Showing papers by "Courant Institute of Mathematical Sciences published in 2016"

Character-aware neural language models

Abstract: We describe a simple neural language model that relies only on character-level inputs. Predictions are still made at the word-level. Our model employs a convolutional neural network (CNN) and a highway network over characters, whose output is given to a long short-term memory (LSTM) recurrent neural network language model (RNN-LM). On the English Penn Treebank the model is on par with the existing state-of-the-art despite having 60% fewer parameters. On languages with rich morphology (Arabic, Czech, French, German, Spanish, Russian), the model outperforms word-level/morpheme-level LSTM baselines, again with fewer parameters. The results suggest that on many languages, character inputs are sufficient for language modeling. Analysis of word representations obtained from the character composition part of the model reveals that the model is able to encode, from characters only, both semantic and orthographic information.

Stereo matching by training a convolutional neural network to compare image patches

Abstract: We present a method for extracting depth information from a rectified image pair. Our approach focuses on the first stage of many stereo algorithms: the matching cost computation. We approach the problem by learning a similarity measure on small image patches using a convolutional neural network. Training is carried out in a supervised manner by constructing a binary classification data set with examples of similar and dissimilar pairs of patches. We examine two network architectures for this task: one tuned for speed, the other for accuracy. The output of the convolutional neural network is used to initialize the stereo matching cost. A series of post-processing steps follow: cross-based cost aggregation, semiglobal matching, a left-right consistency check, subpixel enhancement, a median filter, and a bilateral filter. We evaluate our method on the KITTI 2012, KITTI 2015, and Middlebury stereo data sets and show that it outperforms other approaches on all three data sets.

Learning to Navigate in Complex Environments

Abstract: Learning to navigate in complex environments with dynamic elements is an important milestone in developing AI agents. In this work we formulate the navigation question as a reinforcement learning problem and show that data efficiency and task performance can be dramatically improved by relying on additional auxiliary tasks leveraging multimodal sensory inputs. In particular we consider jointly learning the goal-driven reinforcement learning problem with auxiliary depth prediction and loop closure classification tasks. This approach can learn to navigate from raw sensory input in complicated 3D mazes, approaching human-level performance even under conditions where the goal location changes frequently. We provide detailed analysis of the agent behaviour, its ability to localise, and its network activity dynamics, showing that the agent implicitly learns key navigation abilities.

Learning to Navigate in Complex Environments

Abstract: Learning to navigate in complex environments with dynamic elements is an important milestone in developing AI agents. In this work we formulate the navigation question as a reinforcement learning problem and show that data efficiency and task performance can be dramatically improved by relying on additional auxiliary tasks leveraging multimodal sensory inputs. In particular we consider jointly learning the goal-driven reinforcement learning problem with auxiliary depth prediction and loop closure classification tasks. This approach can learn to navigate from raw sensory input in complicated 3D mazes, approaching human-level performance even under conditions where the goal location changes frequently. We provide detailed analysis of the agent behaviour, its ability to localise, and its network activity dynamics, showing that the agent implicitly learns key navigation abilities.

Entropy-SGD: Biasing Gradient Descent Into Wide Valleys

Abstract: This paper proposes a new optimization algorithm called Entropy-SGD for training deep neural networks that is motivated by the local geometry of the energy landscape. Local extrema with low generalization error have a large proportion of almost-zero eigenvalues in the Hessian with very few positive or negative eigenvalues. We leverage upon this observation to construct a local-entropy-based objective function that favors well-generalizable solutions lying in large flat regions of the energy landscape, while avoiding poorly-generalizable solutions located in the sharp valleys. Conceptually, our algorithm resembles two nested loops of SGD where we use Langevin dynamics in the inner loop to compute the gradient of the local entropy before each update of the weights. We show that the new objective has a smoother energy landscape and show improved generalization over SGD using uniform stability, under certain assumptions. Our experiments on convolutional and recurrent networks demonstrate that Entropy-SGD compares favorably to state-of-the-art techniques in terms of generalization error and training time.

