Linear Algebra Research Articles

Fast polynomial multiplication using matrix multiplication accelerators with applications to NTRU on Apple M1/M3 SoCs

Efficient polynomial multiplication routines are critical to the performance of lattice-based post-quantum cryptography (PQC). As PQC standards only recently started to emerge, CPUs still lack specialized instructions to accelerate such routines. Meanwhile, deep learning has grown immeasurably in importance. Its workloads call for teraflops-level of processing power for linear algebra operations, mainly matrix multiplication. Computer architects have responded by introducing ISA extensions, coprocessors and special-purpose cores to accelerate such operations. In particular, Apple ships an undocumented matrix-multiplication coprocessor, AMX, in hundreds of millions of mobile phones, tablets and personal computers. Our work repurposes AMX to implement polynomial multiplication and applies it to the NTRU cryptosystem, setting new speed records on the Apple M1 and M3 systems-on-chip (SoCs): polynomial multiplication, key generation, encapsulation and decapsulation are sped up by 1.54 – 3.07 × , 1.08 – 1.33 × , 1.11 – 1.50 × and 1.20 – 1.98 × , respectively, over the previous state-of-the-art.

An encoder-decoder ConvLSTM surrogate model for simulating geological CO2 sequestration with dynamic well controls

In Geological Carbon Sequestration (GCS), effectively managing the project requires predicting state variables such as pressure and saturation. However, numerical simulation of multiphase flow in subsurface porous media involves solving large linear algebra systems, resulting in a substantial computational burden. This can make it impractical for real-time history matching or optimization. Meanwhile, deep learning-based surrogate models are emerging as fast and accurate approximators. This study delves into the application of the Encoder-Decoder Convolutional Long Short-Term Memory (ED-ConvLSTM) neural network for predicting the complex evolution of state variables under dynamic CO2 injection schemes. Inception blocks enhanced with light-weighted attention modules are introduced in the encoder to extract high-dimensional input features. ConvLSTM is employed to propagate spatial temporal information in the low-dimensional latent space. Further, progressive upsampling blocks are used to reconstruct the latent features for the desired output. Instead of taking discrete time steps as an input feature, the proposed network captures the dynamic dependencies with the inherent ConvLSTM cell. The network has access to data at only portion of the initial time steps during training stage, while it is used to predict the state variables at unseen time steps during testing stage. Results show that the network can produce excellent predictions for both pressure and saturation, even at unseen future time steps. The remarkable generalizability to different geological permeability fields is also evaluated. ED-ConvLSTM outperforms the standard U-Net by far, especially when predicting beyond the training time period. These numerical experiments demonstrate the advantages of ED-ConvLSTM in terms of prediction accuracy, extrapolability and generalizability. This study highlights the importance of incorporating recurrent connections into the deep neural networks for simulating time-dependent multiphase flow problems. The proposed methodology has great potential in GCS surrogate modeling and offers a possible approach for real-time optimization of CO2 injection.

On the computation of Gröbner bases for matrix-weighted homogeneous systems

In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques.As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.

The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems

Qudits with local dimension d>2 can have unique structure and uses that qubits (d=2) cannot. Qudit Pauli operators provide a very useful basis of the space of qudit states and operators. We study the structure of the qudit Pauli group for any, including composite, d in several ways. To cover composite values of d, we work with modules over commutative rings, which generalize the notion of vector spaces over fields. For any specified set of commutation relations, we construct a set of qudit Paulis satisfying those relations. We also study the maximum size of sets of Paulis that mutually non-commute and sets that non-commute in pairs. Finally, we give methods to find near minimal generating sets of Pauli subgroups, calculate the sizes of Pauli subgroups, and find bases of logical operators for qudit stabilizer codes. Useful tools in this study are normal forms from linear algebra over commutative rings, including the Smith normal form, alternating Smith normal form, and Howell normal form of matrices. Possible applications of this work include the construction and analysis of qudit stabilizer codes, entanglement assisted codes, parafermion codes, and fermionic Hamiltonian simulation.

Formalizing screw theory with 3D geometric algebra

This paper is intended for students and researchers looking for more insight into Screw Theory. It shows how algebraic considerations lead to both physical and geometrical understanding of screws, and how they can connect affine geometry (what the world is) to linear algebra (what we can easily compute). Various formulations of the theory are first reviewed, as each of them highlights a particular aspect of screws. Their respective qualities and defects are also discussed. Subsequently, the powerful framework of Geometric Algebra (GA) is introduced to elucidate the nature of various physical objects commonly associated with screws, and eventually a new formalism based on GA is proposed, in which traditional screws clearly appear as a special case of more general affine objects. This approach generalizes the concept of a screw in a coordinate-free and origin-independent form. A simultaneous proof of Euler’s First and Second Laws is provided to illustrate the use of this formalism.

