New Class Of Methods Research Articles

A new class of third-order iterative methods for multiple roots and their geometric construction

This work aims to provide a class of third-order iterative methods for solving univariate nonlinear equations with multiple roots when the multiplicity is unknown. To obtain the class of methods mentioned above, a univariate nonlinear equation is used that has the same roots as the original equation. However, the roots of this equivalent equation are simple; that is, they have multiplicity one. Therefore, Gander’s theorem can be used to construct a new class of methods. Then the geometric construction of the elements of said class is done, which satisfies the osculance condition. In addition, some families of methods belonging to said class are presented, as well as their geometric construction. Finally, numerical examples are presented where the behavior of some methods belonging to the families within the method class is observed.

Hybrid block formulae whose eigenvalues of Jacobian matrices are close to the imaginary axis of the complex plane

Methods for integrating stiff initial value problems are required to be A-stable. Of great interest are A-stable methods whose Jacobian matrices have their eigenvalues close to the imaginary axis of the complex plane. This class of A-stable methods are very rare. This paper is on the development of a new family of A-stable hybrid block method whose Jacobian matrices possess eigenvalues on the imaginary of the complex plane via interpolation and collocation techniques. The family of methods developed herein are A-stable for order p ≤ 18. Numerical solutions generated by the new method are compared with existing methods in the literature. The numerical results show that the new class of methods are more efficient and accurate.

From thermoeconomics to environomics and beyond

Modern energy systems are more and more based on integrated technologies in a world evolving towards increasing concerns not only for economics but also for environmental issues with a growing focus on resources and emissions. In parallel, information technologies including advanced simulation and optimisation algorithms develop at a rapid pace. Thermoeconomics based on either the First Law or exergy, together with cost factors distributed through its equipment, also evolved during the last 30 years. However modern muti-objective evolutionary optimisation techniques, extending the capacity to tackle optimisation considering a growing number of parameters including the internalization of the costs of emissions of whole systems has also emerged. The latter is called environomics. So far, the supply of energy services was essentially dominated by fossil-based resources with a high focus on operational costs. The new trends towards renewable energies requires a growing need to consider the embedded exergies since renewable energy, like solar energy, is economically free and operational costs are inherently low. While economic factors can vary over a broad spectrum throughout the years of operation, the embedded exergies are more stable values, particularly because the systems are built in a known economic and energetic environment. A new class of methods is only emerging to deal with more complete exergy approaches to formulate more holistic exergy life cycle analyses. Those should provide a lower bound of the expected exergy payback over the lifetime of the systems to be compared with thermoeconomic or environomic optima. It is still a huge challenge ahead to provide practical tools to do it.

Realizing Efficient On-Device Language-based Image Retrieval

Advances in deep learning have enabled accurate language-based search and retrieval, e.g., over user photos, in the cloud. Many users prefer to store their photos in the home due to privacy concerns. As such, a need arises for models that can perform cross-modal search on resource-limited devices. State-of-the-art cross-modal retrieval models achieve high accuracy through learning entangled representations that enable fine-grained similarity calculation between a language query and an image, but at the expense of having a prohibitively high retrieval latency. Alternatively, there is a new class of methods that exhibits good performance with low latency, but requires a lot more computational resources, and an order of magnitude more training data (i.e. large web-scraped datasets consisting of millions of image-caption pairs) making them infeasible to use in a commercial context. From a pragmatic perspective, none of the existing methods are suitable for developing commercial applications for low-latency cross-modal retrieval on low-resource devices. We propose CrispSearch, a cascaded approach that greatly reduces the retrieval latency with minimal loss in ranking accuracy for on-device language-based image retrieval. The idea behind our approach is to combine a light-weight and runtime-efficient coarse model with a fine re-ranking stage. Given a language query, the coarse model effectively filters out many of the irrelevant image candidates. After this filtering, only a handful of strong candidates will be selected and sent to a fine model for re-ranking. Extensive experimental results with two SOTA models for the fine re-ranking stage, on standard benchmark datasets show that CrispSearch results in a speedup of up to 38 times over the SOTA fine methods with negligible performance degradation. Moreover, our method does not require millions of training instances, making it a pragmatic solution to on-device search and retrieval.

A flexible short recurrence Krylov subspace method for matrices arising in the time integration of port-Hamiltonian systems and ODEs/DAEs with a dissipative Hamiltonian

For several classes of mathematical models that yield linear systems, the splitting of the matrix into its Hermitian and skew Hermitian parts is naturally related to properties of the underlying model. This is particularly so for discretizations of dissipative Hamiltonian ODEs, DAEs and port-Hamiltonian systems where, in addition, the Hermitian part is positive definite or semi-definite. It is then possible to develop short recurrence optimal Krylov subspace methods in which the Hermitian part is used as a preconditioner. In this paper, we develop new, right preconditioned variants of this approach which, as their crucial new feature, allow the systems with the Hermitian part to be solved only approximately in each iteration while keeping the short recurrences. This new class of methods is particularly efficient as it allows, for example, to use few steps of a multigrid solver or a (preconditioned) CG method for the Hermitian part in each iteration. We illustrate this with several numerical experiments for large scale systems.

