Successive Refinement Research Articles

A Comparison of Free and Mapped Meshes for Static Structural Analysis

Abstract This study addresses the challenge faced by Finite Element Analysts when choosing between free and mapped meshes, especially in terms of convergence stability and solution accuracy. The investigation focuses on 3D solid models under static structural loading, analyzed using Ansys® and MSC Patran®. Both free and mapped mesh types, employing equivalent 3D solid elements, are used to assess an aircraft structural component under design load conditions, with fixed boundaries. For free meshes, Tet10 elements in Patran (equivalent to Solid 72 in Ansys) are used, whereas for mapped meshes, CPENTA / CHEXA elements in Patran (equivalent to Wed6 / Hex8 in Ansys) are employed. Mesh convergence studies ensure that discretization does not affect the numerical solution. Notably, a significant stress increase is observed with successive refinement of free meshes, while mapped meshes achieve mesh independence at coarser refinement levels. Comparison of fringe plots indicates the same location for maximum deformation and equivalent stress in both free and mapped mesh models. The findings demonstrate that free meshes tend to underpredict maximum deformation and equivalent stress compared to mapped meshes, with both meshes showing deformation and stress at consistent locations. The findings underscore the importance of carefully choosing the appropriate mesh type, particularly when analyzing critical structural components, to ensure reliability and accuracy in FEA simulations.

Irregular IRS-aided SWIPT secure communication system with imperfect CSI

The performance of traditional regular Intelligent Reflecting Surface (IRS) improves as the number of IRS elements increases, but more reflecting elements lead to higher IRS power consumption and greater overhead of channel estimation. The Irregular Intelligent Reflecting Surface (IIRS) can enhance the performance of the IRS as well as boost the system performance when the number of reflecting elements is limited. However, due to the lack of radio frequency chain in IRS, it is challenging for the Base Station (BS) to gather perfect Channel State Information (CSI), especially in the presence of Eavesdroppers (Eves). Therefore, in this paper we investigate the minimum transmit power problem of IIRS-aided Simultaneous Wireless Information and Power Transfer (SWIPT) secure communication system with imperfect CSI of BS-IIRS-Eves links, which is subject to the rate outage probability constraints of the Eves, the minimum rate constraints of the Information Receivers (IRs), the energy harvesting constraints of the Energy Receivers (ERs), and the topology matrix constraints. Afterward, the formulated non-convex problem can be efficiently tackled by employing joint optimization algorithm combined with successive refinement method and adaptive topology design method. Simulation results demonstrate the effectiveness of the proposed scheme and the superiority of IIRS.

Phase quantification using deep neural network processing of XRD patterns.

Mineral identification and quantification are key to the understanding and, hence, the capacity to predict material properties. The method of choice for mineral quantification is powder X-ray diffraction (XRD), generally using a Rietveld refinement approach. However, a successful Rietveld refinement requires preliminary identification of the phases that make up the sample. This is generally carried out manually, and this task becomes extremely long or virtually impossible in the case of very large datasets such as those from synchrotron X-ray diffraction computed tomography. To circumvent this issue, this article proposes a novel neural network (NN) method for automating phase identification and quantification. An XRD pattern calculation code was used to generate large datasets of synthetic data that are used to train the NN. This approach offers significant advantages, including the ability to construct databases with a substantial number of XRD patterns and the introduction of extensive variability into these patterns. To enhance the performance of the NN, a specifically designed loss function for proportion inference was employed during the training process, offering improved efficiency and stability compared with traditional functions. The NN, trained exclusively with synthetic data, proved its ability to identify and quantify mineral phases on synthetic and real XRD patterns. Trained NN errors were equal to 0.5% for phase quantification on the synthetic test set, and 6% on the experimental data, in a system containing four phases of contrasting crystal structures (calcite, gibbsite, dolomite and hematite). The proposed method is freely available on GitHub and allows for major advances since it can be applied to any dataset, regardless of the mineral phases present.