New Accurate Flexural Analysis for Different Types of Plates in a Rectangular Sewage Tank by Utilizing a Unified Analytic Solution Procedure

In the present paper, a modified Fourier series approach is developed for new precise flexural analysis of three different types of concrete plates in a rectangular sewage tank. The bending problems of the bottom plate, side-plate, and the fluid-guiding plate are not easily solved via using the traditional analytic approaches. Based on the Fourier series theory, the present approach provides a unified semi-inverse solving procedure for the above plates by means of choosing three different kinds of Fourier series as the trial functions. Although all the trial functions are quite similar to the classical Navier-form solution, new, precise analytic flexural solutions for plates without Navier-type edge conditions (all edges simply-supported) are achieved, which is mainly attributed to employing the Stoke’s transform technique. For each case, the plate-bending problems are finally altered to deal with linear algebra equations. Furthermore, owing to the orthogonality and completeness of the Fourier series, the obtained solutions perfectly satisfy both the edge conditions and the governing partial differential equation of plates, which paves an easily implemented and rational way for engineers and researchers to provide new, exact designs of plate structures. The main contribution of this study lies in the provision of a unified solution procedure for addressing complex plate-bending problems across diverse boundary conditions. By employing a range of Fourier series types, this approach offers a comprehensive solution framework that accommodates the complexities inherent in plate analysis. The correctness of the present analytic solutions is verified against precise finite element method (FEM) results and ones available in the literature. Finally, the influences of foundation, edge conditions, and aspect ratio on flexural behaviors of plates are discussed in detail.

Dirac’s Approach to Quantum Mechanics in Physics Teacher Education: From Linear to Circular Polarisation

The teaching/learning of quantum mechanics via two-state systems is continuously spreading in secondary schools due to its promising results in physics education research. One possibility is the use of polarisation states of photons. This paper reports on a polarisation-based introduction to quantum mechanics in physics teacher education. A widely used school material prepares teacher trainees for their future work and also improves their conceptual knowledge. This part includes statistical calculations using only secondary school mathematics and a new formulation of the uncertainty relation, using only real numbers. The second step is to prepare the formalism of quantum mechanics using real two-dimensional vectors and matrices. Considering that students may not learn complex linear algebra, we offer a new way to introduce the complete formalism of two-state systems via circular polarisation providing a step-by-step exploration of complex quantum states. This points out the advantage of using complex linear algebra via a physical example, providing the opportunity to reach the elements of advanced quantum physics and quantum computing while deepening the physics background of the secondary school material.

PyDTS: A Python Toolkit for Deep Learning Time Series Modelling.

In this article, the topic of time series modelling is discussed. It highlights the criticality of analysing and forecasting time series data across various sectors, identifying five primary application areas: denoising, forecasting, nonlinear transient modelling, anomaly detection, and degradation modelling. It further outlines the mathematical frameworks employed in a time series modelling task, categorizing them into statistical, linear algebra, and machine- or deep-learning-based approaches, with each category serving distinct dimensions and complexities of time series problems. Additionally, the article reviews the extensive literature on time series modelling, covering statistical processes, state space representations, and machine and deep learning applications in various fields. The unique contribution of this work lies in its presentation of a Python-based toolkit for time series modelling (PyDTS) that integrates popular methodologies and offers practical examples and benchmarking across diverse datasets.

Promoting Linear Algebraic Reasoning among Students: Affordances and Challenges

As students transition from the mathematics they learn in school years, including their first-year calculus courses, to the first course in linear algebra, they experience discontinuities in their perspective of what mathematics is. Their propensity to continue applying the same habits of learning in the face of this change leads to failure and frustration. The failure manifests itself in the quality of understanding basic concepts as well as in the lack of linear algebraic reasoning. Instructional treatments applied in my teaching experiments to foster students’ ability to reason linear algebraically resulted in mixed success – some of the treatments were successful, others less so. The latter are accounted for by the structural complexity of the subject matter and students’ background knowledge. The pedagogical approaches offered in this paper are oriented within a particular theoretical framework for the learning and teaching of mathematics, called DNR. Reflections and broader implications are addressed through the lenses of this framework.

