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Approximation theory plays a central role in numerical analysis, undergoing continuous evolution through a spectrum of methodologies. Notably, Lebesgue, Weierstrass, Fourier, and Chebyshev approximations stand out among these methods. However, each technique possesses inherent limitations, underscoring the critical importance of selecting an appropriate approximation method tailored to specific problem domains. This article delves into the utilization of Chebyshev polynomials at Chebyshev nodes for approximation. For sufficiently smooth functions, the partial sum of Chebyshev series expansion offers optimal polynomial approximation, rendering it a preferred choice in various applications such as digital signal processing and graph filters due to its computational efficiency. In this article, we focus on functions of bounded variation, which find numerous applications across mathematical physics, hyperbolic conservations, and optimization. We present two optimal error estimations associated with Chebyshev polynomial approximations tailored for such functions. To validate our theoretical assertions, we conduct numerical experiments. Additionally, we delineate promising future avenues aligned with this research, particularly within the realms of machine learning and related fields.

The incorporation of generative models as regularisers within variational formulations for inverse problems has proven effective across numerous image reconstruction tasks. However, the resulting optimisation problem is often non-convex and challenging to solve. In this work, we show that score-based generative models (SGMs) can be used in a graduated optimisation framework to solve inverse problems. We show that the resulting graduated non-convexity flow converge to stationary points of the original problem and provide a numerical convergence analysis of a 2D toy example. We further provide experiments on computed tomography image reconstruction, where we show that this framework is able to recover high-quality images, independent of the initial value. The experiments highlight the potential of using SGMs in graduated optimisation frameworks.

We utilize extreme learning machines for the prediction of partial differential equations (PDEs). Our method splits the state space into multiple windows that are predicted individually using a single model. Despite requiring only few data points (in some cases, our method can learn from a single full-state snapshot), it still achieves high accuracy and can predict the flow of PDEs over long time horizons. Moreover, we show how additional symmetries can be exploited to increase sample efficiency and to enforce equivariance.

Characteristic imsets are 0/1-vectors representing directed acyclic graphs whose edges represent direct cause-effect relations between jointly distributed random variables. A characteristic imset (CIM) polytope is the convex hull of a collection of characteristic imsets. CIM polytopes arise as feasible regions of a linear programming approach to the problem of causal disovery, which aims to infer a cause-effect structure from data. Linear optimization methods typically require a hyperplane representation of the feasible region, which has proven difficult to compute for CIM polytopes despite continued efforts. We solve this problem for CIM polytopes that are the convex hull of imsets associated to DAGs whose underlying graph of adjacencies is a tree. Our methods use the theory of toric fiber products as well as the novel notion of interventional CIM polytopes. Our solution is obtained as a corollary of a more general result for interventional CIM polytopes. The identified hyperplanes are applied to yield a linear optimization-based causal discovery algorithm for learning polytree causal networks from a combination of observational and interventional data.

Stochastic approximation is a class of algorithms that update a vector iteratively, incrementally, and stochastically, including, e.g., stochastic gradient descent and temporal difference learning. One fundamental challenge in analyzing a stochastic approximation algorithm is to establish its stability, i.e., to show that the stochastic vector iterates are bounded almost surely. In this paper, we extend the celebrated Borkar-Meyn theorem for stability from the Martingale difference noise setting to the Markovian noise setting, which greatly improves its applicability in reinforcement learning, especially in those off-policy reinforcement learning algorithms with linear function approximation and eligibility traces. Central to our analysis is the diminishing asymptotic rate of change of a few functions, which is implied by both a form of strong law of large numbers and a commonly used V4 Lyapunov drift condition and trivially holds if the Markov chain is finite and irreducible.