Report on the sixth blind test of organic crystal-structure prediction methods

Abstract: The sixth blind test of organic crystal structure prediction (CSP) methods has been held, with five target systems: a small nearly rigid molecule, a polymorphic former drug candidate, a chloride salt hydrate, a co-crystal and a bulky flexible molecule. This blind test has seen substantial growth in the number of participants, with the broad range of prediction methods giving a unique insight into the state of the art in the field. Significant progress has been seen in treating flexible molecules, usage of hierarchical approaches to ranking structures, the application of density-functional approximations, and the establishment of new workflows and `best practices' for performing CSP calculations. All of the targets, apart from a single potentially disordered Z' = 2 polymorph of the drug candidate, were predicted by at least one submission. Despite many remaining challenges, it is clear that CSP methods are becoming more applicable to a wider range of real systems, including salts, hydrates and larger flexible molecules. The results also highlight the potential for CSP calculations to complement and augment experimental studies of organic solid forms.

Helminth infection promotes colonization resistance via type 2 immunity

Abstract: Increasing incidence of inflammatory bowel diseases, such as Crohn's disease, in developed nations is associated with changes to the microbial environment, such as decreased prevalence of helminth colonization and alterations to the gut microbiota. We find that helminth infection protects mice deficient in the Crohn's disease susceptibility gene Nod2 from intestinal abnormalities by inhibiting colonization by an inflammatory Bacteroides species. Resistance to Bacteroides colonization was dependent on type 2 immunity, which promoted the establishment of a protective microbiota enriched in Clostridiales. Additionally, we show that individuals from helminth-endemic regions harbor a similar protective microbiota and that deworming treatment reduced levels of Clostridiales and increased Bacteroidales. These results support a model of the hygiene hypothesis in which certain individuals are genetically susceptible to the consequences of a changing microbial environment.

Introduction to the SPARC Reanalysis Intercomparison Project (S-RIP) and overview of the reanalysis systems

Abstract: The climate research community uses atmospheric reanalysis data sets to understand a wide range of processes and variability in the atmosphere, yet different reanalyses may give very different results for the same diagnostics. The Stratosphere–troposphere Processes And their Role in Climate (SPARC) Reanalysis Intercomparison Project (S-RIP) is a coordinated activity to compare reanalysis data sets using a variety of key diagnostics. The objectives of this project are to identify differences among reanalyses and understand their underlying causes, to provide guidance on appropriate usage of various reanalysis products in scientific studies, particularly those of relevance to SPARC, and to contribute to future improvements in the reanalysis products by establishing collaborative links between reanalysis centres and data users. The project focuses predominantly on differences among reanalyses, although studies that include operational analyses and studies comparing reanalyses with observations are also included when appropriate. The emphasis is on diagnostics of the upper troposphere, stratosphere, and lower mesosphere. This paper summarizes the motivation and goals of the S-RIP activity and extensively reviews key technical aspects of the reanalysis data sets that are the focus of this activity. The special issue "The SPARC Reanalysis Intercomparison Project (S-RIP)" in this journal serves to collect research with relevance to the S-RIP in preparation for the publication of the planned two (interim and full) S-RIP reports.

In silico methods for drug repurposing and pharmacology

Abstract: Data in the biological, chemical, and clinical domains are accumulating at ever-increasing rates and have the potential to accelerate and inform drug development in new ways. Challenges and opportunities now lie in developing analytic tools to transform these often complex and heterogeneous data into testable hypotheses and actionable insights. This is the aim of computational pharmacology, which uses in silico techniques to better understand and predict how drugs affect biological systems, which can in turn improve clinical use, avoid unwanted side effects, and guide selection and development of better treatments. One exciting application of computational pharmacology is drug repurposing-finding new uses for existing drugs. Already yielding many promising candidates, this strategy has the potential to improve the efficiency of the drug development process and reach patient populations with previously unmet needs such as those with rare diseases. While current techniques in computational pharmacology and drug repurposing often focus on just a single data modality such as gene expression or drug-target interactions, we argue that methods such as matrix factorization that can integrate data within and across diverse data types have the potential to improve predictive performance and provide a fuller picture of a drug's pharmacological action. WIREs Syst Biol Med 2016, 8:186-210. doi: 10.1002/wsbm.1337 For further resources related to this article, please visit the WIREs website.

