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    The contagion number: How fast can a disease spread?
    (National Library of Serbia, 2023) Blessley, Misty; Davila, Randy; Hale, Trevor; Pepper, Ryan
    The burning number of a graph models the rate at which a disease, information, or other externality can propagate across a network. The burning number is known to be NP-hard even for a tree. Herein, we define a relative of the burning number that we coin the contagion number (CN). We aver that the CN is a better metric to model disease spread than the burning number as it only counts first time infections (i.e., constrains a node from getting the same disease/same variant/same alarm more than once). This is important because the Centers for Disease Control and Prevention report that COVID-19 reinfections are rare. This paper delineates a method to solve for the contagion number of any tree, in polynomial time, which addresses how fast a disease could spread (i.e., a worst-cast analysis) and then employs simulation to determine the average contagion number (ACN) (i.e., a most-likely analysis) of how fast a disease would spread. The latter is analyzed on scale-free graphs, which are used to model human social networks generated through a preferential attachment mechanism. With CN differing across network structures and almost identical to ACN, our findings advance disease spread understanding and reveal the importance of network structure. In a borderless world without replete resources, understanding disease spread can do much to inform public policy and managerial decision makers’ allocation decisions. Furthermore, our direct interactions with supply chain executives at two COVID-19 vaccine developers provided practical grounding on what the results suggest for achieving social welfare objectives.
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    Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source
    (EDP Sciences, 2023) Masri, Rami; Shen, Boqian; Riviere, Beatrice
    The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.
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    Quantitative unique continuation for the elasticity system with application to the kinematic inverse rupture problem
    (Taylor & Francis, 2023) de Hoop, Maarten V.; Lassas, Matti; Lu, Jinpeng; Oksanen, Lauri
    We obtain explicit estimates on the stability of the unique continuation for a linear system of hyperbolic equations. In particular, our result applies to the elasticity system and also the Maxwell system. As an application, we study the kinematic inverse rupture problem of determining the jump in displacement and the friction force at the rupture surface, and we obtain new features on the stable unique continuation up to the rupture surface.
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    Multi-Patch Epidemic Models with General Exposed and Infectious Periods
    (EDP Sciences, 2023) Pang, Guodong; Pardoux, Étienne
    We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as “from distance” from all the other patches. The migration processes among the patches in either of the four states are assumed to be Markovian, and independent of the exposed and infectious periods. These periods have general distributions, and are not affected by the possible migrations of the individuals. The infection “from distance” aspect introduces a new formulation of the infection process, which, together with the migration processes, brings technical challenges in proving the functional limit theorems. Generalizing the methods in Pang and Pardoux [Ann. Appl. Probab. 32 (2022) 1615–1665], we establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.
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    Conditional Injective Flows for Bayesian Imaging
    (IEEE, 2023) Khorashadizadeh, AmirEhsan; Kothari, Konik; Salsi, Leonardo; Harandi, Ali Aghababaei; de Hoop, Maarten; Dokmanić, Ivan
    Most deep learning models for computational imaging regress a single reconstructed image. In practice, however, ill-posedness, nonlinearity, model mismatch, and noise often conspire to make such point estimates misleading or insufficient. The Bayesian approach models images and (noisy) measurements as jointly distributed random vectors and aims to approximate the posterior distribution of unknowns. Recent variational inference methods based on conditional normalizing flows are a promising alternative to traditional MCMC methods, but they come with drawbacks: excessive memory and compute demands for moderate to high resolution images and underwhelming performance on hard nonlinear problems. In this work, we propose C-Trumpets—conditional injective flows specifically designed for imaging problems, which greatly diminish these challenges. Injectivity reduces memory footprint and training time while low-dimensional latent space together with architectural innovations like fixed-volume-change layers and skip-connection revnet layers, C-Trumpets outperform regular conditional flow models on a variety of imaging and image restoration tasks, including limited-view CT and nonlinear inverse scattering, with a lower compute and memory budget. C-Trumpets enable fast approximation of point estimates like MMSE or MAP as well as physically-meaningful uncertainty quantification.
