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We introduce a nonconforming finite-element method for second order elliptic interface problems. Our approach applies to problems in which discontinuous coefficients and singular sources on the interface may give rise to jump discontinuities in either the solution or its normal derivative. Given a standard background mesh and an interface that passes between elements, the key idea is to construct a singular correction function that satisfies the prescribed jump conditions, providing accurate subgrid resolution of the discontinuities. Utilizing the closest point extension and an implicit interface representation by the signed distance function, an algorithm is established to construct the correction function. The result is a function that is supported only on the interface elements, represented by the regular basis functions, and bounded independently of the interface location with respect to the background mesh. In the particular case of a constant second-order coefficient, our regularization by a singular function is straightforward, and the resulting left-hand side is identical to that of a regular problem without introducing any instability. The influence of the regularization appears solely on the right-hand side, which simplifies the implementation. In the more general case of discontinuous second-order coefficients, a normalization is invoked which introduces a constraint equation on the interface. This results in a problem statement similar to that of a saddle-point problem. We employ two-level iteration as the solution strategy, which exhibits aspects similar to those of iterative preconditioning strategies. nonconforming FEM | elliptic interface problems | interracial singularities | level set methods

Coefficients for translational and rotational diffusion characterize the Brownian motion of particles. Emerging X-ray photon correlation spectroscopy (XPCS) experiments probe a broad range of length scales and time scales and are well-suited for investigation of Brownian motion. While methods for estimating the translational diffusion coefficients from XPCS are well-developed, there are no algorithms for measuring the rotational diffusion coefficients based on XPCS, even though the required raw data are accessible from such experiments. In this paper, we propose angular-temporal cross-correlation analysis of XPCS data and show that this information can be used to design a numerical algorithm (Multi-Tiered Estimation for Correlation Spectroscopy [MTECS]) for predicting the rotational diffusion coefficient utilizing the cross-correlation: This approach is applicable to other wavelengths beyond this regime. We verify the accuracy of this algorithmic approach across a range of simulated data. [ABSTRACT FROM AUTHOR]

We present a model and algorithm for segmentation of images with missing boundaries. In many situations, the human visual system fills in missing gaps in edges and boundaries, building and completing information that is not present. This presents a considerable challenge in computer vision, since most algorithms attempt to exploit existing data. Completion models, which postulate how to construct missing data, are popular but are often trained and specific to particular images. In this paper, we take the following perspective: We consider a reference point within an image as given and then develop an algorithm that tries to build missing information on the basis of the given point of view and the available information as boundary data to the algorithm. We test the algorithm on some standard images, including the classical triangle of Kanizsa and low signal/noise ratio medical images.

Deep convolutional neural networks have been successfully applied to many image-processing problems in recent works. Popular network architectures often add additional operations and connections to the standard architecture to enable training deeper networks. To achieve accurate results in practice, a large number of trainable parameters are often required. Here, we introduce a network architecture based on using dilated convolutions to capture features at different image scales and densely connecting all feature maps with each other. The resulting architecture is able to achieve accurate results with relatively few parameters and consists of a single set of operations, making it easier to implement, train, and apply in practice, and automatically adapts to different problems. We compare results of the proposed network architecture with popular existing architectures for several segmentation problems, showing that the proposed architecture is able to achieve accurate results with fewer parameters, with a reduced risk of overfitting the training data. [ABSTRACT FROM AUTHOR]

Free-electron lasers now have the ability to collect X-ray diffraction patterns from individual molecules; however, each sample is delivered at unknown orientation and may be in one of several conformational states, each with a different molecular structure. Hit rates are often low, typically around 0.1%, limiting the number of useful images that can be collected. Determining accurate structural information requires classifying and orienting each image, accurately assembling them into a 3D diffraction intensity function, and determining missing phase information. Additionally, single particles typically scatter very few photons, leading to high image noise levels. We develop a multitiered iterative phasing algorithm to reconstruct structural information from singleparticle diffraction data by simultaneously determining the states, orientations, intensities, phases, and underlying structure in a single iterative procedure. We leverage real-space constraints on the structure to help guide optimization and reconstruct underlying structure from very few images with excellent global convergence properties. We show that this approach can determine structural resolution beyond what is suggested by standard Shannon sampling arguments for ideal images and is also robust to noise. [ABSTRACT FROM AUTHOR]

Fluctuation X-ray scattering (FXS) is an extension of small- and wide-angle X-ray scattering in which the X-ray snapshots are taken below rotational diffusion times. This technique, performed using a free electron laser or ultrabright synchrotron source, provides significantly more experimental information compared with traditional solution scattering methods. We develop a multitiered iterative phasing algorithm to determine the underlying structure of the scattering object from FXS data. [ABSTRACT FROM AUTHOR]