Genuinely nonabelian partial difference sets

AbstractStrongly regular graphs (SRGs) provide a fertile area of exploration in algebraic combinatorics, integrating techniques in graph theory, linear algebra, group theory, finite fields, finite geometry, and number theory. Of particular interest are those SRGs with a large automorphism group. If an automorphism group acts regularly (sharply transitively) on the vertices of the graph, then we may identify the graph with a subset of the group, a partial difference set (PDS), which allows us to apply techniques from group theory to examine the graph. Much of the work over the past four decades has concentrated on abelian PDSs using the powerful techniques of character theory. However, little work has been done on nonabelian PDSs. In this paper we point out the existence of genuinely nonabelian PDSs, that is, PDSs for parameter sets where a nonabelian group is the only possible regular automorphism group. We include methods for demonstrating that abelian PDSs are not possible for a particular set of parameters or for a particular SRG. Four infinite families of genuinely nonabelian PDSs are described, two of which—one arising from triangular graphs and one arising from Krein covers of complete graphs constructed by Godsil—are new. We also include a new nonabelian PDS found by computer search and present some possible future directions of research.

The Art of Linear Algebra

All teachers know the power of “visual understanding” and the literature on this topic is overwhelming. In this article, we focus on the visual image of linear algebra and elaborate on two related references [3, 4] that are insightful but perhaps not well known. We aim to help the worlds of math education and math research to join forces in order to achieve what we all aim for: to explain mathematics in a way that truly helps to make it attractive and fascinating and worthy of our best efforts over a lifetime .

Fast and exact fixed-radius neighbor search based on sorting.

Fixed-radius near neighbor search is a fundamental data operation that retrieves all data points within a user-specified distance to a query point. There are efficient algorithms that can provide fast approximate query responses, but they often have a very compute-intensive indexing phase and require careful parameter tuning. Therefore, exact brute force and tree-based search methods are still widely used. Here we propose a new fixed-radius near neighbor search method, called SNN, that significantly improves over brute force and tree-based methods in terms of index and query time, provably returns exact results, and requires no parameter tuning. SNN exploits a sorting of the data points by their first principal component to prune the query search space. Further speedup is gained from an efficient implementation using high-level basic linear algebra subprograms (BLAS). We provide theoretical analysis of our method and demonstrate its practical performance when used stand-alone and when applied within the DBSCAN clustering algorithm.

Learning Eigenvalues and Eigenvectors with Online YouTube Resources: A Journey in the Embodied, Proceptual-symbolic, and Formal Worlds of Mathematics

Linear algebra is one of the service mathematics courses that students in many programs take as part of their undergraduate study. Previous studies indicate that linear algebra courses are often challenging for undergraduate students, leading some to seek help from other resources, including online resources such as YouTube. Through a multiple case study, I focused on identifying the opportunities that YouTube resources could offer students to learn about eigenvalues and eigenvectors, which are among the key concepts in introductory linear algebra. I utilized a combination of APOS (Action-Process-Object-Schema) and Tall’s three worlds of mathematics to analyze two highly viewed YouTube videos on this topic. The two analyzed files had different emphases (one more on the embodied world, and the other on the proceptual-symbolic world), catering to students with different levels of understanding of linear algebra. The findings suggest that, overall, these resources could provide students with opportunities to learn about eigen theory in the embodied and proceptual-symbolic worlds, with more focus on action. Lecturers and teacher assistants could benefit from familiarizing themselves with these available resources and considering the possible integration of these resources into their teaching and/or support they provide for students to meet their academic needs and preferences.

Numerical solution of the boundary value problems for the biharmonic equations via quasiseparable representations

AbstractThe paper incorporates new methods of numerical linear algebra for the approximation of the biharmonic equation with potential, namely, numerical solution of the Dirichlet problem for $$ \left( \frac{d}{dx}\right) ^4u(x)+c(x)u(x)=\phi (x),\quad 0<x<1. $$ d dx 4 u ( x ) + c ( x ) u ( x ) = ϕ ( x ) , 0 < x < 1 . High-order discrete finite difference operators are presented, constructed on the basis of discrete Hermitian derivatives, and the associated Discrete Biharmonic Operator (DBO). It is shown that the matrices associated with the discrete operator belong to a class of quasiseparable matrices of low rank matrices. The application of quasiseparable representation of rank structured matrices yields fast and stable algorithm for variable potentials c(x). Numerical examples corroborate the claim of high order accuracy of the algorithm, with optimal complexity O(N).

Hybrid programming-model strategies for GPU offloading of electronic structure calculation kernels.

To address the challenge of performance portability and facilitate the implementation of electronic structure solvers, we developed the basic matrix library (BML) and Parallel, Rapid O(N), and Graph-based Recursive Electronic Structure Solver (PROGRESS) library. The BML implements linear algebra operations necessary for electronic structure kernels using a unified user interface for various matrix formats (dense and sparse) and architectures (CPUs and GPUs). Focusing on density functional theory and tight-binding models, PROGRESS implements several solvers for computing the single-particle density matrix and relies on BML. In this paper, we describe the general strategies used for these implementations on various computer architectures, using OpenMP target functionalities on GPUs, in conjunction with third-party libraries to handle performance critical numerical kernels. We demonstrate the portability of this approach and its performance in benchmark problems.

