Modeling the behavior of biological tissues and organs often necessitates the knowledge of their shape in the absence of external loads. However, when their geometry is acquired in-vivo through imaging techniques, bodies are typically subject to mechanical deformation due to the presence of external forces, and the load-free configuration needs to be reconstructed. This paper addresses this crucial and frequently overlooked topic, known as the inverse elasticity problem (IEP), by delving into both theoretical and numerical aspects, with a particular focus on cardiac mechanics. In this work, we extend Shield's seminal work to determine the structure of the IEP with arbitrary material inhomogeneities and in the presence of both body and active forces. These aspects are fundamental in computational cardiology, and we show that they may break the variational structure of the inverse problem. In addition, we show that the inverse problem might have no solution even in the presence of constant Neumann boundary conditions and a polyconvex strain energy functional. We then present the results of extensive numerical tests to validate our theoretical framework, and to characterize the computational challenges associated with a direct numerical approximation of the IEP. Specifically, we show that this framework outperforms existing approaches both in terms of robustness and optimality, such as Sellier's iterative procedure, even when the latter is improved with acceleration techniques. A notable discovery is that multigrid preconditioners are, in contrast to standard elasticity, not efficient, where a one-level additive Schwarz and generalized Dryja-Smith-Widlund provide a much more reliable alternative. Finally, we successfully address the IEP for a full-heart geometry, demonstrating that the IEP formulation can compute the stress-free configuration in real-life scenarios.
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In longitudinal observational studies, marginal structural models (MSMs) are a class of causal models used to analyze the effect of an exposure on the (survival) outcome of interest while accounting for exposure-affected time-dependent confounding. In the applied literature, inverse probability of treatment weighting (IPTW) has been widely adopted to estimate MSMs. An essential assumption for IPTW-based MSMs is the positivity assumption, which ensures that each individual in the population has a non-zero probability of receiving each exposure level within confounder strata. Positivity, along with consistency, conditional exchangeability, and correct specification of the weighting model, is crucial for valid causal inference through IPTW-based MSMs but is often overlooked compared to confounding bias. Positivity violations can arise from subjects having a zero probability of being exposed/unexposed (strict violations) or near-zero probabilities due to sampling variability (near violations). This article discusses the effect of violations in the positivity assumption on the estimates from IPTW-based MSMs. Building on the algorithms for simulating longitudinal survival data from MSMs by Havercroft and Didelez (2012) and Keogh et al. (2021), systematic simulations under strict/near positivity violations are performed. Various scenarios are explored by varying (i) the size of the confounder interval in which positivity violations arise, (ii) the sample size, (iii) the weight truncation strategy, and (iv) the subject's propensity to follow the protocol violation rule. This study underscores the importance of assessing positivity violations in IPTW-based MSMs to ensure robust and reliable causal inference in survival analyses.
We describe fast algorithms for approximating the connection coefficients between a family of orthogonal polynomials and another family with a polynomially or rationally modified measure. The connection coefficients are computed via infinite-dimensional banded matrix factorizations and may be used to compute the modified Jacobi matrices all in linear complexity with respect to the truncation degree. A family of orthogonal polynomials with modified classical weights is constructed that support banded differentiation matrices, enabling sparse spectral methods with modified classical orthogonal polynomials.
A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented.
We address the problem of the best uniform approximation of a continuous function on a convex domain. The approximation is by linear combinations of a finite system of functions (not necessarily Chebyshev) under arbitrary linear constraints. By modifying the concept of alternance and of the Remez iterative procedure we present a method, which demonstrates its efficiency in numerical problems. The linear rate of convergence is proved under some favourable assumptions. A special attention is paid to systems of complex exponents, Gaussian functions, lacunar algebraic and trigonometric polynomials. Applications to signal processing, linear ODE, switching dynamical systems, and to Markov-Bernstein type inequalities are considered.
We describe a family of iterative algorithms that involve the repeated execution of discrete and inverse discrete Fourier transforms. One interesting member of this family is motivated by the discrete Fourier transform uncertainty principle and involves the application of a sparsification operation to both the real domain and frequency domain data with convergence obtained when real domain sparsity hits a stable pattern. This sparsification variant has practical utility for signal denoising, in particular the recovery of a periodic spike signal in the presence of Gaussian noise. General convergence properties and denoising performance relative to existing methods are demonstrated using simulation studies. An R package implementing this technique and related resources can be found at //hrfrost.host.dartmouth.edu/IterativeFT.
