Hierarchical Matrix Research Articles

Evolution and Development at the Origin of a Phylum.

Quantifying morphological evolution is key to determining the patterns and processes underlying the origin of phyla. We constructed a hierarchical morphological character matrix to characterize the radiation and establishment of echinoderm body plans during the early Paleozoic. This showed that subphylum-level clades diverged gradually through the Cambrian, and the distinctiveness of the resulting body plans was amplified by the extinction of transitional forms and obscured by convergent evolution during the Ordovician. Higher-order characters that define these body plans were not fixed at the origin of the phylum, countering hypotheses regarding developmental processes governing the early evolution of animals. Instead, these burdened characters were flexible, enabling continued evolutionary innovation throughout the clades' history.

Block-Adaptive Cross Approximation of Discrete Integral Operators

Abstract In this article, we extend the adaptive cross approximation (ACA) method known for the efficient approximation of discretisations of integral operators to a block-adaptive version. While ACA is usually employed to assemble hierarchical matrix approximations having the same prescribed accuracy on all blocks of the partition, for the solution of linear systems, it may be more efficient to adapt the accuracy of each block to the actual error of the solution as some blocks may be more important for the solution error than others. To this end, error estimation techniques known from adaptive mesh refinement are applied to automatically improve the blockwise matrix approximation. This allows to interlace the assembling of the coefficient matrix with the iterative solution.

Hierarchical Membrane Dynamics in Phase-Separated Model Membranes

Lipid membranes, the primary matrix of cell membranes, exhibit a hierarchy of dynamics from molecular motions to collective undulations, which control spatiotemporal membrane phenomena and regulate various cellular functions. Thus, understanding how various dynamic modes influence membrane behavior is crucial to uncovering the design rules of cell membranes and implementing them in future technologies. Previous membrane dynamics studies using neutron spin-echo spectroscopy (NSE) have shed light on the dependence of collective membrane fluctuations on viscoelastic membrane properties, demonstrating that thickness fluctuation modes (∼100ns) are dictated by membrane viscosity, which directly governs fast (ps) molecular diffusive motions. Here, we focus on phase-separating lipid membranes, composed of DMPC:DSPC mixtures, to understand the effect of domain formation/growth on hierarchical matrix dynamics using selective lipid deuteration and different neutron spectroscopy techniques. NSE measurements on DMPC:DSPC-d83 (70:30 mol%) show that bending rigidity of the DMPC-rich matrix gradually increases with decreasing temperature - as the fully-fluid membrane approaches the upper phase transition (i.e. with the formation of transient DSPC clusters) - and increases further in the phase coexistence region with the growth of DSPC domains. Comparatively, NSE measurements on DMPC-d54:DSPC-d83 vesicles show pronounced matrix thickness fluctuations in the fully-fluid phase and drastic suppression of this fluctuation mode in the gel-fluid coexistence phase, even when the DMPC-rich matrix is still in its fluid phase. This suppression in fluctuations suggests an increase in the matrix viscosity with the growth of gel DSPC domains, which is further investigated by selectively measuring lipid diffusion in the DMPC-rich matrix using neutron backscattering spectroscopy. Comparisons of the experimental results with coarse-grained MD simulations illustrate the underlying coupling mechanisms. Overall, these observations indicate a clear mechanical and viscoelastic coupling between lipid domains and their host matrix, suggesting that domain formation influences membrane properties beyond local effects.

SU(5) grand unified theory with A4 modular symmetry

We present the first example of a grand unified theory (GUT) with a modular symmetry interpreted as a family symmetry. The theory is based on supersymmetric $SU(5)$ in 6d, where the two extra dimensions are compactified on a $T_2/\mathbb{Z}_2$ orbifold. We have shown that, if there is a finite modular symmetry, then it can only be $A_4$ with an (infinite) discrete choice of moduli, where we focus on $\tau = \omega=e^{i2\pi/3}$, the unique solution with $|\tau|=1$. The fields on the branes respect a generalised CP and flavour symmetry $A_4\ltimes \mathbb{Z}_2$ which is isomorphic to $S_4$ which leads to an effective $\mu-\tau$ reflection symmetry at low energies, implying maximal atmospheric mixing and maximal leptonic CP violation. We construct an explicit model along these lines with two triplet flavons in the bulk, whose vacuum alignments are determined by orbifold boundary conditions, analogous to those used for $SU(5)$ breaking with doublet-triplet splitting. There are two right-handed neutrinos on the branes whose Yukawa couplings are determined by modular weights. The charged lepton and down-type quarks have diagonal and hierarchical Yukawa matrices, with quark mixing due to a hierarchical up-quark Yukawa matrix.