Algebraic aspects of holomorphic quantum modular forms

Matrix-valued holomorphic quantum modular forms are intricate objects associated to 3-manifolds (in particular to knot complements) that arise in successive refinements of the volume conjecture of knots and involve three holomorphic, asymptotic and arithmetic realizations. It is expected that the algebraic properties of these objects can be deduced from the algebraic properties of descendant state integrals, and we illustrate this for the case of the (-2,3,7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-2,3,7)$$\\end{document}-pretzel knot.

Proximal Policy Optimization for RIS-Assisted Full Duplex 6G-V2X Communications

In this work, we consider a novel reconfigurable intelligent surface (RIS)-assisted full-duplex (FD) sixth generation (6G)-vehicle-to-everything (V2X) communication network having a FD base station (BS) simultaneously communicating with a uplink (UL) and a downlink (DL) mobile vehicles with the aide of two RISs, one for each link. We provide an analytical framework to investigate the performance of this network and, consequently, formulate an optimization problem to jointly optimize the phase shift matrices at both the RISs that maximize the achievable sum-rate. Thereafter, due to the non-convex nature of the problem, we propose a low complexity proximal policy optimization (PPO)-based deep reinforcement learning (DRL) algorithm that solves the problem in continuous action spaces by reducing the overall training overhead and provides optimum values of phase-shift matrix at each RIS. Further, we extend the analysis considering multiple transmit and receive antennas at the BS that simultaneously serves multiple UL and DL vehicles. We present exhaustive simulations-based graphical results to validate the effectiveness and accuracy of the proposed algorithm compared to deep deterministic policy gradient (DDPG) and successive refinement (SR)-based solutions. Accordingly, we demonstrate the dominance of the considered FD system over its half-duplex (HD) counterpart. We also discuss the impact of the number of elements at each RIS, each vehicle's velocity and their distance from the RIS and the BS on the performance of the respective links. Moreover, we also highlight the impact of imperfect self-interference cancellation and discuss the trade-off between the UL and DL performances due to this imperfection. Additionally, it is shown that the proposed algorithm is approximately <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$10 \times$</tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$ 40 \times$</tex-math></inline-formula> faster than DDPG and SR, respectively. Finally, for the multiple vehicle case, we demonstrate the effect of co-channel interference on the DL rate and the impact of RIS elements, the number of UL and DL vehicles on the system performance.

Full-waveform reconstruction of micro-seismic events via topological derivative approach

Micro-seismic events, naturally occurring within geological formations and quasi-brittle engineered systems, provide a powerful window into the evolving processes of material degradation and failure. Accurate characterization of these events is critical for a comprehensive understanding of the underpinning fracturing mechanisms and potential implications. In this work, we present an algorithm for the spatial reconstruction and characterization of micro-seismic events in a three-dimensional bounded elastic body (with known geometry and nominal material properties) via combined source location and moment tensor inversion. Assuming availability of the full-waveform “acoustic emission” traces whose spectral content can be exposed via Fourier transform, the inverse solution is based on (i) a time-harmonic (forward) elastodynamic model and (ii) the concept of topological derivative as a framework for robust event reconstruction. On exploiting an equivalence between the elastic wavefield generated by the creation of a new micro-surface and that stemming from a suitable set of dipoles and double-couples (whose strengths are synthesized via the seismic moment tensor), we formulate the inverse problem as that for the real (in-phase) and imaginary (out-of-phase) components of the moment tensor at trial grid locations. In this way the optimal solution is obtained via a combinatorial search over a prescribed grid, inherently allowing for successive refinements of event reconstruction over the region(s) of interest. The analysis is illustrated by numerical experiments highlighting the key features of the inversion scheme including the reconstruction of multiple (i.e. contemporaneous) events, localization of the “off-grid” micro-seismic events, and the ability to handle noisy data. The results in particular highlight the utility of multi-frequency event reconstruction toward reducing the demand on the number of sensing locations.

