Low-complexity structures are central to modern data analysis they are exploited to tame data dimensionality, to rescue ill-posed problems, and to ease and speed up hard numerical computation. In this line, the past decade features remarkable advances in theory and practice of estimating sparse vectors or low-rank matrices from few linear measurements. Looking ahead, there are numerous fundamental problems in data analysis coming with more complex data formation processes.

For example, the dictionary learning and the blind deconvolution problems have intrinsic bilinear structures, whereas the phase retrieval problem and variants pertain to quadratic measurements. Moreover, many of these applications can be naturally formulated as nonconvex optimization problems, which are ruled to be hard by the worst-case theory. In practice, however, simple numerical methods are surprisingly effective in solving them. Partial explanation of this curious gap has started to appear very recently. Novel results on both theoretical and algorithmic sides of exploiting low-complexity structures will be discussed, with an emphasis on addressing the new challenges.

Within this minisymposium we will consider some actual problems of the generalized inverses, generalized invertibility of operators, representations of the Drazin inverse, least squares problem, and computing generalized inverses using gradient neural networks and using database stored procedures. We will develop the relationship between generalized inverses and the linear least squares problem with applications in signal processing. Due to stagnating processor speeds and increasing core counts, the current paradigm of high performance computing is to achieve shorter computing times by increasing the concurrency of computations.

### Junior-to-Senior Level Courses

Sequential time-stepping is a computational bottleneck when attempting to implement highly concurrent algorithms, thus parallel-in-time methods are desirable. This minisymposium will present recent advances in iterative solvers for parallel-in-time integration. This includes methods like parareal, multigrid reduction, and parallel space-time methods, with application to linear and nonlinear PDEs of parabolic and hyperbolic type. Eigenvalue problem is the essential part and the computationally intensivepart in many applications in a variety of areas, including, electronstructure calculation, dynamic systems, machine learning, etc.

In all theseareas, efficient algorithms for solving large-scale eigenvalue problems aredemanding. Recently many novel scalable eigensolvers were developed to meetthis demand.

## Course Offerings

The choice of an eigensolver highly depends on the properties and structure of the application. Thisminisymposium invites eigensolver developers to discuss the applicabilityand performance of their new solvers. The ultimate goal is to assistcomputational specialists with the proper choice of eigensolvers fortheir applications.

Matrix and tensor optimization problems naturally arise from applications that involve two-dimensional or multi-dimensional array data, such as social network analysis, neuroimaging, Netflix recommendation system, and so on. Hence it is significant to keep the matrix and tensor format.

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This minisymposium includes talks about recently proposed models and algorithms with complexity analysis for large-scale matrix and tensor optimization. Low-rank matrix approximations have demonstrated attractive theoretical bounds, both in memory footprint and arithmetic complexity. In fact, they have even become numerical methods of choice when designing high performance applications, especially when looking at the forthcoming exascale era, where systems with billions of threads will be routine resources at hand.

This minisymposium aims at bringing together experts from the field to assess the software adaptation of low-rank matrix computations into HPC applications. The mini-symposium is focused on Structured Matrix Analysis, with the special target of shedding light on Low-Rank and Toeplitz-related Structures. On sufficiently regular domains, certain combinations of such matrix objects weighted with proper diagonal sampling matrices are sufficient for describing in a great generality approximation of integro-differential operators with variable coefficient, by means of virtually any type of discretization technique finite elements, finite differences, isogeometric analysis, finite volumes etc.

The considered topics and the young age of the speakers are aimed at fostering the contacts between PhD students, postdocs and young researchers, with a balanced choice of talks addressing at improving collaborations between analysis and applied research,showing connections among different methodologies,using the applications as a challenge for the search of more advanced algorithms. Machine learning is experiencing a period of rising impact on many areas of the sciences and engineering such as imaging, advertising, genetics, robotics, and speech recognition.

On the other hand, it has deep roots in various aspects in mathematics, from optimization, approximation theory, to statistics, etc. This mini-symposium aims to bring together researchers in different aspects of machine learning for discussions on the state-of-the-art developments in theory and practice. The mini-symposium has a total of four talks, which are about fast algorithms solving linear inequalities, genetic data analysis, theory and practice of deep learning.

Matrix functions are an important tool in many areas of scientific computing. They arise in the solution of differential equations, as the exponential, sine, or cosine; in graph and network analysis, as measurements of communicability and betweenness; and in lattice quantum chromodynamics, as the sign of the Dirac overlap operator. They also have many applications in statistics, theoretical physics, control theory, and machine learning. Methods for computing matrix functions times a vector encompass a variety of numerical linear algebra tools, such as Gauss quadrature, Krylov subspaces, rational and polynomial approximations, and singular value decompositions.

Given the rapid expansion of the literature on matrix functions in the last few years, this seminar fills an ongoing need to present and discuss state-of-the-art techniques pertaining to matrix functions, their analysis, and applications. These problems arise in various applications such as bio-informatics, data analysis, image processing and materials science, and are also abundant in combinatorial optimization.

