Many physical problems involving heterogeneous spatial scales, such as the flow through fractured porous media, the study of fiber-reinforced materials, or the modeling of the small circulation in living tissues -- just to mention a few examples -- can be described as coupled partial differential equations defined in domains of heterogeneous dimensions that are embedded into each other. This formulation is a consequence of geometric model reduction techniques that transform the original problems defined in complex three-dimensional domains into more tractable ones. The definition and the approximation of coupling operators suitable for this class of problems is still a challenge. We develop a general mathematical framework for the analysis and the approximation of partial differential equations coupled by non-matching constraints across different dimensions, focusing on their enforcement using Lagrange multipliers. In this context, we address in abstract and general terms the well-posedness, stability, and robustness of the problem with respect to the smallest characteristic length of the embedded domain. We also address the numerical approximation of the problem and we discuss the inf-sup stability of the proposed numerical scheme for some representative configuration of the embedded domain. The main message of this work is twofold: from the standpoint of the theory of mixed-dimensional problems, we provide general and abstract mathematical tools to formulate coupled problems across dimensions. From the practical standpoint of the numerical approximation, we show the interplay between the mesh characteristic size, the dimension of the Lagrange multiplier space, and the size of the inclusion in representative configurations interesting for applications. The latter analysis is complemented with illustrative numerical examples.
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The current study investigates the asymptotic spectral properties of a finite difference approximation of nonlocal Helmholtz equations with a Caputo fractional Laplacian and a variable coefficient wave number $\mu$, as it occurs when considering a wave propagation in complex media, characterized by nonlocal interactions and spatially varying wave speeds. More specifically, by using tools from Toeplitz and generalized locally Toeplitz theory, the present research delves into the spectral analysis of nonpreconditioned and preconditioned matrix-sequences. We report numerical evidences supporting the theoretical findings. Finally, open problems and potential extensions in various directions are presented and briefly discussed.
We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.
We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $\tilde{O}(s \log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $\tilde{O}(sD)$ or $\tilde{O}(s^2)$ bit complexity.
Under a generalised estimating equation analysis approach, approximate design theory is used to determine Bayesian D-optimal designs. For two examples, considering simple exchangeable and exponential decay correlation structures, we compare the efficiency of identified optimal designs to balanced stepped-wedge designs and corresponding stepped-wedge designs determined by optimising using a normal approximation approach. The dependence of the Bayesian D-optimal designs on the assumed correlation structure is explored; for the considered settings, smaller decay in the correlation between outcomes across time periods, along with larger values of the intra-cluster correlation, leads to designs closer to a balanced design being optimal. Unlike for normal data, it is shown that the optimal design need not be centro-symmetric in the binary outcome case. The efficiency of the Bayesian D-optimal design relative to a balanced design can be large, but situations are demonstrated in which the advantages are small. Similarly, the optimal design from a normal approximation approach is often not much less efficient than the Bayesian D-optimal design. Bayesian D-optimal designs can be readily identified for stepped-wedge cluster randomised trials with binary outcome data. In certain circumstances, principally ones with strong time period effects, they will indicate that a design unlikely to have been identified by previous methods may be substantially more efficient. However, they require a larger number of assumptions than existing optimal designs, and in many situations existing theory under a normal approximation will provide an easier means of identifying an efficient design for binary outcome data.
Mixed-signal neuromorphic processors provide extremely low-power operation for edge inference workloads, taking advantage of sparse asynchronous computation within Spiking Neural Networks (SNNs). However, deploying robust applications to these devices is complicated by limited controllability over analog hardware parameters, as well as unintended parameter and dynamical variations of analog circuits due to fabrication non-idealities. Here we demonstrate a novel methodology for ofDine training and deployment of spiking neural networks (SNNs) to the mixed-signal neuromorphic processor DYNAP-SE2. The methodology utilizes gradient-based training using a differentiable simulation of the mixed-signal device, coupled with an unsupervised weight quantization method to optimize the network's parameters. Parameter noise injection during training provides robustness to the effects of quantization and device mismatch, making the method a promising candidate for real-world applications under hardware constraints and non-idealities. This work extends Rockpool, an open-source deep-learning library for SNNs, with support for accurate simulation of mixed-signal SNN dynamics. Our approach simplifies the development and deployment process for the neuromorphic community, making mixed-signal neuromorphic processors more accessible to researchers and developers.
