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We provide theoretical analysis of the statistical and computational properties of penalized M-estimators that can be formulated as the solution to a possibly nonconvex optimization problem. Many important estimators fall in this category, including least squares regression with nonconvex regularization, generalized linear models with nonconvex regularization and sparse elliptical random design regression. For these problems, it is intractable to calculate the global solution due to the nonconvex formulation. In this paper, we propose an approximate regularization path-following method for solving a variety of learning problems with nonconvex objective functions. Under a unified analytic framework, we simultaneously provide explicit statistical and computational rates of convergence for any local solution attained by the algorithm. Computationally, our algorithm attains a global geometric rate of convergence for calculating the full regularization path, which is optimal among all first-order algorithms. Unlike most existing methods that only attain geometric rates of convergence for one single regularization parameter, our algorithm calculates the full regularization path with the same iteration complexity. In particular, we provide a refined iteration complexity bound to sharply characterize the performance of each stage along the regularization path. Statistically, we provide sharp sample complexity analysis for all the approximate local solutions along the regularization path. In particular, our analysis improves upon existing results by providing a more refined sample complexity bound as well as an exact support recovery result for the final estimator. These results show that the final estimator attains an oracle statistical property due to the usage of nonconvex penalty.

In strained mechanical resonators, the concurrence of tensile stress and geometric nonlinearity dramatically reduces dissipation. This phenomenon, called dissipation dilution, is employed in mirror suspensions of gravitational-wave interferometers and at the nanoscale, where soft clamping and strain engineering have allowed extremely high quality factors. However, these techniques have so far been applied only to amorphous materials, specifically Si 3 N 4 . Crystalline materials exhibit substantially lower intrinsic damping at cryogenic temperatures. Applying dissipation dilution engineering to strained crystalline materials could, therefore, enable extremely low loss nanomechanical resonators, as they combine low internal friction, high intrinsic strain and high yield strength. This potential has not yet been fully exploited. Here we demonstrate that single-crystal strained silicon—a material developed for high-mobility transistors—can be used to realize mechanical resonators with ultralow dissipation. We fabricate strained silicon nanostrings with high aspect ratios supporting megahertz mechanical modes with quality factors exceeding 10 10 at 7 K, a tenfold improvement over values reported in Si 3 N 4 . We estimate a thermal-noise-limited force sensitivity of (5 ± 2) × 10 –20  N Hz –1/2 at 7 K—approaching that of carbon nanotubes—and a heating rate of only 60 quanta per second. The low mass and high quality factors of our nanomechanical resonators make them particularly promising for quantum sensing and transduction. Soft clamping reduces the dissipation of nanomechanical resonators, but this method has been limited to amorphous materials. When applied in crystalline silicon, it enables resonators with quality factors beyond ten billion.

Assume that we observe i.i.d. points lying close to some unknown d-dimensional Ck submanifold M in a possibly high-dimensional space. We study the problem of reconstructing the probability distribution generating the sample. After remarking that this problem is degenerate for a large class of standard losses (Lp, Hellinger, total variation, etc.), we focus on the Wasserstein loss, for which we build an estimator, based on kernel density estimation, whose rate of convergence depends on d and the regularity s≤k-1 of the underlying density, but not on the ambient dimension. In particular, we show that the estimator is minimax and matches previous rates in the literature in the case where the manifold M is a d-dimensional cube. The related problem of the estimation of the volume measure of M for the Wasserstein loss is also considered, for which a minimax estimator is exhibited.

The Spectral Finite Element Technique (SFEM) has Several Applications in the Sciences, Engineering, and Mathematics, which will be Covered in this Review Article. The Spectral Finite Element Method (SFEM) is a Variant of the Traditional Finite Element Method FEM that Makes use of Higher Order Basis Functions (FEM). One of the most Fundamental Numerical Techniques Employed in the Numerical Simulation is the SFEM, which Outperforms Other Techniques in Terms of Faster Convergence, Reduced Diffusion and Dispersion Errors, Simplicity of the Application as well as Shorter time of Computation. The Spectral Finite Element Technique Combines the Characteristics of Approximating Polynomials of Spectral Methods. The Approach to Discretizing the Examined Region Unique to the FEM is a mix of both Approaches. Combining These Techniques Enables Quicker (Spectral) Convergence of Solutions, Higher Approximation Polynomial Order, the Removal of Geometric Constraints on the Examined Areas, and much Lower Discretization Density Requirements. Spectral Element Methods used in Different Applications are Presented Along with a Statistical Overview of Studies During 2010–2022.

