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This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the…
Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be…
We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform…
The compressive learning framework reduces the computational cost of training on large-scale datasets. In a sketching phase, the data is first compressed to a lightweight sketch vector, obtained by mapping the data samples through a…
In this paper we derive a Newton type method to solve the non-linear system formed by combining the Tikhonov normal equations and Morozov's discrepancy principle. We prove that by placing a bound on the step size of the Newton iterations…
This paper describes a method for scheduling the events of a switched system to achieve an optimal performance. The approach has guarantees on convergence and computational complexity that parallel derivative-based iterative optimization…
This paper argues that randomized linear sketching is a natural tool for on-the-fly compression of data matrices that arise from large-scale scientific simulations and data collection. The technical contribution consists in a new algorithm…
Tensor network contraction is a fundamental mathematical operation that generalizes the dot product and matrix multiplication. It finds applications in numerous domains, such as database systems, graph theory, machine learning, probability…
Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to deal with singular and rectangular linear systems. In this article,…
We propose OverSketch, an approximate algorithm for distributed matrix multiplication in serverless computing. OverSketch leverages ideas from matrix sketching and high-performance computing to enable cost-efficient multiplication that is…
There is an increasing body of work exploring the integration of random projection into algorithms for numerical linear algebra. The primary motivation is to reduce the overall computational cost of processing large datasets. A suitably…
Kernel methods are fundamental in machine learning, and faster algorithms for kernel approximation provide direct speedups for many core tasks in machine learning. The polynomial kernel is especially important as other kernels can often be…
Non-linear least squares solvers are used across a broad range of offline and real-time model fitting problems. Most improvements of the basic Gauss-Newton algorithm tackle convergence guarantees or leverage the sparsity of the underlying…
3D Gaussian Splatting (3DGS) has emerged as a promising representation for photorealistic rendering of 3D scenes. However, its high storage requirements pose significant challenges for practical applications. We observe that Gaussians…
3D Gaussian Splatting (3DGS) has become a state-of-the-art framework for real-time, high-fidelity novel view synthesis. However, its substantial storage requirements and inherently unstructured representation pose challenges for deployment…
The Kaczmarz algorithm is a simple iterative scheme for solving consistent linear systems. At each step, the method projects the current iterate onto the solution space of a single constraint. Hence, it requires very low cost per iteration…
The randomized projection (RP) method is a simple iterative scheme for solving linear feasibility problems and has recently gained popularity due to its speed and low memory requirement. This paper develops an accelerated variant of the…
We study some methods of subgradient projections for solving a convex feasibility problem with general (not necessarily hyperplanes or half-spaces) convex sets in the inconsistent case and propose a strategy that controls the relaxation…
In this study, we consider the numerical solution of large systems of linear equations obtained from the stochastic Galerkin formulation of stochastic partial differential equations. We propose an iterative algorithm that exploits the…
We study a version of the randomized Kaczmarz algorithm for solving systems of linear equations where the iterates are confined to the solution space of a selected subsystem. We show that the subspace constraint leads to an accelerated…