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Stefano Aleotti (University of Insubria, Italy)


Title: Preconditioning Strategies for A Nested Primal-dual Method with Application to Image Deblurring


Abstract:

Proximal-gradient methods are widely used in imaging and can be accelerated by adopting variable metrics and/or extrapolation steps. However, one crucial issue is the inexact computation of the proximal operator, often implemented through a nested primal-dual solver. This represents the main computational bottleneck, especially when higher accuracy is required. In this talk, we present preconditioning strategies for a nested primal-dual method aimed at efficiently solving regularized convex optimization problems. For our preconditioner inspired to the Iterated Tikhonov method,we prove the convergence of the iterates sequence towards a solution of the problem. Numerical results confirm that our preconditioned nested primal-dual method effectively accelerates iterations towards the solution of the problem, with reduced computational cost per iteration, especially in image deblurring problems with total variation regularization.




Chenglong Bao (Tsinghua University, China)


Title: Predicting High-resolution Maps of Rare Conformations from Self-supervised Trajectories in Cryo-EM

 

Abstract: 

Cryo-EM has revolutionized structural biology, enabling the efficient determination of structures at near-atomic resolutions. However, a common challenge arises due to the severe imbalance among various conformations of vitrified particles, leading to low-resolution reconstructions in rare conformations because of a lack of particle images in these quasi-stable states. In this talk, we introduce CryoTRANS, a method that predicts high-resolution maps of rare conformations by constructing a self-supervised pseudo-trajectory between density maps of varying resolutions. This trajectory is represented by a neural ODE, with the loss based on the Wasserstein distance. By leveraging a single high-resolution density map, CryoTRANS significantly improves the reconstruction of rare conformations and has been validated on four experimental datasets. Moreover, CryoTRANS can effectively predict high-resolution structures in maps from cryo-ET by using a high-resolution map from cryo-EM.




Giovanni Barbarino (University of Mons, Belgium)


Title: Dual Simplex Volume Maximization for Simplex-Structured Matrix Factorization



Abstract:




Davide Bianchi (Sun Yat-Sen University, China)


Title: A Data-dependent Regularization Method Based on the Graph Laplacian

 

Abstract: 

Differential operators are popular choices for regularizing ill-posed problems in imaging. We investigate a variational method which embeds a graph Laplacian operator in the regularization term. The novelty of this method lies in constructing the graph Laplacian based on a preliminary approximation of the solution, which is obtained using any existing reconstruction method from the literature. As a result, the regularization term is both dependent on and adaptive to the observed data and noise. 

We demonstrate that this method is both regularizing and stable, capable of producing high-quality reconstructions of the ground truth solution. Notably, it yields outstanding results when paired with a deep neural network (DNN), combining both the regularization and stability properties of a variational method and the accuracy of a well-trained DNN.




Matthias Bolten (Bergische Universität Wuppertal, Germany)


Title:Analysis of Multigrid Methods for Multilevel Block Toeplitz Matrices

 

Abstract:

Multilevel block Toeplitz matrices arise in many applications, for instance when higher-order discretizations are used for scalar PDEs or systems of PDEs are to be solved. For large-scale problems multigrid methods are often the method of choice, as they provide an efficient way of solving the associated linear systems.

The analysis of multigrid methods for structured matrices, i.e., Toeplitz matrices or circulant matrices, and traditional multilevel theory is an established technqiue. For scalar problems, including those arising from the discretization of PDEs, it has been studied intensively. Recently, we started transfering these results to the systems case that results in block-Toeplitz matrices or block-circulant matrices [1]. Besides studying higher-order discretizations of scalar PDEs, certain systems of PDEs also fit in this framework. Systems of PDEs that lead to saddle point structure, like the Stokes equations, need another approach. Based on a result by Notay [3] we were able to establish convergence for these matrices, as well [2].

In the talk the analysis technique, the derived sufficient conditions for optimal convergence and numerical results will be presented.

 

[1] M. Bolten, M. Donatelli, P. Ferrari, and I. Furci. A symbol based analysis for multigrid methods for block-circulant and block-Toeplitz systems. SIAM J. Matrix Anal. Appl., 43(1):405–438, 2022.