DeepQMC: An open-source software suite for variational optimization of deep-learning molecular wave functions.

Computing accurate yet efficient approximations to the solutions of the electronic Schrödinger equation has been a paramount challenge of computational chemistry for decades. Quantum Monte Carlo methods are a promising avenue of development as their core algorithm exhibits a number of favorable properties: it is highly parallel and scales favorably with the considered system size, with an accuracy that is limited only by the choice of the wave function Ansatz. The recently introduced machine-learned parametrizations of quantum Monte Carlo Ansätze rely on the efficiency of neural networks as universal function approximators to achieve state of the art accuracy on a variety of molecular systems. With interest in the field growing rapidly, there is a clear need for easy to use, modular, and extendable software libraries facilitating the development and adoption of this new class of methods. In this contribution, the DeepQMC program package is introduced, in an attempt to provide a common framework for future investigations by unifying many of the currently available deep-learning quantum Monte Carlo architectures. Furthermore, the manuscript provides a brief introduction to the methodology of variational quantum Monte Carlo in real space, highlights some technical challenges of optimizing neural network wave functions, and presents example black-box applications of the program package. We thereby intend to make this novel field accessible to a broader class of practitioners from both the quantum chemistry and the machine learning communities.

Locally structure-preserving div-curl operators for high order discontinuous Galerkin schemes

We propose a novel Structure-Preserving Discontinuous Galerkin (SPDG) operator that recovers at the discrete level the algebraic property related to the divergence of the curl of a vector field, which is typically referred to as div-curl problem. A staggered Cartesian grid is adopted in 3D, where the vector field is naturally defined at the corners of the control volume, while its curl is evaluated as a cell-centered quantity. Firstly, the curl operator is rewritten as the divergence of a tensor, hence allowing compatible finite difference schemes to be devised and to be proven to mimic the algebraic div-curl property. Successively, a high order DG divergence operator is built upon integration by parts, so that the structure-preserving finite difference div-curl operator is exactly retrieved for first order discretizations. We further demonstrate that the novel SPDG schemes are capable of obtaining a zero div-curl identity with machine precision from second up to sixth order accuracy. Curl-grad operators follow as a consequence. In a second part, we show the applicability of these SPDG methods by solving the incompressible Navier-Stokes equations written in vortex-stream formulation. This hyperbolic system deals with divergence-free involutions related to the velocity and vorticity field as well as to the stream function, thus it provides an ideal setting for the validation of the novel schemes. A compatible discretization of the numerical viscosity is also proposed in order to maintain the structure-preserving property of the div-curl DG operators even in the presence of artificial or physical dissipative terms. Finally, to overcome the time step restriction dictated by the viscous sub-system, Implicit-Explicit (IMEX) Runge-Kutta time stepping techniques are tailored to handle the SPDG framework. Numerical examples show that the theoretical order of convergence is reached by this new class of methods, and prove the exact preservation of the incompressibility constraint.

Generalized TASE-RK methods for stiff problems

A family of Time-Accurate and Stable Explicit (TASE) methods for the numerical integration of Initial Value Problems in stiff Ordinary Differential Equations (ODEs) y′(t)=f(t,y) was recently introduced in [1]. The main idea was to make local extrapolation of a stabilized Euler method. More recently, in [3] a similar approach by considering the stabilization of arbitrary explicit Runge-Kutta methods (TASE-RK) was analyzed. In this case the explicit Runge-Kutta method integrates a transformed ODE obtained by multiplying the vector field f(t,y) by a certain operator which approximates the identity mapping up to a given order p. The main inconvenience of both approaches is that to reach order p the solution of p2 linear systems plus the evaluation of p derivatives are required per integration step.In order to substantially reduce the computational costs of the former approaches in the linear system solution, but maintaining the good accuracy and stability properties, a new family of TASE-RK methods which allow to introduce a few more free parameters are considered. The formulation of the methods was conceived to be implemented not only in sequential mode but it admits parallelism in a straightforward way. Furthermore, since these methods are linearly implicit, connections to the class of W-methods [19] are properly established. The order conditions for the new class of methods are widely studied by using the rooted tree theory. For p=3,4, new methods with p sequential stages and order p are derived and compared on semidiscrete 1D and 2D Partial Differential Equations (PDEs) to those in [1,3] and other standard Rosenbrock and W-methods in the literature.