The importance of considering contextual probabilities in shaping response patterns within psychological testing is underscored, despite the ubiquitous nature of order effects discussed extensively in methodological literature. Drawing from concepts such as path-dependency, first-order autocorrelation, state-dependency, and hysteresis, the present study is an attempt to address how earlier responses serve as an anchor for subsequent answers in tests, surveys, and questionnaires. Introducing the notion of non-commuting observables derived from quantum physics, I highlight their role in characterizing psychological processes and the impact of measurement instruments on participants' responses. We advocate for the utilization of first-order Markov chain modeling to capture and forecast sequential dependencies in survey and test responses. The employment of the first-order Markov chain model lies in individuals' propensity to exhibit partial focus to preceding responses, with recent items most likely exerting a substantial influence on subsequent response selection. This study contributes to advancing our understanding of the dynamics inherent in sequential data within psychological research and provides a methodological framework for conducting longitudinal analyses of response patterns of test and questionnaire.
The timely detection of disease outbreaks through reliable early warning signals (EWSs) is indispensable for effective public health mitigation strategies. Nevertheless, the intricate dynamics of real-world disease spread, often influenced by diverse sources of noise and limited data in the early stages of outbreaks, pose a significant challenge in developing reliable EWSs, as the performance of existing indicators varies with extrinsic and intrinsic noises. Here, we address the challenge of modeling disease when the measurements are corrupted by additive white noise, multiplicative environmental noise, and demographic noise into a standard epidemic mathematical model. To navigate the complexities introduced by these noise sources, we employ a deep learning algorithm that provides EWS in infectious disease outbreak by training on noise-induced disease-spreading models. The indicator's effectiveness is demonstrated through its application to real-world COVID-19 cases in Edmonton and simulated time series derived from diverse disease spread models affected by noise. Notably, the indicator captures an impending transition in a time series of disease outbreaks and outperforms existing indicators. This study contributes to advancing early warning capabilities by addressing the intricate dynamics inherent in real-world disease spread, presenting a promising avenue for enhancing public health preparedness and response efforts.
We propose a method for obtaining parsimonious decompositions of networks into higher order interactions which can take the form of arbitrary motifs.The method is based on a class of analytically solvable generative models, where vertices are connected via explicit copies of motifs, which in combination with non-parametric priors allow us to infer higher order interactions from dyadic graph data without any prior knowledge on the types or frequencies of such interactions. Crucially, we also consider 'degree--corrected' models that correctly reflect the degree distribution of the network and consequently prove to be a better fit for many real world--networks compared to non-degree corrected models. We test the presented approach on simulated data for which we recover the set of underlying higher order interactions to a high degree of accuracy. For empirical networks the method identifies concise sets of atomic subgraphs from within thousands of candidates that cover a large fraction of edges and include higher order interactions of known structural and functional significance. The method not only produces an explicit higher order representation of the network but also a fit of the network to analytically tractable models opening new avenues for the systematic study of higher order network structures.
Decision making and learning in the presence of uncertainty has attracted significant attention in view of the increasing need to achieve robust and reliable operations. In the case where uncertainty stems from the presence of adversarial attacks this need is becoming more prominent. In this paper we focus on linear and nonlinear classification problems and propose a novel adversarial training method for robust classifiers, inspired by Support Vector Machine (SVM) margins. We view robustness under a data driven lens, and derive finite sample complexity bounds for both linear and non-linear classifiers in binary and multi-class scenarios. Notably, our bounds match natural classifiers' complexity. Our algorithm minimizes a worst-case surrogate loss using Linear Programming (LP) and Second Order Cone Programming (SOCP) for linear and non-linear models. Numerical experiments on the benchmark MNIST and CIFAR10 datasets show our approach's comparable performance to state-of-the-art methods, without needing adversarial examples during training. Our work offers a comprehensive framework for enhancing binary linear and non-linear classifier robustness, embedding robustness in learning under the presence of adversaries.
Artificial neural networks thrive in solving the classification problem for a particular rigid task, acquiring knowledge through generalized learning behaviour from a distinct training phase. The resulting network resembles a static entity of knowledge, with endeavours to extend this knowledge without targeting the original task resulting in a catastrophic forgetting. Continual learning shifts this paradigm towards networks that can continually accumulate knowledge over different tasks without the need to retrain from scratch. We focus on task incremental classification, where tasks arrive sequentially and are delineated by clear boundaries. Our main contributions concern 1) a taxonomy and extensive overview of the state-of-the-art, 2) a novel framework to continually determine the stability-plasticity trade-off of the continual learner, 3) a comprehensive experimental comparison of 11 state-of-the-art continual learning methods and 4 baselines. We empirically scrutinize method strengths and weaknesses on three benchmarks, considering Tiny Imagenet and large-scale unbalanced iNaturalist and a sequence of recognition datasets. We study the influence of model capacity, weight decay and dropout regularization, and the order in which the tasks are presented, and qualitatively compare methods in terms of required memory, computation time, and storage.
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