Field experiments and computer simulations are effective but time-consuming methods of measuring the quality of engineered systems at different settings. To reduce the total time required, experimenters may employ Bayesian optimization, which is parsimonious with measurements, and take measurements of multiple settings simultaneously, in a batch. In practice, experimenters use very few batches, thus, it is imperative that each batch be as informative as possible. Typically, the initial batch in a Batch Bayesian Optimization (BBO) is constructed from a quasi-random sample of settings values. We propose a batch-design acquisition function, Minimal Terminal Variance (MTV), that designs a batch by optimization rather than random sampling. MTV adapts a design criterion function from Design of Experiments, called I-Optimality, which minimizes the variance of the post-evaluation estimates of quality, integrated over the entire space of settings. MTV weights the integral by the probability that a setting is optimal, making it able to design not only an initial batch but all subsequent batches, as well. Applicability to both initialization and subsequent batches is novel among acquisition functions. Numerical experiments on test functions and simulators show that MTV compares favorably to other BBO methods.

We study the fundamental problem of efficiently computing the stationary distribution of general classes of structured Markov processes. In strong contrast with previous work, we consider this problem within the context of quantum computational environments from a mathematical perspective and devise the first quantum algorithms for computing the stationary distribution of structured Markov processes. We derive a mathematical analysis of the computational properties of our quantum algorithms together with related theoretical results, establishing that our quantum algorithms provide the potential for significant computational improvements over that of the best-known classical algorithms in various settings of both theoretical and practical importance. Although motivated by structured Markov processes, our quantum algorithms have the potential for being exploited to address a much larger class of numerical computation problems.

Network slicing has emerged as an integral concept in 5G, aiming to partition the physical network infrastructure into isolated slices, customized for specific applications. We theoretically formulate the key performance metrics of an application, in terms of goodput and delivery delay, at a cost of network resources in terms of bandwidth. We explore an un-coded communication protocol that uses feedback-based repetitions, and a coded protocol, implementing random linear network coding and using coding-aware acknowledgments. We find that coding reduces the resource demands of a slice to meet the requirements for an application, thereby serving more applications efficiently. Coded slices thus free up resources for other slices, be they coded or not. Based on these results, we propose a hybrid approach, wherein coding is introduced selectively in certain network slices. This approach not only facilitates a smoother transition from un-coded systems to coded systems but also reduces costs across all slices. Theoretical findings in this paper are validated and expanded upon through real-time simulations of the network.

The performance bounds of near-field sensing are studied for circular arrays, focusing on the impact of bandwidth and array size. The closed-form Cramer-Rao bound (CRBs) for angle and distance estimation are derived, revealing the scaling laws of the CRBs with bandwidth and array size. Contrary to expectations, enlarging array size does not always enhance sensing performance. Furthermore, the asymptotic CRBs are analyzed under different conditions, unveiling that the derived expressions include the existing results as special cases. Finally, the derived expressions are validated through numerical results.

We settle the pseudo-polynomial complexity of the Demand Strip Packing (DSP) problem: Given a strip of fixed width and a set of items with widths and heights, the items must be placed inside the strip with the objective of minimizing the peak height. This problem has gained significant scientific interest due to its relevance in smart grids[Deppert et al.\ APPROX'21, G\'alvez et al.\ APPROX'21]. Smart Grids are a modern form of electrical grid that provide opportunities for optimization. They are forecast to impact the future of energy provision significantly. Algorithms running in pseudo-polynomial time lend themselves to these applications as considered time intervals, such as days, are small. Moreover, such algorithms can provide superior approximation guarantees over those running in polynomial time. Consequently, they evoke scientific interest in related problems. We prove that Demand Strip Packing is strongly NP-hard for approximation ratios below $5/4$. Through this proof, we provide novel insights into the relation of packing and scheduling problems. Using these insights, we show a series of frameworks that solve both Demand Strip Packing and Parallel Task Scheduling optimally when increasing the strip's width or number of machines. Such alterations to problems are known as resource augmentation. Applications are found when penalty costs are prohibitively large. Finally, we provide a pseudo-polynomial time approximation algorithm for DSP with an approximation ratio of $(5/4+\varepsilon)$, which is nearly optimal assuming $P\neq NP$. The construction of this algorithm provides several insights into the structure of DSP solutions and uses novel techniques to restructure optimal solutions.