Very deep multilingual convolutional neural networks for LVCSR

Abstract: Convolutional neural networks (CNNs) are a standard component of many current state-of-the-art Large Vocabulary Continuous Speech Recognition (LVCSR) systems. However, CNNs in LVCSR have not kept pace with recent advances in other domains where deeper neural networks provide superior performance. In this paper we propose a number of architectural advances in CNNs for LVCSR. First, we introduce a very deep convolutional network architecture with up to 14 weight layers. There are multiple convolutional layers before each pooling layer, with small 3×3 kernels, inspired by the VGG Imagenet 2014 architecture. Then, we introduce multilingual CNNs with multiple untied layers. Finally, we introduce multi-scale input features aimed at exploiting more context at negligible computational cost. We evaluate the improvements first on a Babel task for low resource speech recognition, obtaining an absolute 5.77% WER improvement over the baseline PLP DNN by training our CNN on the combined data of six different languages. We then evaluate the very deep CNNs on the Hub5'00 benchmark (using the 262 hours of SWB-1 training data) achieving a word error rate of 11.8% after cross-entropy training, a 1.4% WER improvement (10.6% relative) over the best published CNN result so far.

Learning with Rejection

Abstract: We introduce a novel framework for classification with a rejection option that consists of simultaneously learning two functions: a classifier along with a rejection function. We present a full theoretical analysis of this framework including new data-dependent learning bounds in terms of the Rademacher complexities of the classifier and rejection families as well as consistency and calibration results. These theoretical guarantees guide us in designing new algorithms that can exploit different kernel-based hypothesis sets for the classifier and rejection functions. We compare and contrast our general framework with the special case of confidence-based rejection for which we devise alternative loss functions and algorithms as well. We report the results of several experiments showing that our kernel-based algorithms can yield a notable improvement over the best existing confidence-based rejection algorithm.

Solving the Optimal Trading Trajectory Problem Using a Quantum Annealer

Abstract: We solve a multi-period portfolio optimization problem using D-Wave Systems’ quantum annealer. We derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. The formulation incorporates transaction costs (including permanent and temporary market impact), and, significantly, the solution does not require the inversion of a covariance matrix. The discrete multi-period portfolio optimization problem we solve is significantly harder than the continuous variable problem. We present insight into how results may be improved using suitable software enhancements and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques.

End-to-end optimization of nonlinear transform codes for perceptual quality

Abstract: We introduce a general framework for end-to-end optimization of the rate-distortion performance of nonlinear transform codes assuming scalar quantization. The framework can be used to optimize any differentiable pair of analysis and synthesis transforms in combination with any differentiable perceptual metric. As an example, we consider a code built from a linear transform followed by a form of multi-dimensional local gain control. Distortion is measured with a state-of-the-art perceptual metric. When optimized over a large database of images, this representation offers substantial improvements in bitrate and perceptual appearance over fixed (DCT) codes, and over linear transform codes optimized for mean squared error.

