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1. Qualitative inverse problems: mapping data to the features of trajectories and parameter values of an ODE model

   https://doi.org/10.1088/1361-6420/acd414  

   Duan, X. ; Rubin, J. E. ; Swigon, D. ( May 2023 , Inverse Problems)

   Our recent work on linear and affine dynamical systems has laid out a general framework for inferring the parameters of a differential equation model from a discrete set of data points collected from a system being modeled. It introduced a new class of inverse problems where qualitative information about the parameters and the associated dynamics of the system is determined for regions of the data space, rather than just for isolated experiments. Rigorous mathematical results have justified this approach and have identified common features that arise for certain classes of integrable models. In this work we present a thorough numerical investigation that shows that several of these core features extend to a paradigmatic linear-in-parameters model, the Lotka–Volterra (LV) system, which we consider in the conservative case as well as under the addition of terms that perturb the system away from this regime. A central construct for this analysis is a concise representation of parameter and dynamical features in the data space that we call theP_n-diagram, which is particularly useful for visualization of the qualitative dependence of the system dynamics on data for low-dimensional (smalln) systems. Our work also exposes some new properties related to non-uniqueness that arise for these LV systems, with non-uniqueness manifesting as a multi-layered structure in the associatedP₂-diagrams.

    

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2. PreQual: An automated pipeline for integrated preprocessing and quality assurance of diffusion weighted MRI images

   https://doi.org/10.1002/mrm.28678  

   Cai, Leon Y. ; Yang, Qi ; Hansen, Colin B. ; Nath, Vishwesh ; Ramadass, Karthik ; Johnson, Graham W. ; Conrad, Benjamin N. ; Boyd, Brian D. ; Begnoche, John P. ; Beason‐Held, Lori L. ; et al ( February 2021 , Magnetic Resonance in Medicine)

   Diffusion weighted MRI imaging (DWI) is often subject to low signal‐to‐noise ratios (SNRs) and artifacts. Recent work has produced software tools that can correct individual problems, but these tools have not been combined with each other and with quality assurance (QA). A single integrated pipeline is proposed to perform DWI preprocessing with a spectrum of tools and produce an intuitive QA document.

   The proposed pipeline, built around the FSL, MRTrix3, and ANTs software packages, performs DWI denoising; inter‐scan intensity normalization; susceptibility‐, eddy current‐, and motion‐induced artifact correction; and slice‐wise signal drop‐out imputation. To perform QA on the raw and preprocessed data and each preprocessing operation, the pipeline documents qualitative visualizations, quantitative plots, gradient verifications, and tensor goodness‐of‐fit and fractional anisotropy analyses.

   Raw DWI data were preprocessed and quality checked with the proposed pipeline and demonstrated improved SNRs; physiologic intensity ratios; corrected susceptibility‐, eddy current‐, and motion‐induced artifacts; imputed signal‐lost slices; and improved tensor fits. The pipeline identified incorrect gradient configurations and file‐type conversion errors and was shown to be effective on externally available datasets.

   The proposed pipeline is a single integrated pipeline that combines established diffusion preprocessing tools from major MRI‐focused software packages with intuitive QA.

    

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3. Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

   https://doi.org/10.1115/1.4046831  

   Patnaik, Sansit ; Semperlotti, Fabio ( September 2020 , Journal of Computational and Nonlinear Dynamics)

   Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems. 

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4. Analysis of an approximation to a fractional extension problem

   https://doi.org/10.1007/s10543-019-00787-y  

   Padgett, Joshua L. ( November 2019 , BIT Numerical Mathematics)

   The purpose of this article is to study an approximation to an abstract Bessel-type problem, which is a generalization of the extension problem associated with fractional powers of the Laplace operator. Motivated by the success of such approaches in the analysis of time-stepping methods for abstract Cauchy problems, we adopt a similar framework herein. The proposed method differs from many standard techniques, as we approximate the true solution to the abstract problem, rather than solve an associated discrete problem. The numerical method is shown to be consistent, stable, and convergent in an appropriate Banach space. These results are built upon well understood results from semigroup theory. Numerical experiments are provided to demonstrate the theoretical results. 

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5. Experimental and modeling studies of IPDI ‐based polyurea elastomers – The role of hard segment fraction

   https://doi.org/10.1002/app.53592  

   Tzelepis, Demetrios A. ; Suzuki, Jorge ; Su, Yi Feng ; Wang, Yiyu ; Lim, Yong Chae ; Zayernouri, Mohsen ; Ginzburg, Valeriy V. ( January 2023 , Journal of Applied Polymer Science)

   Segmented polyureas (PUa) are industrially important class of polymers widely used in coatings, sealant, and adhesive applications. Here, we report synthesis, characterization, and modeling of Isophorone Diisocyanate‐Diethyl‐Toluene‐Diamine‐Polyether amine (IPDI‐DETDA‐PO PUa) with varied hard segment contents of 20, 30, and 40 weight percent. For each of the three materials, we study its structure and phase behavior using FTIR, DSC, and TEM, and clearly show the presence of microphase separation between the hard and soft nanodomains. We then measure the linear viscoelastic response of the PUa‐s using DMA (frequency sweeps at multiple temperatures). The DMA data are shown to obey the time‐temperature superposition. Finally, we develop a new micromechanical model describing the DMA results; the model describes a phase‐separated PUa as two “Fractional‐order Maxwell gels” branches, connected in parallel, with the first FMG branch representing the “percolated hard phase and the second one modeling the “filled soft phase. In agreement with the earlier thermodynamic theories, the volume‐fraction of the percolated hard phase is related to the hard segment weight‐fraction (HSWF), defined as the combined mass of IPDI and DETDA normalized to the total mass of the polymer. The data and model are found to be in a good qualitative and quantitative agreement.

    

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