matlab

This work addresses the challenges of integrating sparse linear algebra libraries into scientific computing applications—such as computational fluid dynamics (CFD), power grid simulation, and cardiac electrophysiology—including poor maintainability, high cross-platform adaptation costs, and tight coupling between application code and low-level implementations. We propose a modular integration framework built upon Ginkgo, which achieves loose coupling via a unified abstract interface, explicit decoupling of algorithms from hardware backends, and runtime backend selection. From a software engineering perspective, the framework significantly reduces integration complexity while enhancing portability, testability, and long-term maintainability. Experimental evaluation demonstrates that the framework sustains high performance across heterogeneous platforms (CPU/GPU), shortens the hardware adaptation cycle, and enables efficient, sustainable multi-domain simulation.

Scientific software deployment faces a productivity–performance gap between high-level prototyping environments (e.g., MATLAB) and production-grade C++ code: scripting languages sacrifice performance, while conventional C++ linear algebra libraries (e.g., BLAS/LAPACK) suffer from verbose interfaces and error-prone manual memory management. This paper introduces a novel C++ linear algebra library built on template metaprogramming and expression templates. It pioneers a compile-time expression optimization framework that enables operator fusion, lazy evaluation, and elimination of temporary objects—delivering MATLAB-like syntactic simplicity without explicit memory management. The design achieves both high code readability and performance competitive with hand-optimized C++. Benchmark results demonstrate speedups of several-fold over conventional C++ implementations. The library has been successfully deployed in production-scale machine learning and signal processing systems.

This work addresses the high memory and computational complexity typically associated with solving three-dimensional partial differential equations on Cartesian grids. By exploiting tensor-product structure, the proposed method decomposes the 3D operator into one-dimensional banded kernels aligned with coordinate axes, thereby avoiding explicit assembly of the global matrix and enabling a matrix-free solution strategy. Within a unified framework that integrates diverse numerical approaches—including Kronecker product algebra, compact finite differences, isogeometric analysis, and direct diagonalization—the study systematically identifies three key techniques: multi-right-hand-side reshaping, sum factorization, and pencil-style MPI decomposition. These innovations collectively enhance hardware affinity and parallel scalability, reducing algorithmic complexity to O(N) and storage requirements to O(Nₓ + Nᵧ + N_z), thus enabling efficient large-scale 3D PDE simulations.

Large-scale, multiphysics, multiscale scientific simulations face significant challenges in achieving both performance portability and software maintainability across heterogeneous hardware (CPUs, GPUs, and accelerators). Method: This project designs and implements an open-source software framework built upon the Kokkos ecosystem. Leveraging C++17+ template metaprogramming and a modular product-domain architecture, it integrates MPI-based distributed parallelism with a unified GPU abstraction layer to enable high-performance computing across diverse architectures. Continuous integration and community-driven development ensure scalability and collaborative evolution. Contribution/Results: The framework has enabled数十 (dozens of) national-level scientific applications, demonstrating efficient scalability on million-core heterogeneous systems. It substantially reduces adaptation effort for emerging hardware, improves code reuse, and enhances long-term software maintainability.

This work addresses the susceptibility of ill-conditioned M-matrices to subtractive cancellation in componentwise high-precision computations by introducing a novel approach based on a triplet representation. By integrating componentwise high-precision arithmetic with the structural properties of M-matrices, the authors present—for the first time—high-precision variants of the GTH algorithm, including non-blocking, recursive, and blocked formulations, all encapsulated within an object-oriented MATLAB interface. The resulting toolbox supports a range of operations such as linear system solving, LU factorization, Schur complement computation, matrix square roots, singular value decomposition, and solutions to nonsymmetric algebraic Riccati equations. Even under severe ill-conditioning, the framework preserves componentwise numerical accuracy, thereby substantially enhancing computational reliability.

This work addresses the challenges of complex boundary condition handling, code redundancy, and poor distributed efficiency in partial differential equation solvers on block-structured grids. The authors propose a unified modeling approach that expresses user-defined boundary conditions as affine sparse linear operators and, for the first time, systematically reformulates them into sparse matrix-vector multiplication (SpMV) form. Leveraging a domain-specific language (DSL) and compiler techniques—combined with multi-stage programming and polyhedral analysis—the framework automatically generates highly optimized matrix-free or sparse matrix kernels while optimizing communication scheduling and reuse. The method achieves substantial performance gains, demonstrating 72%–88% strong scaling efficiency on 1,344 CPU cores, up to 7.6× acceleration in boundary computation kernels, and a reduction of over 70% in code size.

This work addresses the gap between algorithmic prototypes and efficient implementations in scientific research by proposing a lightweight approach to translate statistical and machine learning algorithms—such as kernel ridge regression and stochastic gradient descent matrix factorization—from mathematical formulations into readable, high-performance C++ code. Leveraging the Eigen template library for core linear algebra operations—including kernel matrix construction, regularized solvers, and vectorized updates—the implementation seamlessly integrates into the Python ecosystem via pybind11, enabling efficient interoperability with NumPy arrays. The project provides concise, reproducible code examples that encapsulate common computational patterns in research, significantly lowering the barrier for researchers to adopt C++ for high-performance development while balancing performance, readability, and usability.

This work addresses the interoperability challenges arising from inconsistent interfaces among numerical solvers by proposing and implementing MaRDI—a standardized, open interface tailored for nonlinear optimization. Designed with a modular architecture, MaRDI establishes a generic solver adapter layer that enables seamless integration of diverse optimizers and embeds naturally within physics-informed neural network (PINN) training pipelines. Its efficacy is demonstrated through application to the viscous Burgers equation, where it substantially reduces the development overhead and benchmarking costs associated with solver-specific bindings. By abstracting low-level implementation details, MaRDI allows researchers to focus on core algorithmic innovation while significantly enhancing the efficiency and reproducibility of cross-solver experimentation.

This work addresses the lack of a unified, open-source, and modular platform for collaboratively exploring shape and topology optimization methods in both teaching and research. The authors present an object-oriented, MATLAB-based open-source framework that employs abstract base classes to define core interfaces, enabling seamless integration of parametric and level-set-based shape optimization alongside density-based, level-set, and topological sensitivity approaches to topology optimization. By directly mapping mathematical formulations to executable code, the framework allows users to extend objective functionals or constraints simply by deriving new classes without modifying the core implementation. Highly modular and reproducible, the framework bridges the gap between shape and topology optimization, offering a continuous research pathway. Its effectiveness and flexibility are demonstrated through diverse numerical examples in both educational and research contexts.

This work proposes a unified framework for solving systems of multivariate polynomials and rectangular multiparameter eigenvalue problems, implemented in the open-source MATLAB toolbox MacaulayLab. The approach leverages numerical linear algebra and Macaulay matrix constructions without relying on any specific polynomial basis or monomial ordering. It is the first method capable of efficiently handling both problem classes within a single framework while accurately characterizing positive-dimensional solution components at infinity. Numerical experiments demonstrate that the proposed method matches or surpasses the performance of established software packages such as PHCpack, PNLA, and MultiParEig. To support reproducible research, the authors provide an extensive suite of test cases alongside the toolbox.