A High-Kinetics Sulfur Cathode with a Highly Efficient Mechanism for Superior Room-Temperature Na-S Batteries.

Applications of room-temperature-sodium sulfur (RT-Na/S) batteries are currently impeded by the insulating nature of sulfur, the slow redox kinetics of sulfur with sodium, and the dissolution and migration of sodium polysulfides. Herein, a novel micrometer-sized hierarchical S cathode supported by FeS2 electrocatalyst, which is grown in situ in well-confined carbon nanocage assemblies, is presented. The hierarchical carbon matrix can provide multiple physical entrapment to polysulfides, and the FeS2 nanograins exhibit a low Na-ion diffusion barrier, strong binding energy, and high affinity for sodium polysulfides. Their combination makes it an ideal sulfur host to immobilize the polysulfides and achieve reversible conversion of polysulfides toward Na2 S. Importantly, the hierarchical S cathode is suitable for large-scale production via the inexpensive and green spray-drying method. The porous hierarchical S cathode offers a high sulfur content of 65.5 wt%, and can deliver high reversible capacity (524 mAh g-1 over 300 cycles at 0.1 A g-1 ) and outstanding rate capability (395 mAh g-1 at 1 A g-1 for 850 cycles), holding great promise for both scientific research and real application.

Extending the hierarchical decision matrix to incorporate a dominance ranking score for economic systematic reviews

As the base of clinical evidence grows, it is increasingly common to conduct economic evaluations in addition to clinical evaluations of effectiveness in order to inform health policies. For economic systematic reviews there is currently no agreed-upon quantitative method to obtain a pooled economic effect size. With no suitable quantitative method available, the hierarchical decision matrix stands out as a tool that enables a visual summary of different types of economic studies, but there are limitations with the hierarchical decision matrix. We extended the hierarchical decision matrix with a weighted scoring system (termed dominance ranking score) to allow for useful information of a study design to be incorporated.•The scoring system of the dominance ranking score incorporates weighting factors that are based on sample size and effect size of a study.•The dominance ranking score enables a more differentiating analysis of dominance levels.•For systematic reviews that include partial economic studies, both the hierarchical decision matrix and the dominance ranking score assist to indicate the level of economic potential for a particular intervention, which facilitates the conduct of subsequent full economic studies.

Hierarchical Matrix Approximations of Hessians Arising in Inverse Problems Governed by PDEs

Hessian operators arising in inverse problems governed by partial differential equations (PDEs) play a critical role in delivering efficient, dimension-independent convergence for Newton solution of deterministic inverse problems, as well as Markov chain Monte Carlo sampling of posteriors in the Bayesian setting. These methods require the ability to repeatedly perform operations on the Hessian such as multiplication with arbitrary vectors, solving linear systems, inversion, and (inverse) square root. Unfortunately, the Hessian is a (formally) dense, implicitly defined operator that is intractable to form explicitly for practical inverse problems, requiring as many PDE solves as inversion parameters. Low rank approximations are effective when the data contain limited information about the parameters but become prohibitive as the data become more informative. However, the Hessians for many inverse problems arising in practical applications can be well approximated by matrices that have hierarchically low rank structure. Hierarchical matrix representations promise to overcome the high complexity of dense representations and provide effective data structures and matrix operations that have only log-linear complexity. In this work, we describe algorithms for constructing and updating hierarchical matrix approximations of Hessians, and illustrate them on a number of representative inverse problems involving time-dependent diffusion, advection-dominated transport, frequency domain acoustic wave propagation, and low frequency Maxwell equations, demonstrating up to an order of magnitude speedup compared to globally low rank approximations.

Hm-toolbox: MATLAB Software for HODLR and HSS Matrices

Matrices with hierarchical low-rank structure, including HODLR and HSS matrices, constitute a versatile tool to develop fast algorithms for addressing large-scale problems. While existing software ...