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    Optimized decision support for selection of transoral robotic surgery or (chemo)radiation therapy based on posttreatment swallowing toxicity
    (Wiley, 2023) Hemmati, Mehdi; Barbon, Carly; Mohamed, Abdallah S.R.; van Dijk, Lisanne V.; Moreno, Amy C.; Gross, Neil D.; Goepfert, Ryan P.; Lai, Stephen Y.; Hutcheson, Katherine A.; Schaefer, Andrew J.; Fuller, Clifton D.
    Background A primary goal in transoral robotic surgery (TORS) for oropharyngeal squamous cell cancer (OPSCC) survivors is to optimize swallowing function. However, the uncertainty in the outcomes of TORS including postoperative residual positive margin (PM) and extranodal extension (ENE), may necessitate adjuvant therapy, which may cause significant swallowing toxicity to survivors. Methods A secondary analysis was performed on a prospective registry data with low- to intermediate-risk human papillomavirus–related OPSCC possibly resectable by TORS. Decision trees were developed to model the uncertainties in TORS compared with definitive radiation therapy (RT) and chemoradiation therapy (CRT). Swallowing toxicities were measured by Dynamic Imaging Grade of Swallowing Toxicity (DIGEST), MD Anderson Dysphagia Inventory (MDADI), and the MD Anderson Symptom Inventory–Head and Neck (MDASI-HN) instruments. The likelihoods of PM/ENE were varied to determine the thresholds within which each therapy remains optimal. Results Compared with RT, TORS resulted in inferior swallowing function for moderate likelihoods of PM/ENE (>60% in short term for all instruments, >75% in long term for DIGEST and MDASI) leaving RT as the optimal treatment. Compared with CRT, TORS remained the optimal therapy based on MDADI and MDASI but showed inferior swallowing outcomes based on DIGEST for moderate-to-high likelihoods of PM/ENE (>75% for short-term and >40% for long-term outcomes). Conclusion In the absence of reliable estimation of postoperative PM/ENE concurrent with significant postoperative PM, the overall toxicity level in OPSCC patients undergoing TORS with adjuvant therapy may become more severe compared with patients receiving nonsurgical treatments thus advocating definitive (C)RT protocols.
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    On the Entropy Projection and the Robustness of High Order Entropy Stable Discontinuous Galerkin Schemes for Under-Resolved Flows
    (Frontiers Media S.A., 2022) Chan, Jesse; Ranocha, Hendrik; Rueda-Ramírez, Andrés M.; Gassner, Gregor; Warburton, Tim
    High order entropy stable schemes provide improved robustness for computational simulations of fluid flows. However, additional stabilization and positivity preserving limiting can still be required for variable-density flows with under-resolved features. We demonstrate numerically that entropy stable Discontinuous Galerkin (DG) methods which incorporate an “entropy projection” are less likely to require additional limiting to retain positivity for certain types of flows. We conclude by investigating potential explanations for this observed improvement in robustness.
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    MR-Guided Adaptive Radiotherapy for OAR Sparing in Head and Neck Cancers
    (MDPI, 2022) Mulder, Samuel L.; Heukelom, Jolien; McDonald, Brigid A.; Van Dijk, Lisanne; Wahid, Kareem A.; Sanders, Keith; Salzillo, Travis C.; Hemmati, Mehdi; Schaefer, Andrew; Fuller, Clifton D.