The Model Intercomparison Project on the climatic response to Volcanic forcing (VolMIP) : experimental design and forcing input data for CMIP6

Abstract: The enhancement of the stratospheric aerosol layer by volcanic eruptions induces a complex set of responses causing global and regional climate effects on a broad range of timescales. Uncertainties exist regarding the climatic response to strong volcanic forcing identified in coupled climate simulations that contributed to the fifth phase of the Coupled Model Intercomparison Project (CMIP5). In order to better understand the sources of these model diversities, the Model Intercomparison Project on the climatic response to Volcanic forcing (VolMIP) has defined a coordinated set of idealized volcanic perturbation experiments to be carried out in alignment with the CMIP6 protocol. VolMIP provides a common stratospheric aerosol data set for each experiment to minimize differences in the applied volcanic forcing. It defines a set of initial conditions to assess how internal climate variability contributes to determining the response. VolMIP will assess to what extent volcanically forced responses of the coupled ocean–atmosphere system are robustly simulated by state-of-the-art coupled climate models and identify the causes that limit robust simulated behavior, especially differences in the treatment of physical processes. This paper illustrates the design of the idealized volcanic perturbation experiments in the VolMIP protocol and describes the common aerosol forcing input data sets to be used.

Antibiotic perturbation of the murine gut microbiome enhances the adiposity, insulin resistance, and liver disease associated with high-fat diet.

Abstract: Obesity, type 2 diabetes, and non-alcoholic fatty liver disease (NAFLD) are serious health concerns, especially in Western populations. Antibiotic exposure and high-fat diet (HFD) are important and modifiable factors that may contribute to these diseases. To investigate the relationship of antibiotic exposure with microbiome perturbations in a murine model of growth promotion, C57BL/6 mice received lifelong sub-therapeutic antibiotic treatment (STAT), or not (control), and were fed HFD starting at 13 weeks. To characterize microbiota changes caused by STAT, the V4 region of the 16S rRNA gene was examined from collected fecal samples and analyzed. In this model, which included HFD, STAT mice developed increased weight and fat mass compared to controls. Although results in males and females were not identical, insulin resistance and NAFLD were more severe in the STAT mice. Fecal microbiota from STAT mice were distinct from controls. Compared with controls, STAT exposure led to early conserved diet-independent microbiota changes indicative of an immature microbial community. Key taxa were identified as STAT-specific and several were found to be predictive of disease. Inferred network models showed topological shifts concurrent with growth promotion and suggest the presence of keystone species. These studies form the basis for new models of type 2 diabetes and NAFLD that involve microbiome perturbation.

Global Existence of Weak Solutions of the Nematic Liquid Crystal Flow in Dimension Three

Abstract: For any bounded smooth domain Ω⊂ℝ3 (or Ω=ℝ3), we establish the global existence of a weak solution (u,d):Ω×[0,+∞)→ℝ3×S2 of the initial boundary value (or the Cauchy) problem of the simplified Ericksen-Leslie system LLF modeling the hydrodynamic flow of nematic liquid crystals for any initial and boundary (or Cauchy) data (u0,d0)∈H×H1(Ω,S2), with d0(Ω)⊂S+2 (the upper hemisphere). Furthermore, (u,d) satisfies the global energy inequality.© 2016 Wiley Periodicals, Inc.

Landau Damping: Paraproducts and Gevrey Regularity

Abstract: We give a new, simpler, but also and most importantly more general and robust, proof of nonlinear Landau damping on $${\mathbb {T}}^d$$ in Gevrey $$-\frac{1}{s}$$ regularity ( $$s > 1/3$$ ) which matches the regularity requirement predicted by the formal analysis of Mouhot and Villani [67]. Our proof combines in a novel way ideas from the original proof of Landau damping Mouhot and Villani [67] and the proof of inviscid damping in 2D Euler Bedrossian and Masmoudi [10]. As in Bedrossian and Masmoudi [10], we use paraproduct decompositions and controlled regularity loss along time to replace the Newton iteration scheme of Mouhot and Villani [67]. We perform time-response estimates adapted from Mouhot and Villani [67] to control the plasma echoes and couple them to energy estimates on the distribution function in the style of the work Bedrossian and Masmoudi [10]. We believe the work is an important step forward in developing a systematic theory of phase mixing in infinite dimensional Hamiltonian systems.