Step-adaptive accelerated demodulation algorithm for LFM-pulse-based distributed acoustic sensing

We propose a novel, to our knowledge, step-adaptive cross-correlation algorithm tailored for distributed acoustic sensing systems based on linear frequency modulation pulses, aiming for rapid demodulation. This algorithm adjusts its step length through an adaptive “successive refinement” search strategy, which greatly improves computational efficiency by reducing the number of cross-correlation computations. Experimental results have shown that the demodulation time can be reduced by approximately 15 times compared to the conventional method, while maintaining the same demodulation result.

A Precision Benchmark Suite for Nuclear Reactor Point Kinetics Equations via Converged Accelerated Taylor Series (CATS)

Extreme benchmarks of 10 or more places for the point kinetics equations for time-dependent nuclear reactor power transients are rare. Therefore, to establish an extreme benchmark, we employ a Taylor series (TS) with continuous analytical continuation to solve the ordinary differential equations of point kinetics including feedback. Nonlinear Wynn-epsilon convergence acceleration confirms the highly precise solutions for neutron and precursor densities. Through adaptive partitioning of time intervals, the proposed Converged Accelerated Taylor Series, or CATS algorithm in double precision, automatically performs successive mesh refinement to obtain high-precision initial conditions for each subinterval, with the intent to reduce propagation error. Confirmation of 10 to 12 places comes from comparison to the BEFD (Backward Euler Finite Difference) algorithm in quadruple precision also developed by the author. We report benchmark results for common cases found in the literature including step, ramp, zigzag, and sinusoidal prescribed reactivity insertions and insertions with nonlinear adiabatic Doppler feedback. We also establish a suite of new prescribed reactivity insertions and insertions with feedback, based on reactivities with Taylor series representations as suggested by the CATS algorithm.

Integrating Framing Approaches as a Tool for Managing Complex Transitioning to Renewable Energy (TRE) Projects: The Yatir Wind Farm Case Study

Transitioning to renewable energy is an urgent global goal. Wind energy is a promising renewable source with contentious obstacles. Using the Yatir Wind Farm project in Israel as a case study, we explore the potential of framing in identifying and mitigating obstacles in a wind farm project. The cognitive and strategic frames employed by stakeholders were elicited through 18 semi-structured interviews and more than 100 documents. This analysis highlights three conflicting issues: potential or perceived harm to neighboring residents, protecting birds and bats, and adapting to changing regulations throughout the process. Regarding residents’ opposition to feared harm from the project, initial cautious curiosity was followed by distrust and deep disappointment due to a lack of transparency and a sense of abandonment facing perceived existential threats to their health and homes. This led residents to conduct legal battles, which resulted in their claims being rejected. It also led to the breakdown of relationships among neighbors opposing and promoting the wind farms. In the case of bird and bat protection, stakeholders initially framed concerns emotionally until the dialogue shifted to scientific discourse, resulting in the successful refinement of mutually agreed upon regulatory guidelines. The structural appeal mechanisms effectively addressed evolving regulations, overcoming the lack of mutual understanding and resulting in the adoption of the majority of the new regulatory requirements. The analysis underscores the importance of understanding stakeholders’ frames for effectively working through the complex and transdisciplinary nature of sustainability transitions and achieving successful outcomes. It also reveals the need for formal mechanisms to validate stakeholder needs and integrate them into decision-making processes. Recommendations include early and meaningful public involvement, process improvement for stakeholder engagement, and enhanced transparency in decision-making processes.

New Online Tool to Evaluate Transferable Skills in the European Framework

According to the European classification of skills, competencies, qualifications, and occupations (ESCO), transversal knowledge, skills, and competencies are pertinent to a wide array of occupations and sectors. Transversal knowledge, skills, and competencies are the foundational elements for developing the “hard” skills and competencies necessary for success in the labor market. In this paper, we introduce an online platform for assessing the attainment of transversal and soft skills. This tool allows us to define levels of competency acquisition and measure students’ development. The use of these competency levels helps improve the understanding of these skills and the evaluation process. The tool also enhances coordination among courses and teachers. The evaluation process can be established on three different levels: self-evaluation, peer evaluation, and teacher assessment. Developers, students, and teachers have assessed the tool has been developed following a lifecycle of evolutionary prototypes with successive refinements.