    MR-linac devices offer the potential for advancements in radiotherapy (RT) treatment of head and neck cancer (HNC) by using daily MR imaging performed at the time and setup of treatment delivery. This article aims to present a review of current adaptive RT (ART) methods on MR-Linac devices directed towards the sparing of organs at risk (OAR) and a view of future adaptive techniques seeking to improve the therapeutic ratio. This ratio expresses the relationship between the probability of tumor control and the probability of normal tissue damage and is thus an important conceptual metric of success in the sparing of OARs. Increasing spatial conformity of dose distributions to target volume and OARs is an initial step in achieving therapeutic improvements, followed by the use of imaging and clinical biomarkers to inform the clinical decision-making process in an ART paradigm. Pre-clinical and clinical findings support the incorporation of biomarkers into ART protocols and investment into further research to explore imaging biomarkers by taking advantage of the daily MR imaging workflow. A coherent understanding of this road map for RT in HNC is critical for directing future research efforts related to sparing OARs using image-guided radiotherapy (IGRT).
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    Chromosome size affects sequence divergence between species through the interplay of recombination and selection
    (Wiley, 2022) Tigano, Anna; Khan, Ruqayya; Omer, Arina D.; Weisz, David; Dudchenko, Olga; Multani, Asha S.; Pathak, Sen; Behringer, Richard R.; Aiden, Erez L.; Fisher, Heidi; MacManes, Matthew D.; Center for Theoretical and Biological Physics
    The structure of the genome shapes the distribution of genetic diversity and sequence divergence. To investigate how the relationship between chromosome size and recombination rate affects sequence divergence between species, we combined empirical analyses and evolutionary simulations. We estimated pairwise sequence divergence among 15 species from three different mammalian clades—Peromyscus rodents, Mus mice, and great apes—from chromosome-level genome assemblies. We found a strong significant negative correlation between chromosome size and sequence divergence in all species comparisons within the Peromyscus and great apes clades but not the Mus clade, suggesting that the dramatic chromosomal rearrangements among Mus species may have masked the ancestral genomic landscape of divergence in many comparisons. Our evolutionary simulations showed that the main factor determining differences in divergence among chromosomes of different sizes is the interplay of recombination rate and selection, with greater variation in larger populations than in smaller ones. In ancestral populations, shorter chromosomes harbor greater nucleotide diversity. As ancestral populations diverge, diversity present at the onset of the split contributes to greater sequence divergence in shorter chromosomes among daughter species. The combination of empirical data and evolutionary simulations revealed that chromosomal rearrangements, demography, and divergence times may also affect the relationship between chromosome size and divergence, thus deepening our understanding of the role of genome structure in the evolution of species divergence.
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    Severity of radiation pneumonitis, from clinical, dosimetric and biological features: a pilot study
    (Springer Nature, 2020) Aso, Samantha; Navarro‑Martin, Arturo; Castillo, Richard; Padrones, Susana; Castillo, Edward; Montes, Ana; Martínez, José Ignacio; Cubero, Noelia; López, Rosa; Rodríguez, Laura; Palmero, Ramon; Manresa, Federico; Guerrero, Thomas; Molina, María
    Background and objective: Radiation pneumonitis (RP) could be a lethal complication of lung cancer treatment. No reliable predictors of RP severity have been recognized. This prospective pilot study was performed to identify early predictors of high grade lung toxicity and to evaluate clinical, biological or dosimetric features associated with different grades of toxicity. Method: Sixteen patients with non-small cell lung cancer with indication of concurrent chemoradiotherapy using 60 Gy/2 Gy/fraction starting at cycle one of platinum based chemotherapy were included. Bronchoalveolar lavage (BAL), pulmonary function testing (PFT), and 18F-2-fluoro-2-deoxy-D-glucose positron-emission tomography was performed before radiotherapy (RT), after three weeks of treatment, and two months post-RT. For analysis, patients were grouped by grade (low [G1-G2] vs. high [G3-G5]). The two groups were compared to identify predictors of RP. Protein expression BAL and lung tissue metabolism was evaluated in two patients (RP-G1 vs. RP-G3). Categorical variables such as comorbidities, stages and locations were summarized as percentages. Radiation doses, pulmonary function values and time to RP were summarized by medians with ranges or as means with standard deviation. Longitudinal analysis PFT was performed by a T-test. Results: All 16 patients developed RP, as follows: G1 (5 pts; 31.3%); G2 (5 pts; 31.3%); G3 (5 pts; 31.3%); and G5 (1 pts; 6.1%). Patients with high grade RP presented significant decrease (p = 0.02) in diffusing lung capacity for carbon monoxide (DLCO) after three weeks of RT. No correlation between dosimetric values and RP grades was observed. BAL analysis of the selected patients showed that CXCL-1, CD154, IL-1ra, IL-23, MIF, PAI-1 and IFN-γ were overexpressed in the lungs of the RP-G3 patient, even before treatment. The pre-RT SUVmax value in the RP-G3 patient was non-significantly higher than in the patient with RP-G1. Conclusions: RT induces some degree of RP. Our data suggest that decrease in DLCO% is the most sensitive parameter for the early detection of RP. Moreover, we detect biological differences between the two grades of pneumonitis, highlighting the potential value of some cytokines as a prognostic marker for developing high grade lung toxicity. Further multicenter studies with larger sample size are essential to validate these findings.
