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We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as…
As the performance gains from accelerating quantized matrix multiplication plateau, the softmax operation becomes the critical bottleneck in Transformer inference. This bottleneck stems from two hardware limitations: (1) limited data…
Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and…
Recent technological developments have led to big data processing, which resulted in significant computational difficulties when solving large-scale linear systems or inverting matrices. As a result, fast approximate iterative matrix…
Homomorphic Encryption (HE) enables users to securely outsource both the storage and computation of sensitive data to untrusted servers. Not only does HE offer an attractive solution for security in cloud systems, but lattice-based HE…
In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been largely studied. On the other hand, only few papers have investigated the case of block entries, where…
Matrices with the displacement structures of circulant, Toeplitz, and Hankel types as well as matrices with structures generalizing these types are omnipresent in computations of sciences and engineering. In this paper, we present efficient…
To efficiently solve large scale nonlinear systems, we propose a novel Random Greedy Fast Block Kaczmarz method. This approach integrates the strengths of random and greedy strategies while avoiding the computationally expensive…
Maximal Independent Set (MIS) in a graph is a fundamental problem with applications in resource allocation, scheduling, and network optimization. Although graphs are inherently un-structured and challenging for GPU parallelism due to…
This work presents a convex-optimization-based framework for analysis and control of nonlinear partial differential equations. The approach uses a particular weak embedding of the nonlinear PDE, resulting in a linear equation in the space…
We present the asymptotically fastest known algorithms for some basic problems on univariate polynomial matrices: rank, nullspace, determinant, generic inverse, reduced form. We show that they essentially can be reduced to two computer…
Relaxation processes driven by a Laplacian matrix can be found in many real-world big-data systems, for example, in search engines on the World-Wide-Web and the dynamic load balancing protocols in mesh networks. To numerically implement…
Partial Differential Equations (PDEs) are fundamental for modeling physical systems, yet solving them in a generic and efficient manner using machine learning-based approaches remains challenging due to limited multi-input and multi-scale…
In this paper, we introduce a practical GPU-enhanced matrix-free first-order method for solving large-scale conic programming problems, which we refer to as PDCS, standing for the Primal-Dual Conic Programming Solver. Problems that it…
In the literature, there exist several studies on symbol-based multigrid methods for the solution of linear systems having structured coefficient matrices. In particular, the convergence analysis for such methods has been obtained in an…
Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be…
In this paper we derive new uniform convergence estimates for the V-cycle MGM applied to symmetric positive definite Toeplitz block tridiagonal matrices, by also discussing few connections with previous results. More concretely, the…
We study the bit complexity of inverting diagonally dominant matrices, which are associated with random walk quantities such as hitting times and escape probabilities. Such quantities can be exponentially small, even on undirected…
Transmission Topology Optimization has great potential to improve efficiency and flexibility of grid operations through non-costly switching actions, but previous approaches struggle with runtime performance and scalability. In this work,…
Recently, graphics processors (GPUs) have been increasingly leveraged in a variety of scientific computing applications. However, architectural differences between CPUs and GPUs necessitate the development of algorithms that take advantage…