[2] M. Bolten, M. Donatelli, I. Ferrari, and I. Furci. Symbol based convergence analysis in multigrid methods for saddle point problems. Linear Algebra Appl., 671:67–108, 2023.

[3] Y. Notay. A new algebraic multigrid approach for Stokes problems. Numer. Math., 132(1):51–84, 2016.




Raymond Chan (Lingnan University, China)


Title: Selecting Regularization Parameters for Minimization Problems in Imaging.

 

Abstact: 

Many minimization problems in imaging consist of a data-fitting term and a regularization term. The weights of the two terms are balanced by a regularization parameter. In this talk we discuss how to choose this parameter when it has an L2 data-fitting term and the regularization term is either an L1-or a nuclear-norm term.




Huibin Chang (Tianjin Normal University, China)


Title: Fast Minimization for Curvature Based Regularization Models Based on Bilinear Decomposition


Abstact: 

The curvature based regularization models can generate artifact-free results compared with the traditional total variation regularization model in image processing. However, strong nonlinearity and singularity due to the curvature term pose a great challenge for one to design fast and stable algorithms for the EE model. We propose a new, fast, hybrid alternating minimization (HALM) algorithm based on a bilinear decomposition of the gradient of the underlying image and prove the global convergence of the minimizing sequence generated by the algorithm under mild conditions. A host of numerical experiments are conducted to show that the new algorithm produces good results with much-improved efficiency compared to other state-of-the-art algorithms. As one of the benchmarks, we show that the average running time of the HALM algorithm is at most one-quarter of that of the fast operator-splitting-based Deng-Glowinski-Tai algorithm.  The fast solver for the mean curvature model and  learned method based on the proposed HALM algorithm and deep unrolling are further discussed.




Eric King-Wah Chu (Monash University, Australia)


Title: Numerical Solution to Singular Sylvester Equations

 

Abstract: 




Marco Donatelli (University of Insubria, Italy)


Title: Preconditioning Strategies for Generalized Krylov Subspace Methods for lp-lq Minimization

 

Abstract: 

This paper presents an extension of the Maximization-Minimization Generalized Krylov Subspace (MM-GKS) method for solving  lp-lq  minimization problems, as proposed in [1], by introducing a right preconditioner aimed at accelerating convergence without compromising the quality of the computed solution. The original MM-GKS approach relies on iterative reweighting and projection onto subspaces of increasing dimensions, enabling efficient resolution of minimization problems. Our enhanced method leverages a carefully designed regularizing preconditioner, inspired by Iterated Tikhonov regularization, to address the inherent ill-conditioning of the problem. We demonstrate that our preconditioned MM-GKS method preserves the stability and accuracy of the original MM-GKS method, as validated by numerical results in image deblurring, showing significant reductions in CPU time.

This work is a collaboration with A. Buccini (University of Cagliari), M. Ratto (University of Insubria), and L. Reichel (Kent State University).

 

Reference: 

[1] A. Lanza, S. Morigi, L. Reichel, F. Sgallari, A generalized Krylov subspace method for  lp-lq  minimization. SIAM Journal on Scientific Computing,  (2015), 37(5), S30-S50.




Yuping Duan (Beijing Normal University, China)


Title: Curvature Regularization and Its Applications in Non-Line-of-Sight Imaging

 

Abstract:

Non-line-of-sight (NLOS) imaging aims to reconstruct the three-dimensional hidden scenes by using time-of-flight photon information after multiple diffuse reflections. The under-sampled scanning data can facilitate fast imaging. However, the resulting reconstruction problem becomes a serious ill-posed inverse problem, the solution of which is highly likely to be degraded due to noises and distortions. We proposed novelcurvature regularization methods for fast and accurate NLOS imaging reconstruction.




Michiel Hochstenbach (Eindhoven University of Technology, Netherlands)


Title: Recent Progress in Steplengths in Gradient Methods for Unconstrained Optimization

 

Abstract:

We review recent progress in stepsizes in gradient methods for unconstrained numerical optimization, for both quadratic and general nonlinear problems, including several very valuable contributions from Chinese experts such as Prof. Dai Yu-Hong and colleagues. 