From the past few decade, many manufacturing industries are using heat exchangers for reducing the energy consumption and hence reducing the fuel costs. Most widely used types of heat exchangers in industries are Double Pipe Heat Exchangers and Shell & Tube Heat Exchangers. It is recently that industry people and researchers are becoming more aware of the advantage of using Spiral Heat Exchangers for heat transfer between two different fluids.A Spiral Heat Exchanger is formedby a coiled sheet arrangement with two channels coiled one around the other. The distance between the sheets is kept constant to maintain the area of cross section through out the spiral path of the channels. In this work, flow pattern and heat transfer in a Spiral Heat Exchanger are analyzed using a couterflow model geometry. The results obtained for the fluid flow and heat transfer gives an idea about how we can optimize the flow rate of the fluids thus increasing the efficiency of the heat exchanger.

Injection molds with conformal cooling channels have been deemed increasingly important in the mold manufacturing industry. Selective laser melting (SLM) has proven to be an effective process to fabricate conformal cooling molds. However, the surface quality and morphologies of SLM-fabricated cooling channels are different from that of conventional drilled channels. To investigate the differences in the coolant flow rate and cooling performance between SLM-fabricated and drilled cooling channels, test molds with cooling channel diameters of φ2 mm, φ3 mm, and φ4 mm were fabricated via SLM and the computer numerical control (CNC) process. A test system was designed and set up to investigate the flow rate and cooling performance of two types of test molds. The geometrical shape, dimensional accuracy, and surface morphologies of the SLM-fabricated cooling channels were characterized using optical microscopy and laser microscopy. The results indicated that the SLM-fabricated cooling channel exhibited an elliptic shape due to the lack of support along the building direction. The flow rate of the SLM-fabricated cooling channels was smaller than that of the drilled channels due to the low dimensional accuracy and roughness surface. The cooling performance of the SLM-fabricated cooling channels was also poorer than that of the drilled channels due to the presence of unmolten particles and a loose layer on the SLM-fabricated surface. Theoretical analysis was conducted on the influence of surface roughness on the flow rate and cooling performance of two types of channels and the friction factor; Nusselt number and heat transfer coefficient were obtained.

This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems. The book's three sections look at foundations, multiagent networks, and networks as systems. The authors give an overview of important ideas from graph theory, followed by a detailed account of the agreement protocol and its various extensions, including the behavior of the protocol over undirected, directed, switching, and random networks. They cover topics such as formation control, coverage, distributed estimation, social networks, and games over networks. And they explore intriguing aspects of viewing networks as systems, by making these networks amenable to control-theoretic analysis and automatic synthesis, by monitoring their dynamic evolution, and by examining higher-order interaction models in terms of simplicial complexes and their applications. The book will interest graduate students working in systems and control, as well as in computer science and robotics. It will be a standard reference for researchers seeking a self-contained account of system-theoretic aspects of multiagent networks and their wide-ranging applications. This book has been adopted as a textbook at the following universities: University of Stuttgart, GermanyRoyal Institute of Technology, SwedenJohannes Kepler University, AustriaGeorgia Tech, USAUniversity of Washington, USAOhio University, USA

