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Retrieval-augmented generation (RAG) mitigates hallucination in Large Language Models (LLMs) by using query pipelines to retrieve relevant external information and grounding responses in retrieved knowledge. However, query pipeline optimization for cancer patient question-answering (CPQA) systems requires separately optimizing multiple components with domain-specific considerations. We propose a novel three-aspect optimization approach for the RAG query pipeline in CPQA systems, utilizing public biomedical databases like PubMed and PubMed Central. Our optimization includes: (1) document retrieval, utilizing a comparative analysis of NCBI resources and introducing Hybrid Semantic Real-time Document Retrieval (HSRDR); (2) passage retrieval, identifying optimal pairings of dense retrievers and rerankers; and (3) semantic representation, introducing Semantic Enhanced Overlap Segmentation (SEOS) for improved contextual understanding. On a custom-developed dataset tailored for cancer-related inquiries, our optimized RAG approach improved the answer accuracy of Claude-3-haiku by 5.24% over chain-of-thought prompting and about 3% over a naive RAG setup. This study highlights the importance of domain-specific query optimization in realizing the full potential of RAG and provides a robust framework for building more accurate and reliable CPQA systems, advancing the development of RAG-based biomedical systems.

We study the recovery of one-dimensional semipermeable barriers for a stochastic process in a planar domain. The considered process acts like Brownian motion when away from the barriers and is reflected upon contact until a sufficient but random amount of interaction has occurred, determined by the permeability, after which it passes through. Given a sequence of samples, we wonder when one can determine the location and shape of the barriers. This paper identifies several different recovery regimes, determined by the available observation period and the time between samples, with qualitatively different behavior. The observation period $T$ dictates if the full barriers or only certain pieces can be recovered, and the sampling rate significantly influences the convergence rate as $T\to \infty$. This rate turns out polynomial for fixed-frequency data, but exponentially fast in a high-frequency regime. Further, the environment's impact on the difficulty of the problem is quantified using interpretable parameters in the recovery guarantees, and is found to also be regime-dependent. For instance, the curvature of the barriers affects the convergence rate for fixed-frequency data, but becomes irrelevant when $T\to \infty$ with high-frequency data. The results are accompanied by explicit algorithms, and we conclude by illustrating the application to real-life data.

We propose a method utilizing physics-informed neural networks (PINNs) to solve Poisson equations that serve as control variates in the computation of transport coefficients via fluctuation formulas, such as the Green--Kubo and generalized Einstein-like formulas. By leveraging approximate solutions to the Poisson equation constructed through neural networks, our approach significantly reduces the variance of the estimator at hand. We provide an extensive numerical analysis of the estimators and detail a methodology for training neural networks to solve these Poisson equations. The approximate solutions are then incorporated into Monte Carlo simulations as effective control variates, demonstrating the suitability of the method for moderately high-dimensional problems where fully deterministic solutions are computationally infeasible.

We study nonconvex optimization in high dimensions through Langevin dynamics, focusing on the multi-spiked tensor PCA problem. This tensor estimation problem involves recovering $r$ hidden signal vectors (spikes) from noisy Gaussian tensor observations using maximum likelihood estimation. We study the number of samples required for Langevin dynamics to efficiently recover the spikes and determine the necessary separation condition on the signal-to-noise ratios (SNRs) for exact recovery, distinguishing the cases $p \ge 3$ and $p=2$, where $p$ denotes the order of the tensor. In particular, we show that the sample complexity required for recovering the spike associated with the largest SNR matches the well-known algorithmic threshold for the single-spike case, while this threshold degrades when recovering all $r$ spikes. As a key step, we provide a detailed characterization of the trajectory and interactions of low-dimensional projections that capture the high-dimensional dynamics.

We investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. By adapting mathematical tools from multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. We then propose filtered MLMC (F-MLMC) estimators based on a filtering mechanism similar to the smoothing process of multigrid methods, and we show that the filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. Next, the conclusions of the spectral analysis are experimentally verified with a one-dimensional illustration. Finally, the proposed F-MLMC estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

Long, powerful soft detection forward error correction codes are typically constructed by concatenation of shorter component codes that are decoded through iterative Soft-Input Soft-Output (SISO) procedures. The current gold-standard is Low Density Parity Check (LDPC) codes, which are built from weak single parity check component codes that are capable of producing accurate SO. Due to the recent development of SISO decoders that produce highly accurate SO with codes that have multiple redundant bits, square product code constructions that can avail of more powerful component codes have been shown to be competitive with the LDPC codes in the 5G New Radio standard in terms of decoding performance while requiring fewer iterations to converge. Motivated by applications that require more powerful low-rate codes, in the present paper we explore the possibility of extending this design space by considering the construction and decoding of cubic tensor codes.