Designing and analysing a teaching proposal about linear algebra through the dialogue of two theories

In this paper we propose a new strategy of dialogue between theories, considering a semiotic cognitive analysis and a praxeological analysis. We examine the implementation of a teaching and learning sequence for introducing eigentheory in a linear algebra course taught during the first year of an engineering degree programme. Initially, the semiotic cognitive analysis based on the Action, Production and Communication space (APC) approach is used in the preliminary, a priori and in vivo phases of the didactic engineering process to propose a teaching strategy that incorporates a variety of semiotic resources. In a second moment, the praxeological analysis intervenes to reformulate the preliminary analysis and to progress on the consideration of a reference epistemological model about eigentheory in a first course of mathematics for engineers. This model is then used in the a priori and a posteriori analysis to integrate the semiotic and epistemological dimensions in the design and implementation of the teaching proposal. The results show the productivity of this new networking strategy and the complementarity of connecting the semiotic with the epistemological analysis.

On the diagonal product of special unitary matrices

We study the image of SU(n) under the diagonal product map. Using only elementary tools of linear algebra, analysis and general topology, we find the analytical formula for the boundary of this image and find all special unitary matrices whose diagonal product lies on that boundary.

Canonical‐Cell Tilings and their Atomic Decorations

AbstractThe canonical cell tiling is a geometrical framework that uses four kinds of basic polyhedra, called the canonical cells, to model the packing of atoms and clusters in icosahedral quasicrystals and related periodic approximants. Over the past three decades, it has become increasingly clear that this framework is the most sensible approach to describe related structures, albeit technically much less tractable than the Ammann‐Kramer‐Neri tiling, which is the simplest icosahedral tiling geometry based on the two Ammann rhombohedra. Geometrical arrangements of cells pose a number of combinatorial problems that cannot be handled using simple linear algebra, making it infeasible to determine structures using the standard six‐dimensional scheme. This up‐to‐date review begins with the motivation, definition, and mathematical facts about the canonical cell tiling. Then the reader is taken through the zoo of concrete structures, from smaller periodic approximants to larger ones, along with an overview of the techniques and heuristics used to study them. The recent discovery of a quasiperiodic canonical cell tiling is also briefly illustrated. The latter half of this review surveys the atomistic modeling of real atomic structures in all the three existing structural families based on the decoration concept of the canonical cell tiling.

Applying Laplace Transformation on Epidemiological Models as Caputo Derivatives

This paper delves into the application of fractional calculus, with a focus on Caputo derivatives, in epidemiological models using ordinary differential equations. It highlights the critical role Caputo derivatives play in modeling intricate systems with memory effects and assesses various epidemiological models, including SIR variants, demonstrating how Caputo derivatives capture fractional-order dynamics and memory phenomena found in real epidemics. The study showcases the utility of Laplace transformations for analyzing systems described by ordinary differential equations with Caputo derivatives. This approach facilitates both analytical and numerical methods for system analysis and parameter estimation. Additionally, the paper introduces a tabular representation for epidemiological models, enabling a visual and analytical exploration of variable relationships and dynamics. This matrix-based framework permits the application of linear algebra techniques to assess stability and equilibrium points, yielding valuable insights into long-term behavior and control strategies. In summary, this research underscores the significance of Caputo derivatives, Laplace transformations, and matrix representation in epidemiological modeling. We assume that by using this type of methodology we can get analytic solutions by hand when considering a function as constant in certain cases and it will not be necessary to search for numerical methods.

POAS: a framework for exploiting accelerator level parallelism in heterogeneous environments

In the era of heterogeneous computing, a new paradigm called accelerator level parallelism (ALP) has emerged. In ALP, accelerators are used concurrently to provide unprecedented levels of performance and energy efficiency. To reach that there are many problems to be solved, one of the most challenging being co-execution. In this paper, we present a new scheduling framework called POAS, a general method for providing co-execution to applications. Our proposal consists of four steps: predict, optimize, adapt and schedule. With POAS, an unseen application can be executed concurrently in ALP with little effort. We evaluate POAS on a heterogeneous environment consisting of CPUs, GPUs (CUDA cores), and XPUs (Tensor cores) on two different fields, namely linear algebra (matrix multiplication benchmark) and deep learning (convolution benchmark). Our experiments prove that POAS provides excellent performance and completes the tasks within a time very close to the optimal time for the hardware and applications used, with a negligible execution time overhead. Moreover, the POAS predictor performed exceptionally well, achieving very low RMSE values for both use cases. Therefore, POAS can be a valuable tool for fully exploiting ALP and improving overall performance over offloading in heterogeneous settings.