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    Extubation Failure in Critically Ill COVID-19 Patients: Risk Factors and Impact on In-Hospital Mortality
    (Sage, 2021) Ionescu, Filip; Zimmer, Markie S.; Petrescu, Ioana; Castillo, Edward; Bozyk, Paul; Abbas, Amr; Abplanalp, Lauren; Dogra, Sanjay; Nair, Girish B.
    Purpose:We sought to identify clinical factors that predict extubation failure (reintubation) and its prognostic implications in critically ill COVID-19 patients.Materials and Methods:Retrospective, multi-center cohort study of hospitalized COVID-19 patients. Multivariate competing risk models were employed to explore the rate of reintubation and its determining factors.Results:Two hundred eighty-one extubated patients were included (mean age, 61.0 years [±13.9]; 54.8% male). Reintubation occurred in 93 (33.1%). In multivariate analysis accounting for death, reintubation risk increased with age (hazard ratio [HR] 1.04 per 1-year increase, 95% confidence interval [CI] 1.02 -1.06), vasopressors (HR 1.84, 95% CI 1.04-3.60), renal replacement (HR 2.01, 95% CI 1.22-3.29), maximum PEEP (HR 1.07 per 1-unit increase, 95% CI 1.02 -1.12), paralytics (HR 1.48, 95% CI 1.08-2.25) and requiring more than nasal cannula immediately post-extubation (HR 2.19, 95% CI 1.37-3.50). Reintubation was associated with higher mortality (36.6% vs 2.1%; P < 0.0001) and risk of inpatient death after adjusting for multiple factors (HR 23.2, 95% CI 6.45-83.33). Prone ventilation, corticosteroids, anticoagulation, remdesivir and tocilizumab did not impact the risk of reintubation or death.Conclusions:Up to 1 in 3 critically ill COVID-19 patients required reintubation. Older age, paralytics, high PEEP, need for greater respiratory support following extubation and non-pulmonary organ failure predicted reintubation. Extubation failure strongly predicted adverse outcomes.
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    Functional avoidance-based intensity modulated proton therapy with 4DCT derived ventilation imaging for lung cancer
    (Wiley, 2021) Dougherty, Jingjing M.; Castillo, Edward; Castillo, Richard; Faught, Austin M.; Pepin, Mark; Park, Sean S.; Beltran, Chris J.; Guerrero, Thomas; Grills, Inga; Vinogradskiy, Yevgeniy
    The primary objective is to evaluate the potential dosimetric gains of performing functional avoidance-based proton treatment planning using 4DCT derived ventilation imaging. 4DCT data of 31 patients from a prospective functional avoidance clinical trial were evaluated with intensity modulated proton therapy (IMPT) plans and compared with clinical volumetric modulated arc therapy (VMAT) plans. Dosimetric parameters were compared between standard and functional plans with IMPT and VMAT with one-way analysis of variance and post hoc paired student t-test. Normal Tissue Complication Probability (NTCP) models were employed to estimate the risk of two toxicity endpoints for healthy lung tissues. Dose degradation due to proton motion interplay effect was evaluated. Functional IMPT plans led to significant dose reduction to functional lung structures when compared with functional VMAT without significant dose increase to Organ at Risk (OAR) structures. When interplay effect is considered, no significant dose degradation was observed for the OARs or the clinical target volume (CTV) volumes for functional IMPT. Using fV20 as the dose metric and Grade 2+ pneumonitis as toxicity endpoint, there is a mean 5.7% reduction in Grade 2+ RP with the functional IMPT and as high as 26% in reduction for individual patient when compared to the standard IMPT planning. Functional IMPT was able to spare healthy lung tissue to avoid excess dose to normal structures while maintaining satisfying target coverage. NTCP calculation also shows that the risk of pulmonary complications can be further reduced with functional based IMPT.