End-to-end optimization of nonlinear transform codes for perceptual quality

Abstract: We introduce a general framework for end-to-end optimization of the rate--distortion performance of nonlinear transform codes assuming scalar quantization. The framework can be used to optimize any differentiable pair of analysis and synthesis transforms in combination with any differentiable perceptual metric. As an example, we consider a code built from a linear transform followed by a form of multi-dimensional local gain control. Distortion is measured with a state-of-the-art perceptual metric. When optimized over a large database of images, this representation offers substantial improvements in bitrate and perceptual appearance over fixed (DCT) codes, and over linear transform codes optimized for mean squared error.

Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow

Abstract: In this work we study the long time inviscid limit of the two dimensional Navier–Stokes equations near the periodic Couette flow. In particular, we confirm at the nonlinear level the qualitative behavior predicted by Kelvin’s 1887 linear analysis. At high Reynolds number Re, we prove that the solution behaves qualitatively like two dimensional Euler for times \({{t \lesssim Re^{1/3}}}\), and in particular exhibits inviscid damping (for example the vorticity weakly approaches a shear flow). For times \({{t \gtrsim Re^{1/3}}}\), which is sooner than the natural dissipative time scale O(Re), the viscosity becomes dominant and the streamwise dependence of the vorticity is rapidly eliminated by an enhanced dissipation effect. Afterwards, the remaining shear flow decays on very long time scales \({{t \gtrsim Re}}\) back to the Couette flow. When properly defined, the dissipative length-scale in this setting is \({{\ell_D \sim Re^{-1/3}}}\), larger than the scale \({{\ell_D \sim Re^{-1/2}}}\) predicted in classical Batchelor–Kraichnan two dimensional turbulence theory. The class of initial data we study is the sum of a sufficiently smooth function and a small (with respect to Re−1) L2 function.

Dissipative Euler Flows with Onsager-Critical Spatial Regularity

Abstract: For any ɛ > 0 we show the existence of continuous periodic weak solutions v of the Euler equations that do not conserve the kinetic energy and belong to the space Lt1(Cx1/3−e); namely, x ↦ v (x,t) is ⅓−e-Holder continuous in space at a.e. time t and the integral ∫[ υ(⋅,t) ]1/3−edt is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class Lt∞(Cx1/3−e).© 2016 Wiley Periodicals, Inc.

Experimental design for three interrelated marine ice sheet and ocean model intercomparison projects: MISMIP v. 3 (MISMIP +), ISOMIP v. 2 (ISOMIP +) and MISOMIP v. 1 (MISOMIP1)

Abstract: . Coupled ice sheet–ocean models capable of simulating moving grounding lines are just becoming available. Such models have a broad range of potential applications in studying the dynamics of marine ice sheets and tidewater glaciers, from process studies to future projections of ice mass loss and sea level rise. The Marine Ice Sheet–Ocean Model Intercomparison Project (MISOMIP) is a community effort aimed at designing and coordinating a series of model intercomparison projects (MIPs) for model evaluation in idealized setups, model verification based on observations, and future projections for key regions of the West Antarctic Ice Sheet (WAIS). Here we describe computational experiments constituting three interrelated MIPs for marine ice sheet models and regional ocean circulation models incorporating ice shelf cavities. These consist of ice sheet experiments under the Marine Ice Sheet MIP third phase (MISMIP+), ocean experiments under the Ice Shelf-Ocean MIP second phase (ISOMIP+) and coupled ice sheet–ocean experiments under the MISOMIP first phase (MISOMIP1). All three MIPs use a shared domain with idealized bedrock topography and forcing, allowing the coupled simulations (MISOMIP1) to be compared directly to the individual component simulations (MISMIP+ and ISOMIP+). The experiments, which have qualitative similarities to Pine Island Glacier Ice Shelf and the adjacent region of the Amundsen Sea, are designed to explore the effects of changes in ocean conditions, specifically the temperature at depth, on basal melting and ice dynamics. In future work, differences between model results will form the basis for the evaluation of the participating models.

