Sparse Direct Methods Research Articles

Monocular thermal SLAM with neural radiance fields for 3D scene reconstruction

Visual simultaneous localization and mapping (SLAM) faces significant challenges in environments with variable lighting and smoke. Excelling in such visually degraded settings, thermal imaging captures scene radiance effectively. To address the limitations of traditional thermal SLAM in 3D scene reconstruction, we propose ThermalSLAM-NeRF, a novel integration of thermal SLAM with neural radiance fields (NeRF). This method significantly enhances the quality of high dynamic range thermal images by improving their signal-to-noise ratio, contrast, and detail. It also employs online photometric calibration to ensure grayscale consistency between consecutive frames. We utilize a sparse direct method for pose estimation, selecting keyframes based on photometric error and tracking quality. The NeRF map is reconstructed using a multi-view keyframe sequence. Our evaluations on datasets containing over 15,000 thermal images show that ThermalSLAM-NeRF achieves an average improvement of 59.30% in trajectory accuracy over existing state-of-the-art SLAM methods. This approach uniquely tracks all sequences and constructs comprehensive NeRF maps, enabling robust and precise pose estimation without the need for extensive pre-training.

A Robust Semi-Direct 3D SLAM for Mobile Robot Based on Dense Optical Flow in Dynamic Scenes.

Dynamic objects bring about a large number of error accumulations in pose estimation of mobile robots in dynamic scenes, and result in the failure to build a map that is consistent with the surrounding environment. Along these lines, this paper presents a robust semi-direct 3D simultaneous localization and mapping (SLAM) algorithm for mobile robots based on dense optical flow. First, a preliminary estimation of the robot's pose is conducted using the sparse direct method and the homography matrix is utilized to compensate for the current frame image to reduce the image deformation caused by rotation during the robot's motion. Then, by calculating the dense optical flow field of two adjacent frames and segmenting the dynamic region in the scene based on the dynamic threshold, the local map points projected within the dynamic regions are eliminated. On this basis, the robot's pose is optimized by minimizing the reprojection error. Moreover, a high-performance keyframe selection strategy is developed, and keyframes are inserted when the robot's pose is successfully tracked. Meanwhile, feature points are extracted and matched to the keyframes for subsequent optimization and mapping. Considering that the direct method is subject to tracking failure in practical application scenarios, the feature points and map points of keyframes are employed in robot relocation. Finally, all keyframes and map points are used as optimization variables for global bundle adjustment (BA) optimization, so as to construct a globally consistent 3D dense octree map. A series of simulations and experiments demonstrate the superior performance of the proposed algorithm.

An integrated lot-sizing policy for the inventory management of constrained multi-level supply chains: null-space method

ABSTRACT The null-space method (NSM) for solving the nonlinear programming (NLP) is used to transform an indefinite system into a symmetric positive definite one of a smaller dimension. In this paper, an NSM optimises the constrained NLP model of the inventories in multi-level supply chains (SCs). The presented NSM can give a direct method with a predictable level of fill without pivoting, decreasing the number of taken iterations to find the optimum solution. We transform the nonlinear equations into an NLP model, which NSM can solved. We also investigate the suitability of using null-space-based factorisations to derive sparse direct methods. Accordingly, an integrated lot-sizing model of the multi-level SC is designed and then optimised using the presented NSM. The paper's objectives are to find the optimum number of stockpiles and the economic period length for inventories. Some numerical examples demonstrate the applicability of the presented NSM to optimise the integrated lot-sizing policy of the multi-level SCs. The presented NSM shows satisfactory performance in optimum solutions, the number of iterations, infeasibility, optimality error, and complementarity.

A sparsity-constrained sampling method with applications to communications and inverse scattering

We introduce the sparse direct sampling method (DSM) to estimate properties of a region from signals that probe the region. We demonstrate the sparse-DSM on two separate problems: estimating both the angle-of-arrival of a radio wave impinging on an array and the location and shape of an inhomogeneity from scattered acoustic waves. The sparse-DSM is qualitative in nature, so it does not require the simulation of a forward problem to solve the inverse problem. The method generalizes of two older qualitative methods, one which has low-resolution reconstructions but uses few measurements and one which is high-resolution but has higher measurement cost. The sparse-DSM inherits positive qualities from both. We demonstrate the technique on measured and simulated examples.

Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods

AbstractExtreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to degrees of freedom on a petascale supercomputer using 43,200 processes.

Incremental Pose Map Optimization for Monocular Vision SLAM Based on Similarity Transformation.