Bayesian Nonnegative Matrix Factorization With Dirichlet Process Mixtures

Nonnegative Matrix Factorization (NMF) is valuable in many applications of blind source separation, signal processing and machine learning. A number of algorithms that can infer nonnegative latent factors have been developed, but most of these assume a specific noise kernel. This is insufficient to deal with complex noise in real scenarios. In this paper, we present a hierarchical Dirichlet process nonnegative matrix factorization (DPNMF) model in which the Gaussian mixture model is used to approximate the complex noise distribution. Moreover, the model is cast in the nonparametric Bayesian framework by using Dirichlet process mixture to infer the necessary number of Gaussian components. We derive a mean-field variational inference algorithm for the proposed nonparametric Bayesian model. We first test the model on synthetic data sets contaminated by Gaussian, sparse and mixed noise. We then apply it to extract muscle synergies from the electromyographic (EMG) signal and to select discriminative features for motor imagery single-trial electroencephalogram (EEG) classification. Experimental results demonstrate that DPNMF performs better in extracting the latent nonnegative factors in comparison with state-of-the-art methods.

Risk assessment of landslide-induced surge disaster of river type reservoir in mountainous area

The landslide-induced surge of river type reservoir in the mountainous area has become one of the most important disasters affecting the waterborne transportation and the safety of human life and property in the reservoir area. Based on the field observation of landslide and landslide-induced surge in the Three Gorges Reservoir (TGR) area and relevant research results, this paper develops the basic principles for selecting the risk factors of landslide-induced surge, analyses the risk factors of landslide-induced surge, and gives the main and secondary risk factors and their characteristics. According to the characteristics of landslide-induced surge disaster, the hierarchical structure, judgment matrix, and risk calculation model suitable for landslide-induced surge disaster are established by using analytic hierarchy process method, and the classification standard of the action index of each main evaluation factor is determined. This well-developed evaluation method is then applied to evaluate the risk of Hongyanzi landslide-induced surge disaster in the TGR area. The evaluation result is high risk, which is consistent with the actual disaster situation.

PEEC Modeling of Planar Spiral Resonators

The partial element equivalent circuit (PEEC) method is applied to a frequency-sweep study for investigating the resonant modes of planar spiral resonators. A low-rank approximation algorithm, based on hierarchical matrices, is coupled to the method in order to reduce the computational effort due to the use of dense matrices. The partitioning of the matrices is carefully addressed in order to allow an efficient hierarchical off-diagonal low-rank structure of the system matrix.

A Novel Sign Language Recognition Framework Using Hierarchical Grassmann Covariance Matrix

Visual sign language recognition is an interesting and challenging problem. To create a discriminative representation, a hierarchical Grassmann covariance matrix (HGCM) model is proposed for sign description. Furthermore, a multi-temporal belief propagation (MTBP) based segmentation approach is presented for continuous sequence spotting. Concretely speaking, a sign is represented by multiple covariance matrices, followed by evaluating and selecting their most significant singular vectors. These covariance matrices are transformed into a more compact and discriminative HGCM, which is formulated on the Grassmann manifold. Continuous sign sequences can be recognized frame by frame using the HGCM model, before being optimized by MTBP, which is a carefully designed graphic model. The proposed method is thoroughly evaluated on isolated and synthetic and real continuous sign datasets as well as on HDM05. Extensive experimental results convincingly show the effectiveness of our proposed framework.

On the characterization of interstitial fluid flow in the skeletal muscle endomysium

In this paper, the interstitial fluid flow in skeletal muscle endomysium was examined using an in-situ indentation testing in combination with theoretical modelling. The objective was to understand the transport properties of the three-dimensional and highly hierarchical muscular interstitial matrices, which play important roles in muscle-bone cross-talk and signaling during musculoskeletal development and maintenance. Gastrocnemius muscles from four 3-month old calves were harvested and subjected to a creeping test using a custom-designed device. The experiments, in combination with an anatomy-based theoretical model, were used to capture the spatial-temporal response of the skeletal muscle to external impacts. For the first time, the detailed load-induced interstitial fluid pressurization in the muscle endomyseal space was obtained. The relative contribution from the solid muscle fibers and the interstitial fluid to the temporal loading response was captured. The paper presented herein provides important information regarding the mechanical environment within the muscle tissue, which could help the future study of muscle's response to forces and its subsequent signaling to surrounding tissues in vivo.