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Eigenvalue problems arise in many fields of science and engineering and their mathematical properties and numerical solution methods for standard, linear eigenvalue problems are well understood. Moreover, the nonlinear eigenvalue problem received more and more attention from the numerical linear algebra community during the last decade. So far, the majority of the work has been focused on polynomial eigenvalue problems. In this minisymposium we will address the general nonlinear eigenvalue problem involving nonlinear functions such as exponential, rational, and irrational ones. Recent literature on numerical methods for solving these general nonlinear eigenvalue problems can, roughly speaking, be subdivided into three main classes: Newton-based techniques, Krylov subspace methods applied to linearizations, and contour integration and rational filtering methods.

Within this minisymposium we would like to address all three classes used to solve large-scale nonlinear eigenvalue problems in different application areas. Nonlinear Perron-Frobenius theory addresses problems such as existence, uniqueness and maximality of positive eigenpairs of different types of nonlinear and order-preserving mappings.

In recent years tools from this theory have been successfully exploited to address problems arising from a range of diverse applications and various areas, such as graph and hypergraph analysis, machine learning, signal processing, optimization and spectral problems for nonnegative tensors. This minisymposium sample some recent contributions in this field, covering advances in both the theory and the applications of Perron-Frobenius theory for nonlinear mappings.

Data science is currently one of the hottest research fields in many real applications such as medicine, business, finance, transportation, etc.. Lots of computational problems arise in the process of data modelling and data analysis. Due to the finite dimension property of the data samples, most computational problems can be transformed to linear algebra related problems. To date, numerical linear algebra has played important roles in data science. With the fast development of experimental techniques and growth of internet communications, more and more data are generated nowdays.

The availability of a huge amount of data brings big challenges for traditional computational methods. On one hand, to handle the big data matrices high dimension, big sample size , algorithms having high computational speed and accuracy are in great need.

## Mathematics

This proposes the problem of improving the traditional methods such as SVD methods, conjugate gradient method, matrix preconditioning methods, and so on. On the other hand, with the generation of more data, many new models are proposed. This brings the chances for developing novel algorithms. Taking into account the properties of data to build good models and propose fast and accurate algorithms will accelerate the development of data science greatly.

Numerical linear algebra as the essential technique for numerical algorithm development should be paid more attention. The speakers in this minisymposium will discuss work that arises in data modelling including multiview data learning, data dimension reduction, data approximation, and stochastic data analysis.

The numerical linear algebra methods cover low-dimension projection, matrix splitting, parallel SVD, conjugate gradient method, matrix preconditioning and so on. This minisymposium brings together researchers from different data analysis fields focusing on numerical linear algebra related algorithm development. It will emphasize the importance and strengthen the role of linear algebra in data science, thereby advances the collaborations for researchers from different fields.

Electronic structure theory and first principle calculations are among the most challenging and computationally demanding science and engineering problems. At their core, many of the methods used require the development of efficient and specialized linear algebraic techniques. This minisymposium aims to discuss new developments in the linear algebraic tools, numerical methods, and mathematical analysis used to achieve high levels of accuracy and efficiency in electronic structure theory.

We bring together experts on electronic structure theory representing a broad set of computational approaches used in the field. Riemannian optimization methods are a natural extension of Euclidean optimization methods: the search space is generalized from a Euclidean space to a manifold endowed with a Riemannian structure. This allows for many constrained Euclidean optimization problems to be formulated as unconstrained problems on Riemannian manifolds; the geometric structure can be exploited to provide mathematically elegant and computationally efficient solution methods by using tangent spaces as local linearizations.

Many important structures from linear algebra admit a Riemannian manifold structure, such as matrices with mutually orthogonal columns Stiefel manifold , subspaces of fixed dimension Grassmann manifold , positive definite matrices, or matrices of fixed rank. The first session of this minisymposium will present some applications of the Riemannian optimization framework, such as blind deconvolution, computation of the Karcher mean, and low-rank matrix learning. It will also present novel results on subspace methods in Riemannian optimization.

The second session will be centered on the particular class of low-rank tensor manifolds, which make computations with multiway arrays of large dimension feasible and have attracted particular interest in recent research.

It will present novel results on second-order methods on tensor manifolds, such as trust-region or quasi-Newton methods. It will also present results on dynamical approximation of tensor differential equations. Sparse triangular solve SpTRSV is an important building block in a number of numerical linear algebra routines such as sparse direct solvers and preconditioned sparse iterative solvers.

Compared to dense triangular solve and other sparse basic linear algebra subprograms, SpTRSV is more difficult to parallelize since it is inherently sequential. The set-based methods i. In this proposed minisymposium, we will discuss current challenges and novel algorithms for SpTRSV on shared memory processors with homogeneous architectures such as GPU and Xeon Phi and with heterogeneous architectures such as Sunway and APU , and distributed memory clusters.

## Undergraduate Courses | Mathematical Institute Course Management

The objective of this minisymposium is to explore and discuss how emerging parallel platforms can help next-generation SpTRSV algorithm design. Polynomial and rational matrices have attracted much attention in the last years. Their appearance in numerous modern applications requires revising and improving known as well as developing new theories and algorithms concerning the associated eigenvalue problems, error and perturbation analyses, efficient numerical implementations, etc.