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Multilevel techniques are efficient approaches for solving the large linear systems that arise from discretized partial differential equations and other problems. While geometric multigrid requires detailed knowledge about the underlying…

Neural networks are predominantly trained using gradient-based methods, yet in many applications their final predictions remain far from the accuracy attainable within the model's expressive capacity. We introduce Linearized Subspace…

We describe an approximate rational arithmetic with round-off errors (both absolute and relative) controlled by the user. The rounding procedure is based on the continued fraction expansion of real numbers. Results of computer experiments…

The Logarithmic Linear Relaxation (LLR) algorithm is an efficient method for computing densities of states for systems with a continuous spectrum. A key feature of this method is exponential error reduction, which allows us to evaluate the…

Many scientific and engineering applications require fitting regression models that are nonlinear in the parameters. Advances in computer hardware and software in recent decades have made it easier to fit such models. Relative to fitting…

The question of what can be computed, and how efficiently, are at the core of computer science. Not surprisingly, in distributed systems and networking research, an equally fundamental question is what can be computed in a…

Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear…

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…

This paper presents a novel method for utilizing fine-tuned Large Language Models (LLMs) to minimize data requirements in load profile analysis, demonstrated through the restoration of missing data in power system load profiles. A two-stage…

The reasoning capabilities of Large Language Models (LLMs) play a critical role in many downstream tasks, yet depend strongly on the quality of training data. Despite various proposed data construction methods, their practical utility in…

Adapting large language models (LLMs) to specific domains often faces a critical bottleneck: the scarcity of high-quality, human-curated data. While large volumes of unchecked data are readily available, indiscriminately using them for…

Classical probabilistic rounding error analysis is particularly well suited to stochastic rounding (SR), and it yields strong results when dealing with floating-point algorithms that rely heavily on summation. For many numerical linear…

The high computational complexity associated with training deep neural networks limits online and real-time training on edge devices. This paper proposed an end-to-end training and inference scheme that eliminates multiplications by…

Recent advances in generative AI have led to remarkable interest in using systems that rely on large language models (LLMs) for practical applications. However, meaningful evaluation of these systems in real-world scenarios comes with a…

The high inference demands of transformer-based Large Language Models (LLMs) pose substantial challenges in their deployment. To this end, we introduce Neural Block Linearization (NBL), a novel framework for accelerating transformer model…

Large Language Models (LLMs) have the unique capability to understand and generate human-like text from input queries. When fine-tuned, these models show enhanced performance on domain-specific queries. OpenAI highlights the process of…

Imposing constraints on machine translation systems presents a challenging issue because these systems are not trained to make use of constraints in generating adequate, fluent translations. In this paper, we leverage the capabilities of…

Machine learning can provide deep insights into data, allowing machines to make high-quality predictions and having been widely used in real-world applications, such as text mining, visual classification, and recommender systems. However,…

The vast use of computers on scientific numerical computation makes the awareness of the limited precision that these machines are able to provide us an essential matter. A limited and insufficient precision allied to the truncation and…

Large language models (LLMs) have been explored in a variety of reasoning tasks including solving of mathematical problems. Each math dataset typically includes its own specially designed evaluation script, which, while suitable for its…