Rates and mechanisms of bacterial mutagenesis from maximum-depth sequencing

Abstract: In 1943, Luria and Delbruck used a phage-resistance assay to establish spontaneous mutation as a driving force of microbial diversity. Mutation rates are still studied using such assays, but these can only be used to examine the small minority of mutations conferring survival in a particular condition. Newer approaches, such as long-term evolution followed by whole-genome sequencing, may be skewed by mutational ‘hot’ or ‘cold’ spots. Both approaches are affected by numerous caveats. Here we devise a method, maximum-depth sequencing (MDS), to detect extremely rare variants in a population of cells through error-corrected, high-throughput sequencing. We directly measure locus-specific mutation rates in Escherichia coli and show that they vary across the genome by at least an order of magnitude. Our data suggest that certain types of nucleotide misincorporation occur 10(4)-fold more frequently than the basal rate of mutations, but are repaired in vivo. Our data also suggest specific mechanisms of antibiotic-induced mutagenesis, including downregulation of mismatch repair via oxidative stress, transcription–replication conflicts, and, in the case of fluoroquinolones, direct damage to DNA.

Hierarchical looping of zigzag nucleosome chains in metaphase chromosomes

Abstract: The architecture of higher-order chromatin in eukaryotic cell nuclei is largely unknown. Here, we use electron microscopy-assisted nucleosome interaction capture (EMANIC) cross-linking experiments in combination with mesoscale chromatin modeling of 96-nucleosome arrays to investigate the internal organization of condensed chromatin in interphase cell nuclei and metaphase chromosomes at nucleosomal resolution. The combined data suggest a novel hierarchical looping model for chromatin higher-order folding, similar to rope flaking used in mountain climbing and rappelling. Not only does such packing help to avoid tangling and self-crossing, it also facilitates rope unraveling. Hierarchical looping is characterized by an increased frequency of higher-order internucleosome contacts for metaphase chromosomes compared with chromatin fibers in vitro and interphase chromatin, with preservation of a dominant two-start zigzag organization associated with the 30-nm fiber. Moreover, the strong dependence of looping on linker histone concentration suggests a hierarchical self-association mechanism of relaxed nucleosome zigzag chains rather than longitudinal compaction as seen in 30-nm fibers. Specifically, concentrations lower than one linker histone per nucleosome promote self-associations and formation of these looped networks of zigzag fibers. The combined experimental and modeling evidence for condensed metaphase chromatin as hierarchical loops and bundles of relaxed zigzag nucleosomal chains rather than randomly coiled threads or straight and stiff helical fibers reconciles aspects of other models for higher-order chromatin structure; it constitutes not only an efficient storage form for the genomic material, consistent with other genome-wide chromosome conformation studies that emphasize looping, but also a convenient organization for local DNA unraveling and genome access.

Fixed Energy Universality for Generalized Wigner Matrices

Abstract: We prove the Wigner-Dyson-Mehta conjecture at fixed energy in the bulk of the spectrum for generalized symmetric and Hermitian Wigner matrices. Previous results concerning the universality of random matrices either require an averaging in the energy parameter or they hold only for Hermitian matrices if the energy parameter is fixed. We develop a homogenization theory of the Dyson Brownian motion and show that microscopic universality follows from mesoscopic statistics.© 2015 Wiley Periodicals, Inc.

Super-resolution of point sources via convex programming

Abstract: Recent work has shown that convex programming allows to recover a superposition of point sources exactly from low-resolution data as long as the sources are separated by 2/fc, where fc is the cut-off frequency of the sensing process. The proof relies on the construction of a certificate whose existence implies exact recovery. This certificate has since been used to establish that the approach is robust to noise and to analyze related problems such as compressed sensing off the grid and the super-resolution of splines from moment measurements. In this work we construct a new certificate that allows to extend all these results to signals with minimum separations above 1.26/fc. This is close to 1/fc, the threshold at which the problem becomes inherently ill posed, in the sense that signals with a smaller minimum separation may have low-pass projections with negligible energy.

