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Aristotelous A.C.,Statistical and Applied Mathematical science Institute SAMSI | Aristotelous A.C.,Duke University | Haider M.A.,North Carolina State University
International Journal for Numerical Methods in Biomedical Engineering

Macroscopic models accounting for cellular effects in natural or engineered tissues may involve unknown constitutive terms that are highly dependent on interactions at the scale of individual cells. Hybrid discrete models, which represent cells individually, were used to develop and apply techniques for modeling diffusive nutrient transport and cellular uptake to identify a nonlinear nutrient loss term in a macroscopic reaction-diffusion model of the system. Flexible and robust numerical methods were used, based on discontinuous Galerkin finite elements in space and a Crank-Nicolson temporal discretization. Scales were bridged via averaging operations over a complete set of subdomains yielding data for identification of a macroscopic nutrient loss term that was accurately captured via a fifth-order polynomial. Accuracy of the identified macroscopic model was demonstrated by direct, quantitative comparisons of the tissue and cellular scale models in terms of three error norms computed on a mesoscale mesh. © 2014 John Wiley & Sons, Ltd. Source

Gordon-Wright R.,North Carolina State University | Gremaud P.A.,Statistical and Applied Mathematical science Institute SAMSI | Martens E.G.H.J.,TU Eindhoven | Novak V.,Beth Israel Deaconess Medical Center
1st International Workshop on Innovative Simulation for Health Care, IWISH 2012, Held at the International Multidisciplinary Modeling and Simulation Multiconference, I3M 2012

There is a pressing need for noninvasive and continuous measurements of cerebral blood flow (CBF) in several areas of medicine. Transcranial Doppler (TCD) technology is clinically used for measurements of blood flow velocities (BFV). It is assumed that perfusion and vasoreactivity in a vascular territory can be inferred from BFV measurements in the corresponding stem artery. However, only very modest correlations have been found between TCD-based and magnetic resonance imaging (MRI)-based CBF measurements. Several factors, whose values are uncertain, such as vessel diameter, hematocrit and insonation angle, affect the BFV - CBF relationship. Their influence on CBF and vasoreactivity estimates has not been rigorously analyzed but cannot be ignored. We present initial work toward a subject specific computational and experimental model to both quantify and reduce the uncertainty attached to CBF and vasoreactivity estimates based on Doppler ultrasound. Source

Thai D.H.,University of Gottingen | Thai D.H.,Statistical and Applied Mathematical science Institute SAMSI | Gottschlich C.,University of Gottingen
Eurasip Journal on Image and Video Processing

We consider the task of image decomposition, and we introduce a new model coined directional global three-part decomposition (DG3PD) for solving it. As key ingredients of the DG3PD model, we introduce a discrete multi-directional total variation norm and a discrete multi-directional G-norm. Using these novel norms, the proposed discrete DG3PD model can decompose an image into two or three parts. Existing models for image decomposition by Vese and Osher (J. Sci. Comput. 19(1–3):553–572, 2003), by Aujol and Chambolle (Int. J. Comput. Vis. 63(1):85–104, 2005), by Starck et al. (IEEE Trans. Image Process. 14(10):1570–1582, 2005), and by Thai and Gottschlich are included as special cases in the new model. Decomposition of an image by DG3PD results in a cartoon image, a texture image, and a residual image. Advantages of the DG3PD model over existing ones lie in the properties enforced on the cartoon and texture images. The geometric objects in the cartoon image have a very smooth surface and sharp edges. The texture image yields oscillating patterns on a defined scale which are both smooth and sparse. Moreover, the DG3PD method achieves the goal of perfect reconstruction by summation of all components better than the other considered methods. Relevant applications of DG3PD are a novel way of image compression as well as feature extraction for applications such as latent fingerprint processing and optical character recognition. © 2016, Thai and Gottschlich. Source

Srivastava S.,Duke University | Srivastava S.,Statistical and Applied Mathematical science Institute SAMSI | Cevher V.,Ecole Polytechnique Federale de Lausanne | Tran-Dinh Q.,Ecole Polytechnique Federale de Lausanne | Dunson D.B.,Duke University
Journal of Machine Learning Research

The promise of Bayesian methods for big data sets has not fully been realized due to the lack of scalable computational algorithms. For massive data, it is necessary to store and process subsets on different machines in a distributed manner. We propose a simple, general, and highly efficient approach, which first runs a posterior sampling algorithm in parallel on different machines for subsets of a large data set. To combine these subset posteriors, we calculate the Wasserstein barycenter via a highly efficient linear program. The resulting estimate for the Wasserstein posterior (WASP) has an atomic form, facilitating straightforward estimation of posterior summaries of functionals of interest. The WASP approach allows posterior sampling algorithms for smaller data sets to be trivially scaled to huge data. We provide theoretical justification in terms of posterior consistency and algorithm efficiency. Examples are provided in complex settings including Gaussian process regression and nonparametric Bayes mixture models. Copyright 2015 by the authors. Source

Collins J.,North Carolina State University | Gremaud P.,North Carolina State University | Gremaud P.,Statistical and Applied Mathematical science Institute SAMSI
Mathematics and Computers in Simulation

A simple mathematical model of laser drilling is proposed. Assuming axi-symmetry of the process around the axis of the laser beam, a one-dimensional formulation is obtained after cross-sectional averaging. The novelty of the approach relies on the fact that even after dimension reduction, the shape of the hole can still be described. The model is derived, implemented and validated for drilling using lasers with intensities in the GW/cm2 range and microsecond pulses. © 2010 IMACS. Source

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