Cerebral aneurysms are a serious clinical challenge, with ∼half resulting in death or disability. Treatment via endovascular coiling significantly reduces the chances of rupture, but the techniquehas failure rates of ∼20 %. This presents a pressing need to develop a method fordetermining optimal coildeploymentstrategies. Quantification of the hemodynamics of coiled aneurysms using computational fluid dynamics (CFD) has the potential to predict post-treatment outcomes, but representing the coil mass in CFD simulations remains a challenge. We use the Finite Element Method (FEM) for simulating patient-specific coil deployment for n = 4 ICA aneurysms for which 3D printed in vitro models were also generated, coiled, and scanned using ultra-high resolution synchrotron micro-CT. The physical and virtual coil geometries were voxelized onto a binary structured grid and porosity maps were generated for geometric comparison. The average binary accuracy score is 0.8623 and the average error in porosity map is 4.94 %. We then conduct patient-specific CFD simulations of the aneurysm hemodynamics using virtual coils geometries, micro-CT generated oil geometries, and using the porous medium method to represent the coil mass. Hemodynamic parameters including Neck Inflow Rate (Qneck) and Wall Shear Stress (WSS) were calculated for each of the CFD simulations. The average relative error in Qneck and WSS from CFD using FEM geometry were 6.6 % and 21.8 % respectively, while the error from CFD using a porous media approximation resulted in errors of 55.1 % and 36.3 % respectively; demonstrating a marked improvement in the accuracy of CFD simulations using FEM generated coil geometries.

MR imaging, a noninvasive radiation-free imaging modality commonly used during clinical follow-up, has been widely utilized to reconstruct realistic 3D vascular models for patient-specific analysis. In recent work, we used patient-specific hemodynamic analysis of the circle of Willis to noninvasively assess stroke risk in pediatric Moyamoya disease (MMD)—a progressive steno-occlusive cerebrovascular disorder that leads to recurrent stroke. The objective was to identify vascular regions with critically high wall shear rate (WSR) that signifies elevated stroke risk. However, sources of error such as insufficient resolution of MR images can negatively impact vascular model accuracy, especially in areas of severe pathological narrowing, and thus diminish clinical relevance of simulation results, as local hemodynamics are sensitive to vessel geometry. To improve the accuracy of MR-derived vascular models, we have developed a novel method for adjusting model vessel geometry utilizing 2D X-ray angiography (XA), which is considered the gold standard for clinically assessing vessel caliber. In this workflow, “virtual angiographies” (VAs) of 3D MR-derived vascular models are conducted, producing 2D projections that are compared with corresponding XA images to guide the local adjustment of modeled vessels. This VA-comparison-adjustment loop is iterated until the two agree, as confirmed by an expert neuroradiologist. Using this method, we generated models of the circle of Willis of two patients with a history of unilateral stroke. Blood flow simulations were performed using a Navier–Stokes solver within an isogeometric analysis framework, and WSR distributions were quantified. Results for one patient show as much as 45% underestimation of local WSR in the stenotic left anterior cerebral artery (LACA), and up to a 56% underestimation in the right anterior cerebral artery when using the initial MR-derived model compared to the XA-adjusted model. To evaluate whether XA-based adjustment improves model accuracy, vessel cross-sectional areas of the pre- and post-adjustment models were compared to those seen in 3D CTA images of the same patient. CTA has superior resolution and signal-to-noise ratio compared to MR imaging but is not commonly used in the clinic due to radiation exposure concerns, especially in pediatric patients. While the vessels in the initial model had normalized root mean squared deviations (NRMSDs) ranging from 26 to 182 and 31 to 69% in two patients with respect to CTA, the adjusted vessel NRMSDs were comparatively smaller (32–53% and 11–42%). In the mildly stenotic LACA of patient 1, the NRMSDs for the pre- and post-adjusted models were 49% and 32%, respectively. These findings suggest that our XA-based adjustment method can considerably improve the accuracy of vascular models, and thus, stroke-risk prediction. An accurate, individualized assessment of stroke risk would be of substantial help in guiding the timing of preventive surgical interventions in pediatric MMD patients.

We study localization occurring during high-speed shear deformations of metals leading to the formation of shear bands. The localization instability results from the competition between Hadamard instability (caused by softening response) and the stabilizing effects of strain rate hardening. We consider a hyperbolic–parabolic system that expresses the above mechanism and construct self-similar solutions of localizing type that arise as the outcome of the above competition. The existence of self-similar solutions is turned, via a series of transformations, into a problem of constructing a heteroclinic orbit for an induced dynamical system. The dynamical system is in four dimensions but has a fast–slow structure with respect to a small parameter capturing the strength of strain rate hardening. Geometric singular perturbation theory is applied to construct the heteroclinic orbit as a transversal intersection of two invariant manifolds in the phase space.