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There have been several modifications of how basic calculus has been taught, but very few of these modifications have considered the computational tools available at our disposal. Here, we present a few tools that are easy to develop and…
We present a prototypical linear algebra compiler that automatically exploits domain-specific knowledge to generate high-performance algorithms. The input to the compiler is a target equation together with knowledge of both the structure of…
Large Language Models (LLMs) have demonstrated impressive zero-shot capabilities and versatility in NLP tasks, however they sometimes fail to maintain crucial invariances for specific tasks. One example is permutation sensitivity, where…
This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and…
Efficiently solving sparse linear algebraic equations is an important research topic of numerical simulation. Commonly used approaches include direct methods and iterative methods. Compared with the direct methods, the iterative methods…
We consider the problem of efficiently solving large-scale linear least squares problems that have one or more linear constraints that must be satisfied exactly. Whilst some classical approaches are theoretically well founded, they can face…
Recent work showed that ML-based attacks on Learning with Errors (LWE), a hard problem used in post-quantum cryptography, outperform classical algebraic attacks in certain settings. Although promising, ML attacks struggle to scale to more…
Open-ended tasks, such as coding problems that are common in computer science education, provide detailed insights into student knowledge. However, training large language models (LLMs) to simulate and predict possible student errors in…
Inverse reinforcement learning (IRL) is computationally challenging, with common approaches requiring the solution of multiple reinforcement learning (RL) sub-problems. This work motivates the use of potential-based reward shaping to reduce…
Recovering the digital input of a time-discrete linear system from its (noisy) output is a significant challenge in the fields of data transmission, deconvolution, channel equalization, and inverse modeling. A variety of algorithms have…
This Ph.D. thesis contains original contributions to several areas within the disciplines of disordered systems, numerical linear algebra, and scientific computing: (1) Theoretical and numerical study of the errors caused by using certain…
We showcase applications of nonlinear algebra in the sciences and engineering. Our review is organized into eight themes: polynomial optimization, partial differential equations, algebraic statistics, integrable systems, configuration…
Large Language Models (LLMs) have upended decades of pedagogy in computing education. Students previously learned to code through \textit{writing} many small problems with less emphasis on code reading and comprehension. Recent research has…
Integer linear programming (ILP) encompasses a very important class of optimization problems that are of great interest to both academia and industry. Several algorithms are available that attempt to explore the solution space of this class…
This paper discusses several linear algebra activities designed to help enhance students' skills in collaborating, exploring mathematics, and linking together abstract and visual ways of approaching mathematics. Most of these activities are…
Neural networks allow solving many ill-posed inverse problems with unprecedented performance. Physics informed approaches already progressively replace carefully hand-crafted reconstruction algorithms in real applications. However, these…
We explore the automatic generation of interactive, scenario-based lessons designed to train novice human tutors who teach middle school mathematics online. Employing prompt engineering through a Retrieval-Augmented Generation approach with…
Distilling data into compact and interpretable analytic equations is one of the goals of science. Instead, contemporary supervised machine learning methods mostly produce unstructured and dense maps from input to output. Particularly in…
Quick interaction between a human teacher and a learning machine presents numerous benefits and challenges when working with web-scale data. The human teacher guides the machine towards accomplishing the task of interest. The learning…
Finding model parameters from data is an essential task in science and engineering, from weather and climate forecasts to plasma control. Previous works have employed neural networks to greatly accelerate finding solutions to inverse…