A Bayesian approach to estimate uncertainty for full-waveform inversion using a priori information from depth migration

Abstract: Full-waveform inversion (FWI) enables us to obtain high-resolution subsurface images; however, estimating model uncertainties associated with this technique is still a challenging problem. We have used a Bayesian inference framework to estimate model uncertainties associated with FWI. The uncertainties were assessed based on an a posteriori covariance operator, evaluated at the maximum a posteriori model. For the prior distribution, we have used a spatially nonstationary covariance operator based on a plane-wave construction with local dips measured from migrated images. Preconditioned frequency-domain FWI was used to estimate the maximum a posteriori model. Efficient manipulation of the posterior covariance was based on a low-rank approximation of the data misfit Hessian preconditioned by the prior covariance operator. The strong decay of the singular values indicated that data were mostly informative about a low-dimensional subspace of model parameters. To reduce computational cost of the random...

Optimal Transport Methods in Economics

Abstract: Optimal Transport Methods in Economics is the first textbook on the subject written especially for students and researchers in economics. Optimal transport theory is used widely to solve problems in mathematics and some areas of the sciences, but it can also be used to understand a range of problems in applied economics, such as the matching between job seekers and jobs, the determinants of real estate prices, and the formation of matrimonial unions. This is the first text to develop clear applications of optimal transport to economic modeling, statistics, and econometrics. It covers the basic results of the theory as well as their relations to linear programming, network flow problems, convex analysis, and computational geometry. Emphasizing computational methods, it also includes programming examples that provide details on implementation. Applications include discrete choice models, models of differential demand, and quantile-based statistical estimation methods, as well as asset pricing models. Authoritative and accessible, Optimal Transport Methods in Economics also features numerous exercises throughout that help you develop your mathematical agility, deepen your computational skills, and strengthen your economic intuition.

An immersed boundary method for rigid bodies

Abstract: We develop an immersed boundary (IB) method for modeling flows around fixed or moving rigid bodies that is suitable for a broad range of Reynolds numbers, including steady Stokes flow. The spatio-temporal discretization of the fluid equations is based on a standard staggered-grid approach. Fluid-body interaction is handled using Peskin's IB method; however, unlike existing IB approaches to such problems, we do not rely on penalty or fractional-step formulations. Instead, we use an unsplit scheme that ensures the no-slip constraint is enforced exactly in terms of the Lagrangian velocity field evaluated at the IB markers. Fractionalstep approaches, by contrast, can impose such constraints only approximately, which can lead to penetration of the flow into the body, and are inconsistent for steady Stokes flow. Imposing no-slip constraints exactly requires the solution of a large linear system that includes the fluid velocity and pressure as well as Lagrange multiplier forces that impose the motion of the body. The principal contribution of this paper is that it develops an efficient preconditioner for this exactly constrained IB formulation which is based on an analytical approximation to the Schur complement. This approach is enabled by the near translational and rotational invariance of Peskin's IB method. We demonstrate that only a few cycles of a geometric multigrid method for the fluid equations are required in each application of the preconditioner, and we demonstrate robust convergence of the overall Krylov solver despite the approximations made in the preconditioner. We empirically observe that to control the condition number of the coupled linear system while also keeping the rigid structure impermeable to fluid, we need to place the immersed boundary markers at a distance of about two grid spacings, which is significantly larger from what has been recommended in the literature for elastic bodies. We demonstrate the advantage of our monolithic solver over split solvers by computing the steady state flow through a two-dimensional nozzle at several Reynolds numbers. We apply the method to a number of benchmark problems at zero and finite Reynolds numbers, and we demonstrate first-order convergence of the method to several analytical solutions and benchmark computations.