Refining shape and size of silver nanoparticles using ion irradiation for enhanced and homogeneous SERS activity

We present green synthesis of silver nanoparticles in water using unirradiated and Ag15+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{15+}$$\\end{document} ion irradiated phytoextracts of Bergenia Ciliata leaf, Eupatorium adenophorum leaf, Rhododendron arboreum leaf and flower. The use of different plant extracts and their subsequent ion irradiation allow for successful refinement of nanoparticle size and morphology. Due to changes in reducing and capping agents the nanoparticle surface functionalization also varies which not only controls the morphology but also allows for surface oxidation and aggregation processes. In this work, we have synthesized silver nanoparticles which exhibit sizes in the range from 13 to 24 nm and having shapes like spherical, quasispherical, trigonal, hexagonal, cylindrical, dendritic assemblies, and porous nanoparticles. Owing to changes in the size and shape of the nanoparticles, their direct bandgap (2.05 eV - 2.48 eV) and local surface plasmon resonance (420 nm - 490 nm) could also be tuned. These nanoparticles are examined as SERS substrates, where their enhancement factors, limit of detection for methylene blue, and SERS substrate homogeneity have been tested. It has been observed the nanoparticles synthesized using unirradiated plant extracts present an enhancement factor of 106\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^6$$\\end{document} with a limit of detection 10-8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{-8}$$\\end{document} M. Whereas nanoparticles with refined morphology and shapes upon irradiation present high enhancement factors of >107\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^7$$\\end{document} and detection limit down to 10-9\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{-9}$$\\end{document} M. In addition, uniformity in Raman spectra over the SERS substrates has been obtained for selected Ag NPs substrates synthesized using irradiated extracts with minimum relative standard deviation in enhancement factor < 12%.

A Product-form Network for Systems with Job Stealing Policies

In queueing networks, product-form solutions are of fundamental importance to efficiently compute performance metrics in complex models of computer systems. The product-form property entails that the steady-state probabilities of the joint stochastic process underlying the network can be expressed as the normalized product of functions that only depend on the local state of the components. In many relevant cases, product-forms are the only way to perform exact quantitative analyses of large systems. In this work, we introduce a novel class of product-form queueing networks where servers are always busy. Applications include model of systems where successive refinements on jobs improve the processes quality but are not strictly required to obtain a result. To this aim, we define a job movement policy that admits instantaneous migrations of jobs from non-empty waiting buffers to empty ones. Thus, the resulting routing scheme is state-dependent. This class of networks maximizes the system throughput. This model can be implemented with arbitrary topology, including feedback, and both in an open and closed setting. As far as closed systems are concerned, we give a convolution algorithm and the corresponding mean value analysis to compute expected performance indices for closed models.

Characterisation of cotadutide's dual GLP-1/glucagon receptor agonistic effects on glycaemic control using an in vivo human glucose regulation quantitative systems pharmacology model.

Cotadutide is a dual GLP-1 and glucagon receptor agonist with balanced agonistic activity at each receptor designed to harness the advantages on promoting liver health, weight loss and glycaemic control. We characterised the effects of cotadutide on glucose, insulin, GLP-1, GIP, and glucagon over time in a quantitative manner using our glucose dynamics systems model (4GI systems model), in combination with clinical data from a multiple ascending dose/Phase 2a (MAD/Ph2a) study in overweight and obese subjects with a history of Type 2 diabetes mellitus (NCT02548585). The cotadutide PK-4GI systems model was calibrated to clinical data by re-estimating only food related parameters. In vivo cotadutide efficacy was scaled based on in vitro potency. The model was used to explore the effect of weight loss on insulin sensitivity and predict the relative contribution of the GLP-1 and glucagon receptor agonistic effects on glucose. Cotadutide MAD/Ph2a clinical endpoints were successfully predicted. The 4GI model captured a positive effect of weight loss on insulin sensitivity and showed that the stimulating effect of glucagon on glucose production counteracts the GLP-1 receptor-mediated decrease in glucose, resulting in a plateau for glucose decrease around a 200-μg cotadutide dose. The 4GI quantitative systems pharmacology model was able to predict the clinical effects of cotadutide on glucose, insulin, GLP-1, glucagon and GIP given known in vitro potency. The analyses demonstrated that the quantitative systems pharmacology model, and its successive refinements, will be a valuable tool to support the clinical development of cotadutide and related compounds.