X-ray nanocrystallography allows the structure of a macromolecule to be determined from a large ensemble of nanocrystals. However, several parameters, including crystal sizes, orientations, and incident photon flux densities, are initially unknown and images are highly corrupted with noise. Autoindexing techniques, commonly used in conventional crystallography, can determine orientations using Bragg peak patterns, but only up to crystal lattice symmetry. This limitation results in an ambiguity in the orientations, known as the indexing ambiguity, when the diffraction pattern displays less symmetry than the lattice and leads to data that appear twinned if left unresolved. Furthermore, missing phase information must be recovered to determine the imaged object's structure. We present an algorithmic framework to determine crystal size, incident photon flux density, and orientation in the presence of the indexing ambiguity. We show that phase information can be computed from nanocrystallographic diffraction using an iterative phasing algorithm, without extra experimental requirements, atomicity assumptions, or knowledge of similar structures required by current phasing methods. The feasibility of this approach is tested on simulated data with parameters and noise levels common in current experiments. [ABSTRACT FROM AUTHOR]

Cells and multicellular structures can mechanically align and concentrate fibers in their ECM environment and can sense and respond to mechanical cues by differentiating, branching, or disorganizing. Here we show that mammary acini with compromised structural integrity can interconnect by forming long collagen lines. These collagen lines then coordinate and accelerate transition to an invasive phenotype. Interacting acini begin to disorganize within 12.5 ± 4.7 h in a spatially coordinated manner, whereas acini that do not interact mechanically with other acini disorganize more slowly (in 21.8 ± 4.1 h) and to a lesser extent (P < 0.0001). When the directed mechanical connections between acini were cut with a laser, the acini reverted to a slowly disorganizing phenotype. When acini were fully mechanically isolated from other acini and also from the bulk gel by box-cuts with a side length <900 μm, transition to an invasive phenotype was blocked in 20 of 20 experiments, regardless of waiting time. Thus, pairs or groups of mammary acini can interact mechanically over long distances through the collagen matrix, and these directed mechanical interactions facilitate transition to an invasive phenotype. [ABSTRACT FROM AUTHOR]

We introduce a numerical framework, the Voronoi Implicit Interface Method for tracking multiple interacting and evolving regions (phases) whose motion is determined by complex physics (fluids, mechanics, elasticity, etc.), intricate jump conditions, internal constraints, and boundary conditions. The method works in two and three dimensions, handles tens of thousands of interfaces and separate phases, and easily and automatically handles multiple junctions, triple points, and quadruple points in two dimensions, as well as triple lines, etc., in higher dimensions. Topological changes occur naturally, with no surgery required. The method is first-order accurate at junction points/lines, and of arbitrarily high-order accuracy away from such degeneracies. The method uses a single function to describe all phases simultaneously, represented on a fixed Eulerian mesh. We test the method's accuracy through convergence tests, and demonstrate its applications to geometric flows, accurate prediction of von Neumann's law for multiphase curvature flow, and robustness under complex fluid flow with surface tension and large shearing forces. [ABSTRACT FROM AUTHOR]

We introduce a nonconforming finite-element method for second order elliptic interface problems. Our approach applies to problems in which discontinuous coefficients and singular sources on the interface may give rise to jump discontinuities in either the solution or its normal derivative. Given a standard background mesh and an interface that passes between elements, the key idea is to construct a singular correction function that satisfies the prescribed jump conditions, providing accurate subgrid resolution of the discontinuities. Utilizing the closest point extension and an implicit interface representation by the signed distance function, an algorithm is established to construct the correction function. The result is a function that is supported only on the interface elements, represented by the regular basis functions, and bounded independently of the interface location with respect to the background mesh. In the particular case of a constant second-order coefficient, our regularization by a singular function is straightforward, and the resulting left-hand side is identical to that of a regular problem without introducing any instability. The influence of the regularization appears solely on the right-hand side, which simplifies the implementation. In the more general case of discontinuous second- order coefficients, a normalization is invoked which introduces a constraint equation on the interface. This results in a problem statement similar to that of a saddle-point problem. We employ two-level iteration as the solution strategy, which exhibits aspects similar to those of iterative preconditioning strategies. [ABSTRACT FROM AUTHOR]

The global automotive industry sprayed over 2.6 billion liters of paint in 2018, much of which through electrostatic rotary bell atomization, a highly complex process involving the fluid mechanics of rapidly rotating thin films tearing apart into micrometer-thin filaments and droplets. Coating operations account for 65% of the energy usage in a typical automotive assembly plant, representing 10,000s of gigawatt-hours each year in the United States alone. Optimization of these processes would allow for improved robustness, reduced material waste, increased throughput, and significantly reduced energy usage. Here, we introduce a high-fidelity mathematical and algorithmic framework to analyze rotary bell atomization dynamics at industrially relevant conditions. Our approach couples laboratory experiment with the development of robust non-Newtonian fluid models; devises high-order accurate numerical methods to compute the coupled bell, paint, and gas dynamics; and efficiently exploits high-performance supercomputing architectures. These advances have yielded insight into key dynamics, including i) parametric trends in film, sheeting, and filament characteristics as a function of fluid rheology, delivery rates, and bell speed; ii) the impact of nonuniform film thicknesses on atomization performance; and iii) an understanding of spray composition via primary and secondary atomization. These findings result in coating design principles that are poised to improve energy- and cost-efficiency in a wide array of industrial and manufacturing settings.