Untargeted metabolomic profiling through liquid chromatography-mass spectrometry (LC-MS) measures a vast array of metabolites within biospecimens, advancing drug development, disease diagnosis, and risk prediction. However, the low throughput of LC-MS poses a major challenge for biomarker discovery, annotation, and experimental comparison, necessitating the merging of multiple datasets. Current data pooling methods encounter practical limitations due to their vulnerability to data variations and hyperparameter dependence. Here we introduce GromovMatcher, a flexible and user-friendly algorithm that automatically combines LC-MS datasets using optimal transport. By capitalizing on feature intensity correlation structures, GromovMatcher delivers superior alignment accuracy and robustness compared to existing approaches. This algorithm scales to thousands of features requiring minimal hyperparameter tuning. Applying our method to experimental patient studies of liver and pancreatic cancer, we discover shared metabolic features related to patient alcohol intake, demonstrating how GromovMatcher facilitates the search for biomarkers associated with lifestyle risk factors linked to several cancer types.
Although deep neural networks yield high classification accuracy given sufficient training data, their predictions are typically overconfident or under-confident, i.e., the prediction confidences cannot truly reflect the accuracy. Post-hoc calibration tackles this problem by calibrating the prediction confidences without re-training the classification model. However, current approaches assume congruence between test and validation data distributions, limiting their applicability to out-of-distribution scenarios. To this end, we propose a novel meta-set-based cascaded temperature regression method for post-hoc calibration. Our method tailors fine-grained scaling functions to distinct test sets by simulating various domain shifts through data augmentation on the validation set. We partition each meta-set into subgroups based on predicted category and confidence level, capturing diverse uncertainties. A regression network is then trained to derive category-specific and confidence-level-specific scaling, achieving calibration across meta-sets. Extensive experimental results on MNIST, CIFAR-10, and TinyImageNet demonstrate the effectiveness of the proposed method.
Following initial work by JaJa and Ahlswede/Cai, and inspired by a recent renewed surge in interest in deterministic identification via noisy channels, we consider the problem in its generality for memoryless channels with finite output, but arbitrary input alphabets. Such a channel is essentially given by (the closure of) the subset of its output distributions in the probability simplex. Our main findings are that the maximum number of messages thus identifiable scales super-exponentially as $2^{R\,n\log n}$ with the block length $n$, and that the optimal rate $R$ is upper and lower bounded in terms of the covering (aka Minkowski, or Kolmogorov, or entropy) dimension $d$ of the output set: $\frac14 d \leq R \leq d$. Leading up to the general case, we treat the important special case of the so-called Bernoulli channel with input alphabet $[0;1]$ and binary output, which has $d=1$, to gain intuition. Along the way, we show a certain Hypothesis Testing Lemma (generalising an earlier insight of Ahlswede regarding the intersection of typical sets) that implies that for the construction of a deterministic identification code, it is sufficient to ensure pairwise reliable distinguishability of the output distributions. These results are then shown to generalise directly to classical-quantum channels with finite-dimensional output quantum system (but arbitrary input alphabet), and in particular to quantum channels on finite-dimensional quantum systems under the constraint that the identification code can only use tensor product inputs.
The volumetric representation of human interactions is one of the fundamental domains in the development of immersive media productions and telecommunication applications. Particularly in the context of the rapid advancement of Extended Reality (XR) applications, this volumetric data has proven to be an essential technology for future XR elaboration. In this work, we present a new multimodal database to help advance the development of immersive technologies. Our proposed database provides ethically compliant and diverse volumetric data, in particular 27 participants displaying posed facial expressions and subtle body movements while speaking, plus 11 participants wearing head-mounted displays (HMDs). The recording system consists of a volumetric capture (VoCap) studio, including 31 synchronized modules with 62 RGB cameras and 31 depth cameras. In addition to textured meshes, point clouds, and multi-view RGB-D data, we use one Lytro Illum camera for providing light field (LF) data simultaneously. Finally, we also provide an evaluation of our dataset employment with regard to the tasks of facial expression classification, HMDs removal, and point cloud reconstruction. The dataset can be helpful in the evaluation and performance testing of various XR algorithms, including but not limited to facial expression recognition and reconstruction, facial reenactment, and volumetric video. HEADSET and its all associated raw data and license agreement will be publicly available for research purposes.
This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of polynomials based on derivative samples is introduced, which is different from the Euler-Frobenius polynomials for the multiplicity $r>1$. A complete characterization of uniform sampling with derivatives is given using Laurent operators. The rate of approximation of a signal (not necessarily continuous) by the derivative sampling expansions in shift-invariant spaces generated by compactly supported functions is established in terms of $L^p$- average modulus of smoothness. Finally, several typical examples illustrating the various problems are discussed in detail.
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