The novel contribution of this paper is to propose an incremental pose map optimization for monocular vision simultaneous localization and mapping (SLAM) based on similarity transformation, which can effectively solve the scale drift problem of SLAM for monocular vision and eliminate the cumulative error by global optimization. With the method of mixed inverse depth estimation based on a probability graph, the problem of the uncertainty of depth estimation is effectively solved and the robustness of depth estimation is improved. Firstly, this paper proposes a method combining the sparse direct method based on histogram equalization and the feature point method for front-end processing, and the mixed inverse depth estimation method based on a probability graph is used to estimate the depth information. Then, a bag-of-words model based on the mean initialization K-means is proposed for closed-loop feature detection. Finally, the incremental pose map optimization method based on similarity transformation is proposed to process the back end to optimize the pose and depth information of the camera. When the closed loop is detected, global optimization is carried out to effectively eliminate the cumulative error of the system. In this paper, indoor and outdoor environmental experiments are carried out using open data sets, such as TUM and KITTI, which fully proves the effectiveness of this method. Closed-loop detection experiments using hand-held cameras verify the importance of closed-loop detection. This method can effectively solve the scale drift problem of monocular vision SLAM and has strong robustness.

QpSWIFT: A Real-Time Sparse Quadratic Program Solver for Robotic Applications

In this letter, we present qpSWIFT, a real-time quadratic program (QP) solver. Motivated by the need for a robust embedded QP solver in robotic applications, qpSWIFT employs standard primal-dual interior-point method, along with Mehrotra predictor–corrector steps and Nesterov–Todd scaling. The sparse structure of the resulting Karush–Kuhn–Tucker linear system in the QP formulation is exploited, and sparse direct methods are utilized to solve the linear system of equations. To further accelerate the factorization process, we only modify the corresponding rows of the matrix factors that change during iterations and cache the nonzero Cholesky pattern. qpSWIFT is library free, written in ANSI-C and its performance is benchmarked through standard problems that could be cast as QP. Numerical results show that qpSWIFT outperforms state-of-the-art solvers for small scale problems. To evaluate the performance of the solver, a real-time implementation of the solver in the model predictive control framework through experiments on a quadrupedal robot are presented.

Real-Time Photometric Calibrated Monocular Direct Visual SLAM

To solve the illumination sensitivity problems of mobile ground equipment, an enhanced visual SLAM algorithm based on the sparse direct method was proposed in this paper. Firstly, the vignette and response functions of the input sequences were optimized based on the photometric formation of the camera. Secondly, the Shi–Tomasi corners of the input sequence were tracked, and optimization equations were established using the pixel tracking of sparse direct visual odometry (VO). Thirdly, the Levenberg–Marquardt (L–M) method was applied to solve the joint optimization equation, and the photometric calibration parameters in the VO were updated to realize the real-time dynamic compensation of the exposure of the input sequences, which reduced the effects of the light variations on SLAM’s (simultaneous localization and mapping) accuracy and robustness. Finally, a Shi–Tomasi corner filtered strategy was designed to reduce the computational complexity of the proposed algorithm, and the loop closure detection was realized based on the oriented FAST and rotated BRIEF (ORB) features. The proposed algorithm was tested using TUM, KITTI, EuRoC, and an actual environment, and the experimental results show that the positioning and mapping performance of the proposed algorithm is promising.

Direct solution method for the equilibrium problem for elastic stents

SummaryWe are interested in numerical methods for approximating vector‐valued functions on a metric graph. As a model problem, we formulate and analyze a numerical method for the solution of the stationary problem for the one‐dimensional elastic stent model. The approximation is built using the mixed finite element method. The discretization matrix is a symmetric saddle‐point matrix, and we discuss sparse direct methods for the fast and robust solution of the associated equilibrium system. The convergence of the numerical method is proven and the error estimate is obtained. Numerical examples confirm the theoretical estimates.

On Exploiting Sparsity of Multiple Right-Hand Sides in Sparse Direct Solvers

Le cout des resolutions triangulaires des solveurs creux directs est parfois critique. Ce cout peut depasser celui de la factorisation dans les applications qui necessitent la resolution de plusieurs milliers de seconds membres. Cette etude se concentre sur les cas ou les seconds membres sont multiples, creux et n’ont pas tous la meme structure. Etant donnee la factorisation A = LU d’une matrice creuse A, l’etude met surtout l’accent sur la resolution du premier systeme triangulaire LY = B, ou L est triangulaire inferieure. Dans ce type de problemes, le creux dans la matrice B peut etre exploite de deux manieres. Premierement, on evite le calcul de certaines lignes, ce qui correspond a elaguer certains noeuds de l’arbre d’elimination (qui represente les dependances entre les calculs de la resolution). Deuxiemement, on reduit, pour chaque noeud, le calcul sur des sous-ensembles de colonnes de B plutot que sur la matrice complete. Dans ce cas, un probleme combinatoire doit etre resolu afin de trouver une permutation des colonnes de B. S’appuyant d’abord sur l’algorithme de dissection emboitee applique a un domaine regulier, un premier algorithme est propose pour contruire une permutation des colonnes de B. Puis une nouvelle approche permet de poursuivre la reduction du nombre d’operations grâce a la creation de blocs. Pour preserver la flexibilite de l’implementation ainsi que l’efficacite des operations de type BLAS 3, un nombre minimal de groupe est cree. Inspires d’abord par des observations geometriques, ces nouveaux algorithmes ont ete etendus algebriquement pour n’utiliser que des informations provenant de la structure des seconds membres et des arbres d’elimination. Ils permettent ainsi une convergence rapide vers le nombre minimal d’operations. Les resultats experimentaux demontrent le gain obtenu par rapport a d’autres approches classiques. Enfin, les applications et extensions possibles de ce travail sont presentees.