Analyzing Tourism Online Reviews

Tourism managers are increasingly turning to the online sphere to gain relevant customer insights. However, current approaches to analyzing vast and rapidly changing user-generated content (UGC) face several limitations. Supervised approaches require significant effort to provide pre-tagged training data and cannot dynamically identify topics mentioned in UGC. On the other hand, unsupervised approaches typically do not support different abstraction levels or enable a successive refinement of analysis in a drill-down manner, which is often expected as a practical requirement of tourism and destination management. Our research objective is, therefore, to extend current supervised approaches for identifying predefined topics by adopting unsupervised approaches using cluster analysis. The results emphasize that unsupervised approaches can (1) detect non-predefined topics dynamically with an accuracy similar to supervised approaches, thus demonstrating the potential to replace them and avoid the necessity of providing pre-tagged training data. (2) To build a topic hierarchy, unsupervised approaches sense more fine-grained topics as an enhancement of predefined topics on a lower level of abstraction, enabling more powerful drill-down-like analyses. Overall, the proposed extended approach to topic detection promises to support tourism management by meaningfully analyzing the increasing mass of visitors’ online feedback.

Outage Probability Analysis of Cooperative NOMA With Successive Refinement

Outage Probability Analysis of Cooperative NOMA With Successive Refinement

DoA Estimation for Hybrid Receivers: Full Spatial Coverage and Successive Refinement

DoA Estimation for Hybrid Receivers: Full Spatial Coverage and Successive Refinement

Comparison of different pilot point parameterization strategies when measurements are unevenly distributed in space

The parameterization of spatially distributed hydraulic properties is one of the most crucial steps in groundwater modeling. A common approach is to estimate hydraulic properties at a set of pilot points and interpolate the values at each model cell. Despite the popularity of this method, several questions remain about the optimum number and distribution of pilot points, which are determining factors for the efficiency of the method. This study proposes a strategy for optimal pilot point parameterization that minimizes the number of parameters while maximizing the assimilation of an observed dataset unevenly distributed in space. The performance of different pilot point distributions has been compared with a synthetic groundwater model, considering regular grids of pilot points with different spacings and adaptive grids with different refinement criteria. This work considered both prior and iterative refinements, with a parameter estimation step between successive refinements. The parameter estimation was conducted with the Gauss–Levenberg–Marquardt algorithm, and the strategies were ranked according to the number of model calls to reach the target objective function. The strategy leading to the best fit with the measurement dataset at the minimum computational burden is an adaptive grid of pilot points with prior refinement based on measurement density. This strategy was successfully implemented on a regional, multilayered groundwater flow model in the south-western geological basin of France.

Microstructural and Thermal Characterization of AZ31 and Ca-modified AZ31 Magnesium Alloys: Insight into Dynamic Recrystallization and Discontinuous Precipitation Phenomena

Magnesium alloys are highly desirable for weight critical applications owing to their high weight to strangth ratio. However, their poor formability at room temperature limits their widespread use in industrial applications. In this study, we invstigate the hot deformation behaviour of AZ31 and AZ31-0.7% Ca magnesium alloys and explore their microstructural and thermal properties. Our findings reveal that dynamic recrystallization during hot deformation leads to successful grain refinement in the AZ31 alloy, resulting in a normal grain size distribution. In contrast, the AZ31-0.7% Ca alloy shows bimodal grain size distribution due to the addition of calcium. Additionally, the number and size of β-Mg17Al12 particles were found to increase with the addition of a small amount of calcium. These particles are responsible for the discontinuous precipitation phenomenon, which strongly influences microstructural changes during hot rolling. Our study provides valuable insights into the dynamic recrystallization and discontinuous precipitation phenomena of magnesium alloys, which can aid in the development of novel alloys with improved formability and mechanical properties.