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    Chromatin architecture transitions from zebrafish sperm through early embryogenesis
    (Cold Spring Harbor Laboratory Press, 2021) Wike, Candice L.; Guo, Yixuan; Tan, Mengyao; Nakamura, Ryohei; Shaw, Dana Klatt; Díaz, Noelia; Whittaker-Tademy, Aneasha F.; Durand, Neva C.; Aiden, Erez Lieberman; Vaquerizas, Juan M.; Grunwald, David; Takeda, Hiroyuki; Cairns, Bradley R.; Center for Theoretical Biological Physics
    Chromatin architecture mapping in 3D formats has increased our understanding of how regulatory sequences and gene expression are connected and regulated in a genome. The 3D chromatin genome shows extensive remodeling during embryonic development, and although the cleavage-stage embryos of most species lack structure before zygotic genome activation (pre-ZGA), zebrafish has been reported to have structure. Here, we aimed to determine the chromosomal architecture in paternal/sperm zebrafish gamete cells to discern whether it either resembles or informs early pre-ZGA zebrafish embryo chromatin architecture. First, we assessed the higher-order architecture through advanced low-cell in situ Hi-C. The structure of zebrafish sperm, packaged by histones, lacks topological associated domains and instead displays “hinge-like” domains of ∼150 kb that repeat every 1–2 Mbs, suggesting a condensed repeating structure resembling mitotic chromosomes. The pre-ZGA embryos lacked chromosomal structure, in contrast to prior work, and only developed structure post-ZGA. During post-ZGA, we find chromatin architecture beginning to form at small contact domains of a median length of ∼90 kb. These small contact domains are established at enhancers, including super-enhancers, and chemical inhibition of Ep300a (p300) and Crebbpa (CBP) activity, lowering histone H3K27ac, but not transcription inhibition, diminishes these contacts. Together, this study reveals hinge-like domains in histone-packaged zebrafish sperm chromatin and determines that the initial formation of high-order chromatin architecture in zebrafish embryos occurs after ZGA primarily at enhancers bearing high H3K27ac.