Robust and scalable domain decomposition solvers for unfitted finite element methods

Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and are thus not well suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose a {customization} of the classical BDDC methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner. These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments show the good performance of the proposed preconditioners.

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

SummaryNull‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.

The State-of-the-Art of Preconditioners for Sparse Linear Least-Squares Problems

In recent years, a variety of preconditioners have been proposed for use in solving large sparse linear least-squares problems. These include simple diagonal preconditioning, preconditioners based on incomplete factorizations, and stationary inner iterations used with Krylov subspace methods. In this study, we briefly review preconditioners for which software has been made available, then present a numerical evaluation of them using performance profiles and a large set of problems arising from practical applications. Comparisons are made with state-of-the-art sparse direct methods.

A survey of direct methods for sparse linear systems

Wilkinson defined a sparse matrix as one with enough zeros that it pays to take advantage of them.1This informal yet practical definition captures the essence of the goal of direct methods for solving sparse matrix problems. They exploit the sparsity of a matrix to solve problems economically: much faster and using far less memory than if all the entries of a matrix were stored and took part in explicit computations. These methods form the backbone of a wide range of problems in computational science. A glimpse of the breadth of applications relying on sparse solvers can be seen in the origins of matrices in published matrix benchmark collections (Duff and Reid 1979a, Duff, Grimes and Lewis 1989a, Davis and Hu 2011). The goal of this survey article is to impart a working knowledge of the underlying theory and practice of sparse direct methods for solving linear systems and least-squares problems, and to provide an overview of the algorithms, data structures, and software available to solve these problems, so that the reader can both understand the methods and know how best to use them.

Robust Memory-Aware Mappings for Parallel Multifrontal Factorizations

We study the memory scalability of the parallel multifrontal factorization of sparse matrices. In particular, we are interested in controlling the active memory specific to the multifrontal factorization. We illustrate why commonly used mapping strategies (e.g., the proportional mapping) cannot provide a high memory efficiency, which means that they tend to let the memory usage of the factorization grow when the number of processes increases. We propose memory-aware algorithms that aim at maximizing the granularity of parallelism while respecting memory constraints. These algorithms provide accurate memory estimates prior to the factorization and can significantly enhance the robustness of a multifrontal code. We illustrate our approach with experiments performed on large matrices. 1. Introduction. We consider the memory scalability of sparse direct methods for the solution of linear systems on distributed-memory architectures. We focus on the multifrontal method [6, 7]. In this method, the consumed memory space consists of the factors to be computed (e.g., the L U factors of a given matrix A) and some temporary data that we call the active memory and that we describe in detail in the next section. Both the size of the factors and of the active memory depend on the input matrix permutation [12] (nested dissection methods are commonly used on large size problems); in this work we focus on studying the memory consumption resulting from different mapping techniques for a fixed matrix permutation. The active memory can represent a significant fraction of the total memory, and, as we will show in Section 2, it does not naturally scale when the number of processes increases. This means that the total memory usage of the factorization can dangerously grow with the number of processes. This highlights the need for mapping and scheduling algorithms that are able to control the active memory. In Section 3, we suggest memory-aware mapping algorithms that aim at maximizing the granularity of parallelism under a given memory constraint (the maximum memory usage of a process). We illustrate our approach in Section 4 with experiments carried out on large matrices using up to 256 processes. Although we focus on the multifrontal method, our study can be applied to any application with a tree-shaped workflow graph since, as we recall in the next subsection , the multifrontal method relies on the so-called elimination tree [24]. A similar problem has also been recently explored from a more theoretical point of view [8].

A Novel Factorized Sparse Approximate Inverse Preconditioner with Supernodes

Abstract Krylov methods preconditioned by Factorized Sparse Approximate Inverses (FSAI) are an efficient approach for the solution of symmetric positive definite linear systems on massively parallel computers. However, FSAI often suffers from a high set-up cost, especially in ill-conditioned problems. In this communication we propose a novel algorithm for the FSAI computation that makes use of the concept of supernode borrowed from sparse LU factorizations and direct methods.