An Iteration-Free Domain Decomposition Method for the Fast Finite Element Analysis of Electromagnetic Problems

An iteration-free finite element domain decomposition method (DDM) is proposed for the simulation of 3-D electromagnetic problems. The finite element tearing and interconnecting (FETI) with Robin-type transmission condition is introduced to construct a domain decomposition framework. Based on hierarchical ( $\mathcal {H}$ -) matrix algorithm, the numerical Green’s function of each subdomain can be represented by a data-sparse matrix form. Subsequently, the original 3-D problem is reduced to a 2-D problem on the skeleton. Derived from the finite element discretization of pseudo-differential operator, the skeleton matrix can also be efficiently approximated with an $\mathcal {H}$ -matrix by assembling all the numerical Green’s functions of subdomains. Based on $\mathcal {H}$ -LU factorization algorithm, the computational complexity and memory requirement for the direct solution of the skeleton equation can be reduced to be logarithmic linear. The resulting direct FETI (D-FETI) method completely depends on direct method without any iterative process. Hence, it does not suffer from the issue of convergence rate, and avoids redundant computations for multiple right-hand side problems. Numerical examples demonstrate the effectiveness of the proposed method.

Hierarchical N-doping germanium/carbon nanofibers as anode for high-performance lithium-ion and sodium-ion batteries

Germanium (Ge) has gained a great deal of attention as an anode material for sodium ion batteries (SIBs) and lithium ion batteries (LIBs) for its high theoretical capacity and ion diffusivity. Unfortunately, Ge particle pulverization triggered by huge volume expansion during the alloying and dealloying processes can cause rapid capacity fade. Herein we report a facile method for the preparation of ultrafine Ge nanoparticles embedded in hierarchical N-doped multichannel carbon fibers (denoted as Ge-NMCFs) by electrospinning. The hierarchical carbon matrix not only provides sufficient internal void space to accommodate the large volume expansion of Ge nanoparticles, but also provides numerous open channels for the easy access of electrolyte and Na/Li ions. As half-cell tests revealed, the composite provides discharge capacity of 303 mA h g−1 (1st cycle) and 160 mA h g−1 (700th cycle) for SIBs, 1146.7 mA h g−1 (1st cycle) and 600 mA h g−1 (500th cycle) for LIBs at a current density of 500 mA g−1 (all the presented capacity based on the total weight of Ge/C composites). Density functional theory calculation suggests that N-doped in carbon can enhance the Na/Li ion storage and improve the electrochemical performance. This demonstration is an important step towards the development of SIBs and LIBs with much higher specific energy capacity and longer cycle stability.

MOF-derived yolk-shell Ni@C@ZnO Schottky contact structure for enhanced microwave absorption

Metal–organic frameworks (MOF) derivative used as microwave absorber has been attracting extensive interest due to the diversity of host-guest coordination and unique structure at micro/nano-meters level. However, the accurate adjustment of metal-semiconductor interfaces construct remains a great challenge. Here, a novel hierarchical multi-interfacial Ni@C@ZnO microsphere with special Schottky contact structure was successfully fabricated after annealed the bimetallic Ni-Zn-MOF precursor. The unique yolk-shell microsphere was assembled together by the core-shell Ni@C micro-units and ZnO flakes. The uniformly dispersed ZnO flakes were anchored inside the hierarchical conductive carbon matrix. The yolk-shell Ni@C@ZnO materials displayed high-performance microwave absorption after the micro-nano structure and interfacial design were optimized. In particular, with a low mass additive amount of only 25%, the maximum reflection loss (RL) reached −55.8 dB at 2.5 mm, and the effective absorption bandwidth (RL ≤ −10 dB) covered as wide as 4.1 GHz. The excellent microwave absorption performance could be attributed to the polarized interfaces of Ni-C-ZnO, which contributes a metallic/semiconductor barrier. Thus, both magnetic-dielectric synergetic effect and interfacial polarization might lead to an evident absorption. This novel lightweight magnetic MOF-derived composites promise great potential in the practical microwave absorption fields.