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    Quantifying pulmonary perfusion from noncontrast computed tomography
    (Wiley, 2021) Castillo, Edward; Nair, Girish; Turner‐Lawrence, Danielle; Myziuk, Nicholas; Emerson, Scott; Al‐Katib, Sayf; Westergaard, Sarah; Castillo, Richard; Vinogradskiy, Yevgeniy; Quinn, Thomas; Guerrero, Thomas; Stevens, Craig
    Purpose: Computed tomography (CT)-derived ventilation methods compute respiratory induced volume changes as a surrogate for pulmonary ventilation. Currently, there are no known methods to derive perfusion information from noncontrast CT. We introduce a novel CT-Perfusion (CT-P) method for computing the magnitude mass changes apparent on dynamic noncontrast CT as a surrogate for pulmonary perfusion. Methods: CT-Perfusion is based on a mass conservation model which describes the unknown mass change as a linear combination of spatially corresponding inhale and exhale HU estimated voxel densities. CT-P requires a deformable image registration (DIR) between the inhale/exhale lung CT pair, a preprocessing lung volume segmentation, and an estimate for the Jacobian of the DIR transformation. Given this information, the CT-P image, which provides the magnitude mass change for each voxel within the lung volume, is formulated as the solution to a constrained linear least squares problem defined by a series of subregional mean magnitude mass change measurements. Similar to previous robust CT-ventilation methods, the amount of uncertainty in a subregional sample mean measurement is related to measurement resolution and can be characterized with respect to a tolerance parameter. Spatial Spearman correlation between single photon emission CT perfusion (SPECT-P) and the proposed CT-P method was assessed in two patient cohorts via a parameter sweep of . The first cohort was comprised of 15 patients diagnosed with pulmonary embolism (PE) who had SPECT-P and 4DCT imaging acquired within 24 h of PE diagnosis. The second cohort was comprised of 15 nonsmall cell lung cancer patients who had SPECT-P and 4DCT images acquired prior to radiotherapy. For each test case, CT-P images were computed for 30 different uncertainty parameter values, uniformly sampled from the range [0.01, 0.125], and the Spearman correlation between the SPECT-P and the resulting CT-P images were computed. Results: The median correlations between CT-P and SPECT-P taken over all 30 test cases ranged between 0.49 and 0.57 across the parameter sweep. For the optimal tolerance τ = 0.0385, the CT-P and SPECT-P correlations across all 30 test cases ranged between 0.02 and 0.82. A one-sample sign test was applied separately to the PE and lung cancer cohorts. A low Spearmen correlation of 15% was set as the null median value and two-sided alternative was tested. The PE patients showed a median correlation of 0.57 (IQR = 0.305). One-sample sign test was statistically significant with 96.5 % confidence interval: 0.20–0.63, P < 0.00001. Lung cancer patients had a median correlation of 0.57(IQR = 0.230). Again, a one-sample sign test for median was statistically significant with 96.5 percent confidence interval: 0.45–0.71, P < 0.00001. Conclusion: CT-Perfusion is the first mechanistic model designed to quantify magnitude blood mass changes on noncontrast dynamic CT as a surrogate for pulmonary perfusion. While the reported correlations with SPECT-P are promising, further investigation is required to determine the optimal CT acquisition protocol and numerical method implementation for CT-P imaging.
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    Reciprocity-gap misfit functional for distributed acoustic sensing, combining data from passive and active sources
    (Society of Exploration Geophysicists, 2021) Faucher, Florian; de Hoop, Maarten V.; Scherzer, Otmar
    Quantitative imaging of subsurface earth properties in elastic media is performed from distributed acoustic sensing data. A new misfit functional based upon the reciprocity gap is designed, taking crosscorrelations of displacement and strain, and these products further associate an observation with a simulation. In comparison with other misfit functionals, this functional has the advantage of only requiring little a priori information on the exciting sources. In particular, the misfit criterion enables the use of data from regional earthquakes (teleseismic events can be included as well), followed by exploration data to perform a multiresolution reconstruction. The data from regional earthquakes contain the low-frequency content that is missing in the exploration data, allowing for the recovery of the long spatial wavelength, even with very few sources. These data are used to build prior models for the subsequent reconstruction from the higher frequency exploration data. This results in the elastic full reciprocity-gap waveform inversion method, and we illustrate its performance with a pilot experiment for elastic isotropic reconstruction.
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    Clustering earthquake signals and background noises in continuous seismic data with unsupervised deep learning
    (Springer Nature, 2020) Seydoux, Léonard; Balestriero, Randall; Poli, Piero; de Hoop, Maarten; Campillo, Michel; Baraniuk, Richard
    The continuously growing amount of seismic data collected worldwide is outpacing our abilities for analysis, since to date, such datasets have been analyzed in a human-expert-intensive, supervised fashion. Moreover, analyses that are conducted can be strongly biased by the standard models employed by seismologists. In response to both of these challenges, we develop a new unsupervised machine learning framework for detecting and clustering seismic signals in continuous seismic records. Our approach combines a deep scattering network and a Gaussian mixture model to cluster seismic signal segments and detect novel structures. To illustrate the power of the framework, we analyze seismic data acquired during the June 2017 Nuugaatsiaq, Greenland landslide. We demonstrate the blind detection and recovery of the repeating precursory seismicity that was recorded before the main landslide rupture, which suggests that our approach could lead to more informative forecasting of the seismic activity in seismogenic areas.