We derive a new adaptive leverage score sampling strategy for solving the Column Subset Selection Problem (CSSP). The resulting algorithm, called Adaptive Randomized Pivoting, can be viewed as a randomization of Osinsky's recently proposed deterministic algorithm for CSSP. It guarantees, in expectation, an approximation error that matches the optimal existence result in the Frobenius norm. Although the same guarantee can be achieved with volume sampling, our sampling strategy is much simpler and less expensive. To show the versatility of Adaptive Randomized Pivoting, we apply it to select indices in the Discrete Empirical Interpolation Method, in cross/skeleton approximation of general matrices, and in the Nystroem approximation of symmetric positive semi-definite matrices. In all these cases, the resulting randomized algorithms are new and they enjoy bounds on the expected error that match -- or improve -- the best known deterministic results. A derandomization of the algorithm for the Nystroem approximation results in a new deterministic algorithm with a rather favorable error bound.

The widespread adoption of digital distribution channels both enables and forces more and more logistical service providers to manage booking processes actively to maintain competitiveness. As a result, their operational planning is no longer limited to solving vehicle routing problems. Instead, demand management decisions and vehicle routing decisions are optimized integratively with the aim of maximizing revenue and minimizing fulfillment cost. The resulting integrated demand management and vehicle routing problems (i-DMVRPs) can be formulated as Markov decision process models and, theoretically, can be solved via the well-known Bellman equation. Unfortunately, the Bellman equation is intractable for realistic-sized instances. Thus, in the literature, i-DMVRPs are often addressed via decomposition-based solution approaches involving an opportunity cost approximation as a key component. Despite its importance, to the best of our knowledge, there is neither a technique to systematically analyze how the accuracy of the opportunity cost approximation translates into overall solution quality nor are there general guidelines on when to apply which class of approximation approach. In this work, we address this research gap by proposing an explainability technique that quantifies and visualizes the magnitude of approximation errors, their immediate impact, and their relevance in specific regions of the state space. Exploiting reward decomposition, it further yields a characterization of different types of approximation errors. Applying the technique to a generic i-DMVRP in a full-factorial computational study and comparing the results with observations in existing literature, we show that the technique contributes to better explaining algorithmic performance and provides guidance for the algorithm selection and development process.

Conjugate heat transfer (CHT) analyses are vital for the design of many energy systems. However, high-fidelity CHT numerical simulations are computationally intensive, which limits their applications such as design optimization, where hundreds to thousands of evaluations are required. In this work, we develop a modular deep encoder-decoder hierarchical (DeepEDH) convolutional neural network, a novel deep-learning-based surrogate modeling methodology for computationally intensive CHT analyses. Leveraging convective temperature dependencies, we propose a two-stage temperature prediction architecture that couples velocity and temperature fields. The proposed DeepEDH methodology is demonstrated by modeling the pressure, velocity, and temperature fields for a liquid-cooled cold-plate-based battery thermal management system with variable channel geometry. A computational mesh and CHT formulation of the cold plate is created and solved using the finite element method (FEM), generating a dataset of 1,500 simulations. Our performance analysis covers the impact of the novel architecture, separate DeepEDH models for each field, output geometry masks, multi-stage temperature field predictions, and optimizations of the hyperparameters and architecture. Furthermore, we quantify the influence of the CHT analysis' thermal boundary conditions on surrogate model performance, highlighting improved temperature model performance with higher heat fluxes. Compared to other deep learning neural network surrogate models, such as U-Net and DenseED, the proposed DeepEDH architecture for CHT analyses exhibits up to a 65% enhancement in the coefficient of determination $R^{2}$. (*Due to the notification of arXiv "The Abstract field cannot be longer than 1,920 characters", the appeared Abstract is shortened. For the full Abstract, please download the Article.)

In large-scale systems there are fundamental challenges when centralised techniques are used for task allocation. The number of interactions is limited by resource constraints such as on computation, storage, and network communication. We can increase scalability by implementing the system as a distributed task-allocation system, sharing tasks across many agents. However, this also increases the resource cost of communications and synchronisation, and is difficult to scale. In this paper we present four algorithms to solve these problems. The combination of these algorithms enable each agent to improve their task allocation strategy through reinforcement learning, while changing how much they explore the system in response to how optimal they believe their current strategy is, given their past experience. We focus on distributed agent systems where the agents' behaviours are constrained by resource usage limits, limiting agents to local rather than system-wide knowledge. We evaluate these algorithms in a simulated environment where agents are given a task composed of multiple subtasks that must be allocated to other agents with differing capabilities, to then carry out those tasks. We also simulate real-life system effects such as networking instability. Our solution is shown to solve the task allocation problem to 6.7% of the theoretical optimal within the system configurations considered. It provides 5x better performance recovery over no-knowledge retention approaches when system connectivity is impacted, and is tested against systems up to 100 agents with less than a 9% impact on the algorithms' performance.