Harnessing Photoluminescent Properties of Carbon Nitride Nanosheets in a Hierarchical Matrix

AbstractRapid screening of polycyclic aromatic hydrocarbons (PAHs), a large class of toxic and carcinogenic compounds, is highly desirable for environmental and food safety. In principle, a photoluminescent (PL) assay is a potential route, but demands highly efficient and stable, low cost, and environment‐friendly photoactive materials. 2D carbon nitride nanosheets (CNNS) that have triggered increasing interest may meet all these requirements. However, conventional applications of CNNS are restricted in aqueous solution, in which the long‐term dispersibility is difficult to be addressed. Here, the harnessing of unique PL properties of CNNS is reported by dispersing them in a hierarchical polymeric matrix, driven by electrostatic interactions. Such a configuration not only enables a solid‐state host to perpetually maintain homogeneous dispersibility of CNNS but also processes free channels for sufficient mass transfer. Moreover, the nanohybrid inherits the unique PL properties of CNNS. Further using β‐CD as the molecular recognition unit, an economic, highly selective and sensitive paper‐based PL sensor for PAHs is constructed via an inner‐filter effect. This work opens a new avenue in harnessing unique PL properties of CNNS in solid state with superior stability for a variety of potential sensing applications, especially for point‐of‐care testing and/or in developing countries.

MOF-derived Co9S8/MoS2 embedded in tri-doped carbon hybrids for efficient electrocatalytic hydrogen evolution

Abstract Metal–organic framework (MOF) derived hybrid materials have been developed as an efficient non-noble-metal electrocatalysts for clean energy conversion systems. In this work, a Co-based MOF containing nitrogen and oxygen heteroatoms (Co-NOMOF) mixed with the thiomolybdate [Mo3S13]2− nanoclusters was used to prepare the N, S, O-doped carbon encapsulating Co9S8 and MoS2 (Co9S8/MoS2@NSOC) nanocomposite by one-step pyrolysis. The Co9S8/MoS2@NSOC nanocomposite exhibited remarkable catalytic performance for hydrogen evolution reaction (HER) with overpotential of 194 and 233 mV in 1 M KOH and 0.5 M H2SO4 solution under 10 mA cm−2, respectively, which was ascribed to the multiheteroatom-doped hierarchical porous carbon matrix and the synergistic effect of intrinsic activity of Co9S8 and MoS2. This work provides new opportunity for developing highly efficient non-precious metal electrochemical catalysts.

Scalable Gaussian Process Computations Using Hierarchical Matrices

We present a kernel-independent method that applies hierarchical matrices to the problem of maximum likelihood estimation for Gaussian processes. The proposed approximation provides natural and scalable stochastic estimators for its gradient and Hessian, as well as the expected Fisher information matrix, that are computable in quasilinear complexity for a large range of models. To accomplish this, we (i) choose a specific hierarchical approximation for covariance matrices that enables the computation of their exact derivatives and (ii) use a stabilized form of the Hutchinson stochastic trace estimator. Since both the observed and expected information matrices can be computed in quasilinear complexity, covariance matrices for maximum likelihood estimators (MLEs) can also be estimated efficiently. In this study, we demonstrate the scalability of the method, show how details of its implementation effect numerical accuracy and computational effort, and validate that the resulting MLEs and confidence intervals based on the inverse Fisher information matrix faithfully approach those obtained by the exact likelihood. Supplementary materials for this article are available online.

A Hierarchical VLSM-Based Demand Response Strategy for Coordinative Voltage Control Between Transmission and Distribution Systems

This paper presents a two-stage hierarchical voltage-load sensitivity matrix (VLSM)-based demand response (DR) algorithm for regulating the voltage at distribution feeders while fulfilling the transmission system DR requests. This allows distributed energy resources to be aggregated to provide transmission-level services without causing operational issues in the distribution grids. At the beginning of each operation interval, a transmission system controller will issue DR commands to distribution system controllers. In the first stage, the distribution system controller uses a VLSM-based dispatch algorithm to dispatch the available DR resources on the distribution feeder with an objective to minimize the total voltage deviations at the lowest cost. Then, power flow studies are conducted assuming that the DR commands have been executed. If voltage violations are detected, a second-stage VLSM-based DR dispatch is performed to remove those violations. After that, the upper and lower DR limits calculated for the next operation interval are sent back to the transmission system controller so the transmission optimization algorithm can use them as operational constraints to make subsequent decisions. The DR resources include smart photovoltaic (PV) inverters that can curtail real power and provide reactive power support, controllable loads and capacitor banks. The IEEE 123-bus test system with 5-min PV and load data is used to evaluate the performance of the algorithm. Simulation results show that the proposed algorithm can fulfill the transmission-level DR requests while maintaining the voltage in the distribution system within limits.