Recovering Short Generators of Principal Ideals in Cyclotomic Rings

Abstract: A handful of recent cryptographic proposals rely on the conjectured hardness of the following problem in the ring of integers of a cyclotomic number field: given a basis of a principal ideal that is guaranteed to have a "rather short" generator, find such a generator. Recently, Bernstein and Campbell-Groves-Shepherd sketched potential attacks against this problem; most notably, the latter authors claimed a polynomial-time quantum algorithm. Alternatively, replacing the quantum component with an algorithm of Biasse and Fieker would yield a classical subexponential-time algorithm. A key claim of Campbell et al. is that one step of their algorithm--namely, decoding the log-unit lattice of the ring to recover a short generator from an arbitrary one--is classically efficient whereas the standard approach on general lattices takes exponential time. However, very few convincing details were provided to substantiate this claim. In this work, we clarify the situation by giving a rigorous proof that the log-unit lattice is indeed efficiently decodable, for any cyclotomic of prime-power index. Combining this with the quantum algorithm from a recent work of Biasse and Song confirms the main claim of Campbell et al. Our proof consists of two main technical contributions: the first is a geometrical analysis, using tools from analytic number theory, of the standard generators of the group of cyclotomic units. The second showsthat for a wide class of typical distributions of the short generator, a standard lattice-decoding algorithm can recover it, given any generator. By extending our geometrical analysis, as a second main contribution we obtain an efficient algorithm that, given any generator of a principal ideal in a prime-power cyclotomic, finds a $$2^{\tilde{O}\sqrt{n}}$$ -approximate shortest vector in the ideal. Combining this with the result of Biasse and Song yields a quantum polynomial-time algorithm for the $$2^{\tilde{O}\sqrt{n}}$$ -approximate Shortest Vector Problem on principal ideal lattices.

Hydrodynamics of suspensions of passive and active rigid particles: A rigid multiblob approach

Abstract: Author(s): Usabiaga, FB; Kallemov, B; Delmotte, B; Bhalla, APS; Griffith, BE; Donev, A | Abstract: © 2016 Mathematical Sciences Publishers. We develop a rigid multiblob method for numerically solving the mobility problem for suspensions of passive and active rigid particles of complex shape in Stokes flow in unconfined, partially confined, and fully confined geometries. As in a number of existing methods, we discretize rigid bodies using a collection of minimally resolved spherical blobs constrained to move as a rigid body, to arrive at a potentially large linear system of equations for the unknown Lagrange multipliers and rigid-body motions. Here we develop a block-diagonal preconditioner for this linear system and show that a standard Krylov solver converges in a modest number of iterations that is essentially independent of the number of particles. Key to the efficiency of the method is a technique for fast computation of the product of the blob-blob mobility matrix and a vector. For unbounded suspensions, we rely on existing analytical expressions for the Rotne-Prager-Yamakawa tensor combined with a fast multipole method (FMM) to obtain linear scaling in the number of particles. For suspensions sedimented against a single no-slip boundary, we use a direct summation on a graphical processing unit (GPU), which gives quadratic asymptotic scaling with the number of particles. For fully confined domains, such as periodic suspensions or suspensions confined in slit and square channels, we extend a recently developed rigid-body immersed boundary method by B. Kallemov, A. P. S. Bhalla, B. E. Griffith, and A. Donev (Commun. Appl. Math. Comput. Sci. 11 (2016), no. 1, 79-141) to suspensions of freely moving passive or active rigid particles at zero Reynolds number. We demonstrate that the iterative solver for the coupled fluid and rigid-body equations converges in a bounded number of iterations regardless of the system size. In our approach, each iteration only requires a few cycles of a geometric multigrid solver for the Poisson equation, and an application of the block-diagonal preconditioner, leading to linear scaling with the number of particles. We optimize a number of parameters in the iterative solvers and apply our method to a variety of benchmark problems to carefully assess the accuracy of the rigid multiblob approach as a function of the resolution. We also model the dynamics of colloidal particles studied in recent experiments, such as passive boomerangs in a slit channel, as well as a pair of non-Brownian active nanorods sedimented against a wall.