Wind Tunnel and Grid Resolution Effects in Large-Eddy Simulations of the High-Lift Common Research Model

This paper describes the application of a compressible large-eddy simulation (LES) solver to the flow over the High-Lift Common Research Model (CRM-HL) in landing configuration, a complex external aerodynamic flow configuration with deployed slats, flaps, a flow-through nacelle, and the associated brackets/fairings on the high-lift devices. The bulk Mach number is (0.2) and the mean-aerodynamic-chord-based Reynolds number is typical of a wind tunnel experiment (5.49E6). A key development in these simulations relative to previous work is that this work serves to establish robustness of LES methods to Reynolds number and aircraft configuration. Previous studies focused on the Japan Aerospace Exploration Agency Standard Model, which featured a nearly 3× lower Reynolds number than the present CRM-HL investigation and less aggressive wing/slat leading-edge curvature than the CRM-HL geometry, which led to lower leading-edge pressure suction peak magnitudes, both of which led to less stringent grid resolution requirements. In these LES simulations, an algebraic equilibrium wall modeling approach is employed along with a dynamic implementation of the Smagorinsky subgrid-scale model. The calculations are carried out in both a free air setting and one that includes the wind tunnel facility at seven angles of attack at five grid resolution levels, ranging from ≈10 to 1500 million control volumes. In free air, the solutions show decreasing sensitivity to the grid with each successive refinement level and systematically approach the experimental lift coefficient data as the grid is refined, with the 1.5 billion control volume case showing excellent agreement with the corrected experimental data. The simulations in both free air and in the wind tunnel predict a stall mechanism featuring a large inboard juncture stall and a nose-down break in the pitching moment curve, both of which agree with the experimental observations. The accuracy of the simulations is assessed via comparisons of integrated forces/moments, surface pressures, and surface skin friction visualizations. Graphics processing unit (GPU) acceleration of the charLES solver results in tractable turnaround times that make LES a useful tool in the aerospace industry design cycle. Recent GPU acceleration of the flow solver has made LES solutions that are highly accurate in lift/drag/moment for relevant high-lift aircraft flows achievable within about 5 h of wall time on 600 GPU cores.

Exact and Soft Successive Refinement of the Information Bottleneck.

The information bottleneck (IB) framework formalises the essential requirement for efficient information processing systems to achieve an optimal balance between the complexity of their representation and the amount of information extracted about relevant features. However, since the representation complexity affordable by real-world systems may vary in time, the processing cost of updating the representations should also be taken into account. A crucial question is thus the extent to which adaptive systems can leverage the information content of already existing IB-optimal representations for producing new ones, which target the same relevant features but at a different granularity. We investigate the information-theoretic optimal limits of this process by studying and extending, within the IB framework, the notion of successive refinement, which describes the ideal situation where no information needs to be discarded for adapting an IB-optimal representation's granularity. Thanks in particular to a new geometric characterisation, we analytically derive the successive refinability of some specific IB problems (for binary variables, for jointly Gaussian variables, and for the relevancy variable being a deterministic function of the source variable), and provide a linear-programming-based tool to numerically investigate, in the discrete case, the successive refinement of the IB. We then soften this notion into a quantification of the loss of information optimality induced by several-stage processing through an existing measure of unique information. Simple numerical experiments suggest that this quantity is typically low, though not entirely negligible. These results could have important implications for (i) the structure and efficiency of incremental learning in biological and artificial agents, (ii) the comparison of IB-optimal observation channels in statistical decision problems, and (iii) the IB theory of deep neural networks.