Parallel eigenvalue calculation based on multiple shift–invert Lanczos and contour integral based spectral projection method

We discuss the possibility of using multiple shift–invert Lanczos and contour integral based spectral projection method to compute a relatively large number of eigenvalues of a large sparse and symmetric matrix on distributed memory parallel computers. The key to achieving high parallel efficiency in this type of computation is to divide the spectrum into several intervals in a way that leads to optimal use of computational resources. We discuss strategies for dividing the spectrum. Our strategies make use of an eigenvalue distribution profile that can be estimated through inertial counts and cubic spline fitting. Parallel sparse direct methods are used in both approaches. We use a simple cost model that describes the cost of computing k eigenvalues within a single interval in terms of the asymptotic cost of sparse matrix factorization and triangular substitutions. Several computational experiments are performed to demonstrate the effect of different spectrum division strategies on the overall performance of both multiple shift–invert Lanczos and the contour integral based method. We also show the parallel scalability of both approaches in the strong and weak scaling sense. In addition, we compare the performance of multiple shift–invert Lanczos and the contour integral based spectral projection method on a set of problems from density functional theory (DFT).

An adaptive staggered grid finite difference method for modeling geodynamic Stokes flows with strongly variable viscosity

Here we describe a new staggered grid formulation for discretizing incompressible Stokes flow which has been specifically designed for use on adaptive quadtree‐type meshes. The key to our new adaptive staggered grid (ASG) stencil is in the form of the stress‐conservative finite difference constraints which are enforced at the “hanging” velocity nodes between resolution transitions within the mesh. The new ASG discretization maintains a compact stencil, thus preserving the sparsity within the matrix which both minimizes the computational cost and enables the discrete system to be efficiently solved via sparse direct factorizations or iterative methods. We demonstrate numerically that the ASG stencil (1) is stable and does not produce spurious pressure oscillations across regions of grid refinement, which intersect discontinuous viscosity structures, and (2) possesses the same order of accuracy as the classical nonadaptive staggered grid discretization. Several pragmatic error indicators that are used to drive adaptivity are introduced in order to demonstrate the superior performance of the ASG stencil over traditional nonadaptive grid approaches. Furthermore, to demonstrate the potential of this new methodology, we present geodynamic examples of both lithospheric and planetary scales models.

Adaptive AMG with coarsening based on compatible weighted matching

We introduce a new composite adaptive Algebraic Multigrid (composite \(\alpha \)AMG) method to solve systems of linear equations without a-priori knowledge or assumption on characteristics of near-null components of the AMG preconditioned problem referred to as algebraic smoothness. Our version of \(\alpha \)AMG is a composite solver built through a bootstrap strategy aimed to obtain a desired convergence rate. The coarsening process employed to build each new solver component relies on a pairwise aggregation scheme based on weighted matching in a graph, successfully exploited for reordering algorithms in sparse direct methods to enhance diagonal dominance, and compatible relaxation. The proposed compatible matching process replaces the commonly used characterization of strength of connection in both the coarse space selection and in the interpolation scheme. The goal is to design a method leading to scalable AMG for a wide class of problems that go beyond the standard elliptic Partial Differential Equations (PDEs). In the present work, we introduce the method and demonstrate its potential when applied to symmetric positive definite linear systems arising from finite element discretization of highly anisotropic elliptic PDEs on structured and unstructured meshes. We also report on some preliminary tests for 2D and 3D elasticity problems as well as on problems from the University of Florida Sparse Matrix Collection.

Updated sparse cholesky factors for corotational elastodynamics

We present warp-canceling corotation , a nonlinear finite element formulation for elastodynamic simulation that achieves fast performance by making only partial or delayed changes to the simulation's linearized system matrices. Coupled with an algorithm for incremental updates to a sparse Cholesky factorization, the method realizes the stability and scalability of a sparse direct method without the need for expensive refactorization at each time step. This finite element formulation combines the widely used corotational method with stiffness warping so that changes in the per-element rotations are initially approximated by inexpensive per-node rotations. When the errors of this approximation grow too large, the per-element rotations are selectively corrected by updating parts of the matrix chosen according to locally measured errors. These changes to the system matrix are propagated to its Cholesky factor by incremental updates that are much faster than refactoring the matrix from scratch. A nested dissection ordering of the system matrix gives rise to a hierarchical factorization in which changes to the system matrix cause limited, well-structured changes to the Cholesky factor. We show examples of simulations that demonstrate that the proposed formulation produces results that are visually comparable to those produced by a standard corotational formulation. Because our method requires computing only partial updates of the Cholesky factor, it is substantially faster than full refactorization and outperforms widely used iterative methods such as preconditioned conjugate gradients. Our method supports a controlled trade-off between accuracy and speed, and unlike most iterative methods its performance does not slow for stiffer materials but rather it actually improves.