We will review Barzilai—Borwein steps and secant equations, and show developments in harmonic stepsizes and homogeneous secant equations. The key idea is that the steplength is selected as the inverse Ritz value of an average Hessian. 

For limited memory methods, we will address several challenges, including how to avoid negative or non-real stepsize candidates. 

We will also present some open problems. Part of the talk reflects joint work with Giulia Ferrandi.




Yu Sing Sean Hon (Hong Kong Baptist University, China)


Title: Optimal Preconditioners for Nonsymmetric Multilevel Toeplitz Systems with Application to Solving Non-local Evolutionary PDEs

 

Abstract: 

In this talk, we present a new preconditioning method for nonsymmetric multilevel Toeplitz systems, including those from evolutionary PDEs. We propose a symmetric positive definite multilevel Tau preconditioner that is efficient and optimal, ensuring mesh-independent convergence with the preconditioned generalized minimal residual method. Numerical examples highlight our method's effectiveness, particularly for non-local, time-dependent PDEs solved in parallel. This is joint work with Yuan-Yuan Huang, Lot-Kei Chou, and Siu-Long Lei.




Bangti Jin (The Chinese University of Hong Kong, China)


Title: Conductivity Imaging Using Deep Neural Networks

 

Abstract: 

Conductivity imaging from various observational data represents one fundamental task in medical imaging. In this talk, we discuss numerical methods for identifying the conductivity parameters in elliptic PDEs. Commonly, a regularized formulation consists of a data fidelity and a regularizer is employed, and then it is discretized using finite difference method, finite element methods or deep neural networks in practical computation. One key issue is to establish a priori error estimates for the recovered conductivity distribution. In this talk, we discuss our recent findings on using deep neural networks for this class of problems, by effectively utilizing relevant stability results, and presents computational results for both isotropic and anistropic conductivities distributions.




Malena Sabaté Landman (Emory University, USA)


Title: Inner Product Free Krylov Methods for Large-scale Inverse Problems

 

Abstract:

Inverse problems focus on reconstructing hidden objects from indirect, often noisy measurements, and are prevalent in numerous scientific and engineering disciplines. These reconstructions are typically highly sensitive to perturbations such as measurement errors, making regularization essential.

In this presentation, I will discuss Krylov subspace methods that avoid inner-product computations and are specifically designed to efficiently address large-scale linear inverse problems.  In particular, I will highlight their regularization capabilities and present computational results that demonstrate the effectiveness of these methods in different scenarios.




Shuai Lu (Fudan University, China)


Title: Function and Derivative Approximation by Shallow Neural Networks

 

Abstract:

We investigate a Tikhonov regularization scheme specifically tailored for shallow neural networks within the context of solving a classic inverse problem: approximating an unknown function and its derivatives within a unit cubic domain based on noisy measurements. The proposed Tikhonov regularization scheme incorporates a penalty term that takes three distinct yet intricately related network (semi)norms: the extended Barron norm, the variation norm, and the Radon-BV seminorm. These choices of the penalty term are contingent upon the specific architecture of the neural network being utilized. We establish the connection between various network norms and particularly trace the dependence of the dimensionality index, aiming to deepen our understanding of how these norms interplay with each other. We revisit the universality of function approximation through various norms, establish rigorous error-bound analysis for the Tikhonov regularization scheme, and explicitly elucidate the dependency of the dimensionality index, providing a clearer understanding of how the dimensionality affects the approximation performance and how one designs a neural network with diverse approximating tasks. It is a joint work with Yuanyuan Li (Fudan University).




Ronald Lok Ming Lui (The Chinese University of Hong Kong, China)


Title: Shape Prior Segmentation Guided by Harmonic Beltrami Signature

 

Abstract: 

This talk presents a novel image segmentation method that incorporates the Harmonic Beltrami Signature (HBS) as shape prior knowledge. The HBS represents 2D simply-connected shapes that are invariant to translation, rotation, and scaling. By leveraging the HBS, the proposed method enables direct shape similarity measurement using the $L^2$ distance between signatures, while also encoding shapes robustly against perturbations. The method integrates the HBS into a baseline Beltrami coefficient segmentation framework. It utilizes reference shape boundaries and computes corresponding HBS as prior knowledge. This HBS prior guides the segmentation of partially damaged or occluded objects towards the reference shape(s), ensuring their similarity. Experimental results on synthetic and natural images validate these benefits, and comparisons with baseline segmentation models show significant improvements.  This work is supported by HKRGC GRF (Project ID: 14307622).