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    Convergence of a Class of Stationary Iterative Methods for Saddle Point Problems
    (Springer, 2019) Zhang, Yin
    A unified convergence theory is derived for a class of stationary iterative methods for solving linear equality constrained quadratic programs or saddle point problems. This class is constructed from essentially all possible splittings of the submatrix residing in the (1,1)-block of the augmented saddle point matrix that would produce non-expansive iterations. The classic augmented Lagrangian method and alternating direction method of multipliers are two special members of this class.
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    An accelerated Poisson solver based on multidomain spectral discretization
    (Springer, 2018) Babb, Tracy; Gillman, Adrianna; Hao, Sijia; Martinsson, Per-Gunnar
    This paper presents a numerical method for variable coefficient elliptic PDEs with mostly smooth solutions on two dimensional domains. The method works best for domains that can readily be mapped onto a rectangle, or a collection of nonoverlapping rectangles. The PDE is discretized via a multi-domain spectral collocation method of high local order (order 30 and higher have been tested and work well). Local mesh refinement results in highly accurate solutions even in the presence of local irregular behavior due to corner singularities, localized loads, etc. The system of linear equations attained upon discretization is solved using a direct (as opposed to iterative) solver with O(N1.5)O(N1.5) complexity for the factorization stage and O(NlogN)O(Nlog⁡N) complexity for the solve. The scheme is ideally suited for executing the elliptic solve required when parabolic problems are discretized via time-implicit techniques. In situations where the geometry remains unchanged between time-steps, very fast execution speeds are obtained since the solution operator for each implicit solve can be pre-computed.
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    On discretely entropy conservative and entropy stable discontinuous Galerkin methods
    (Elsevier, 2018) Chan, Jesse
    High order methods based on diagonal-norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic PDEsᅠ[1],ᅠ[2]. These methods can also be interpreted as nodal discontinuous Galerkin methods with diagonal mass matricesᅠ[3],ᅠ[4],ᅠ[5],ᅠ[6]. In this work, we describe how use flux differencing, quadrature-based projections, and SBP-like operators to construct discretely entropy conservative schemes for DG methods under more arbitrary choices of volume and surface quadrature rules. The resulting methods are semi-discretely entropy conservative or entropy stable with respect to the volume quadrature rule used. Numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible Euler equations in one and two dimensions.
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    Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion
    (Elsevier, 2018) Chan, Jesse; Evans, John A.
    We present a class of spline finite element methods for time-domain wave propagation which are particularly amenable to explicit time-stepping. The proposed methods utilize a discontinuous Galerkin discretization to enforce continuity of the solution field across geometric patches in a multi-patch setting, which yields a mass matrix with convenient block diagonal structure. Over each patch, we show how to accurately and efficiently invert mass matrices in the presence of curved geometries by using a weight-adjusted approximation of the mass matrix inverse. This approximation restores a tensor product structure while retaining provable high order accuracy and semi-discrete energy stability. We also estimate the maximum stable timestep for spline-based finite elements and show that the use of spline spaces results in less stringent CFL restrictions than equivalentᅠC0ᅠor discontinuous finite element spaces. Finally, we explore the use of optimal knot vectors based onᅠL2n-widths. We show how the use of optimal knot vectors can improve both approximation properties and the maximum stable timestep, and present a simple heuristic method for approximating optimal knot positions. Numerical experiments confirm the accuracy and stability of the proposed methods.