a primal dual active set with continuation algorithm for solving the ℓ0ℓ0-regularized least-squares problem that frequently arises in compressed sensing. The algorithm couples the primal dual active set method with a continuation strategy on the regularization parameter. At each inner iteration, it first identifies the active set from bo...

We investigate an inverse problem of identifying a Robin coefficient with a sparse structure in the Laplace equation from noisy boundary measurements. The sparse structure of the Robin coefficient γ is understood as a small perturbation of a reference profile γ 0 in the sense that their difference γ−γ 0 has a small support. This problem is formulat...

We present a finite element analysis of electrical impedance tomography for reconstructing the conductivity distribution from electrode voltage measurements by means of Tikhonov regularization. Two popular choices of the penalty term, i.e., the H¹(Ω)-norm smoothness penalty and total variation seminorm penalty, are considered. A piecewise linear fi...