Nicola Mastronardi (Istituto per le Applicazioni del Calcolo, Italy)


Title: On the Computation of  the Zeros of Multiple and Sobolev Orthogonal Polynomials.


Abstract:





Serena Morigi (University of Bologna, Italy)


Title: Nonlinear Compressive Sensing for Ill-Posed Image Reconstruction


Abstract:

While the majority of research on Compressive Sensing (CS) has focused on linear models, many real-world applications, particularly in physics and biomedical sciences, exhibit inherent nonlinearities that render these models inadequate. This paper introduces a proof-of-concept study exploring the application of CS techniques to the numerical solution of nonlinear, ill-posed image reconstruction problems. Specifically, we propose a sparsity-aware model for solving the inverse problem in Electrical Impedance Tomography through a variational formulation.




James G. Nagy (Emory University, USA)


Title: Exploiting Kronecker Products in Image Reconstruction

 

Abstract:

In image reconstruction and restoration applications it is necessary to solve a large-scale inverse problem; that is, to determine quantities defining an object or a system through indirect measurements. Inverse problems are sensitive to errors (e.g., noise) in the measured data, so additional constraints and mathematical tools (often referred to as regularization) are needed to stabilize the numerical methods. The type and amount of regularization is problem dependent, requiring the algorithms to be able to easily adapt to user and/or problem specifications. In this talk we focus on applications where we can exploit Kronecker product structure in the matrices associated with equations that make up the inverse problem. Exploiting these structures allows for more efficient algorithms.




Lucas W. Onisk (Emory University, USA)


Title: Mixed Precision Iterative Refinement for Linear Inverse Problems

 

Abstract:

We investigate the iterative refinement method applied to the solution of linear discrete inverse problems by considering its application to the Tikhonov problem in mixed precision. Previous works on mixed precision iterative refinement methods for the solution of symmetric positive definite linear systems and least-squares problems have shown regularization to be a key requirement when computing low precision factorizations. For problems that are naturally severely ill-posed, we formulate the iterates of iterative refinement in mixed precision as a filtered solution using the preconditioned Landweber method with a Tikhonov-type preconditioner. Through numerical examples simulating various mixed precision choices, we showcase the filtering properties of the method and the achievement of comparable or superior accuracy compared to results computed in double precision as well as another approximate method.




Margherita Porcelli (Università degli Studi di Firenze, Italy)


Title: Regularized Methods Via Cubic Model Subspace Minimization for Nonconvex Optimization

 

Abstract:

Adaptive cubic regularization methods for solving nonconvex problems need the efficient computation of the trial step, involving the minimization of a cubic model. We present a new approach in which this model is minimized in a low dimensional subspace that, in contrast to classic approaches, is reused for a number of iterations. Whenever the trial step produced by the low-dimensional minimization process is unsatisfactory, we employ a regularized Newton step whose regularization parameter is a by-product of the model minimization over the low-dimensional subspace. We show that the worst-case complexity of classic cubic regularized methods is preserved, despite the possible regularized Newton steps. We focus on the large class of problems for which (sparse) direct linear system solvers are available and provide several experimental results showing the very large gains of our new approach when compared to standard implementations of adaptive cubic regularization methods based on direct linear solvers.

This is a joint work with S. Bellavia (UNIFI), D. Palitta (UNIBO) and  V. Simoncini (UNIBO)



 

Meiyue Shao (Fudan University, China) 


Title: Structure-preserving Algorithms for the Bethe--Salpeter Eigenvalue Problem


Abstract:

In a molecular system the excitation of an electron is obtained by solving the so-called Bethe--Salpeter equation (BSE). Discretization of the Bethe--Salpeter equation leads to a dense non-Hermitian matrix eigenvalue problem with a special 2-by-2 block structure. In principle all excitation energies, i.e., all positive eigenvalues of the BSE Hamiltonian, are of interest. This is challenging in practice because the dimension of the BSE Hamiltonian depends quadratically on the number of electrons in the system. We developed a parallel structure preserving algorithm that computes all eigenpairs of the BSE Hamiltonian efficiently and accurately. In some circumstances, instead of computing each individual eigenpair, we need to compute the optical absorption spectrum, which is a frequency dependent matrix functional of the BSE Hamiltonian. We developed a Lanczos-type algorithm to efficiently compute the absorption spectrum without diagonalizing the BSE Hamiltonian. Parallel implementations of these algorithms are available in the software package BSEPACK.



       

Jungong Xue (Fudan University, China)  


Title: Efficient Computation of Wiener-Hopf Factorization of Markov-modulated Brownian Motion

 

Abstract:

The Wiener-Hopf factorization, which is characterized as special solutions to a pair of nonlinear matrix equations, plays a crucial role in analysis of the Markov-modulated Brownian motion (MMBM). This paper deals with the general case where the diffusion coefficients are nonzero for some but not all states of the governing Markov chain. Based on a novel regrouping of the unknowns in the Wiener-Hopf factorization and a new form for the initialization phase, a doubling algorithm is proposed to solve the pair of nonlinear matrix equations simultaneously. With the parameters of the MMBM as input, this doubling algorithm is implemented in a subtraction-free manner to compute the  Wiener-Hopf factorization to high entrywise relative accuracy. Numerical examples are presented to demonstrate and confirm our claims.



  

Ke Ye (Chinese Academy of Sciences, China)    


Title: The Tensor Method for Fast Matrix Operations


Abstract:

In this talk, we discuss the acceleration of matrix operations by tensor structures. We first introduce the classical method which relates algorithms of structured matrix multiplication to decompositions of a structured tensor. As an example, we present an application of our method to signal processing. Next, we explore the possibility of accelerating the matrix inversion by field extension. We provide both theoretical analysis and numerical examples to exhibit the efficiency of our method. If time permits, we also discuss the stability of tensor ranks under field extensions, which plays, somewhat to our surprise, an essential role in additive combinatorics.




Junfeng Yin (Tongji University, China)  


Title:  Adaptively Bregman-Kaczmarz Methods for Linear Inverse Problem

 

Abstract:

Bregman-Kaczmarz method plays an important role for the solution of linear inverse problem, which usually arises from many practical applications including of compressed sensing, image process and machine learning. By making the use of the residual adaptively, a class of residual-based surrogate hyperplane Bregman-Kaczmarz methods are presented and their convergence performances are established. In particular, the residual-based surrogate hyperplane Kaczmarz method and the residual-based surrogate hyperplane sparse Kaczmarz method are studied for finding the least norm solution and the sparse solution, respectively. When the data is contaminated by the independent noise, an adaptive version of the proposed method is developed where the adaptive relaxation parameter is derived for optimizing the bound on the expectation error. Numerical experiments show that the proposed approaches are efficient and better than the existing solvers.



   

Leihong Zhang (Soochow University, China)


Title: Multivariate Confluent Vandermonde with G-Arnoldi and Applications


Abstract:

The Vandermonde matrices are a class of traditional and important structure matrices arising frequently from approximation theory, and notoriously known as the extremely ill-conditioning. In the least-squares fitting framework, the Vandermonde with Arnoldi (V+A) method presented in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405–415] is an effective approach to compute a polynomial that approximates an underlying univariate function f(x). Extensions of V+A include its multivariate version and the univariate confluent V+A; the latter enables us to use the information of the derivative of f(x) in obtaining the approximation polynomial. In this talk, we shall extend V+A further to the multivariate confluent V+A. Besides the technical generalization of the univariate confluent V+A, we also introduce a general and application-dependent G-orthogonalization in the Arnoldi process. We shall demonstrate with several applications that, by specifying an application-related G-inner product, the desired approximate multivariate polynomial as well as its certain partial derivatives can be computed accurately from a well-conditioned least-squares problem whose coefficient matrix is orthonormal. We demonstrate its flexibility by applying it to solve the multivariate Hermite least-squares problem and PDEs with various boundary conditions in irregular domains.





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