Jacobian-free High Order Local Linearization methods for large systems of initial value problems

In this paper, the class of Jacobian-free High Order Local Linearization (HOLL) methods is introduced for integrating large systems of initial value problems. Unlike other high order exponential integrators, this new class of methods involves the approximation of just a single phi-function times vector and does not require the evaluation and storing of Jacobian matrices, which result more efficient in terms of flops operations and computer memory. Specifically, the new Jacobian-free integrators are constructed by approximating the products of Jacobian matrix times vector that appear in the ordinary HOLL methods. A general result on the convergence rate of the new methods is derived in terms of the convergence rate of the mentioned approximations, resulting in a simple order condition that preserves the order of the ordinary HOLL methods. To design effective integrators of the new class, a novel Matrix-free Krylov-Padé approximation to the product of phi-function times vector is also proposed as well as an adaptive strategy for the automatic selection of the Krylov dimension and Padé order. As a particular instance, a class of Jacobian-free Locally Linearized Runge-Kutta schemes is developed, and schemes of third to fifth order explicitly constructed. Numerical simulations are provided illustrating the theoretical findings concerning the convergence rate of the introduced methods as well as their performance in comparison with other Jacobian-free integrators.

Inexact GMRES Policy Iteration for Large-Scale Markov Decision Processes

Policy iteration enjoys a local quadratic rate of contraction, but its iterations are computationally expensive for Markov decision processes (MDPs) with a large number of states. In light of the connection between policy iteration and the semismooth Newton method and taking inspiration from the inexact variants of the latter, we propose inexact policy iteration, a new class of methods for large-scale finite MDPs with local contraction guarantees. We then design an instance based on the deployment of the generalized minimal residual method (GMRES) for the approximate policy evaluation step, which we call inexact GMRES policy iteration. Finally, we demonstrate the superior practical performance of inexact GMRES policy iteration on an MDP with 10000 states, where it achieves a × 5.8 and × 2.2 speedup with respect to policy iteration and optimistic policy iteration, respectively.

Reducing the complexity of chemical networks via interpretable autoencoders

In many astrophysical applications, the cost of solving a chemical network represented by a system of ordinary differential equations (ODEs) grows significantly with the size of the network and can often represent a significant computational bottleneck, particularly in coupled chemo-dynamical models. Although standard numerical techniques and complex solutions tailored to thermochemistry can somewhat reduce the cost, more recently, machine learning algorithms have begun to attack this challenge via data-driven dimensional reduction techniques. In this work, we present a new class of methods that take advantage of machine learning techniques to reduce complex data sets (autoencoders), the optimization of multiparameter systems (standard backpropagation), and the robustness of well-established ODE solvers to to explicitly incorporate time dependence. This new method allows us to find a compressed and simplified version of a large chemical network in a semiautomated fashion that can be solved with a standard ODE solver, while also enabling interpretability of the compressed, latent network. As a proof of concept, we tested the method on an astrophysically relevant chemical network with 29 species and 224 reactions, obtaining a reduced but representative network with only 5 species and 12 reactions, and an increase in speed by a factor 65.

Conditional calibration for false discovery rate control under dependence

We introduce a new class of methods for finite-sample false discovery rate (FDR) control in multiple testing problems with dependent test statistics where the dependence is known. Our approach separately calibrates a data-dependent p-value rejection threshold for each hypothesis, relaxing or tightening the threshold as appropriate to target exact FDR control. In addition to our general framework, we propose a concrete algorithm, the dependence-adjusted Benjamini–Hochberg (dBH) procedure, which thresholds the BH-adjusted p-value for each hypothesis. Under positive regression dependence, the dBH procedure uniformly dominates the standard BH procedure, and in general it uniformly dominates the Benjamini–Yekutieli (BY) procedure (also known as BH with log correction), which makes a conservative adjustment for worst-case dependence. Simulations and real data examples show substantial power gains over the BY procedure, and competitive performance with knockoffs in settings where both methods are applicable. When the BH procedure empirically controls FDR (as it typically does in practice), the dBH procedure performs comparably.

Error bounds in the computation of outer inverses with generalized Schultz iterative methods and its use in computing of Moore-Penrose inverse

An error bound in computing of outer inverses is established in each iteration of the generalized Schultz iterative methods. With this error bound, we built class of iterative method for the calculation of the Moore-Penrose inverse, the class of methods uses these error bounds to generate monotonic inclusion interval matrices which congerges to Moore-Penrose inverse, this process using intervals prevents that round-off errors cause the divergence of the method. Theorems with the error bounds as well as the convergence of the new iterative scheme are proved. Numerical examples are presented to demonstrate the efficacy of the new class of methods.

Subdomain solution decomposition method for nonstationary problems

The reduction of computational costs in the numerical solution of nonstationary problems is achieved through splitting schemes. In this case, solving a set of less computationally complex problems provides the transition to a new level in time. The traditional construction approach of splitting schemes is based on an additive representation of the problem operator(s) and uses explicit-implicit approximations for individual terms. Recently (Efendiev and Vabishchevich (2021) [22]), a new class of methods of approximate solution of nonstationary problems has been introduced based on decomposition not of operators but of the solution itself. This new approach with subdomain solution selection is used in this paper to construct domain decomposition schemes. The boundary value problem for a second-order parabolic equation in a rectangle with a difference approximation in space is typical. Two and three-level schemes for decomposition of the domain with and without overlapping subdomains are investigated. Our numerical experiments complement the theoretical results.

Estimating social influence in a social network using potential outcomes.

Social influence occurs when an individual's outcome is affected by another individual's actions. Current approaches in psychology that seek to examine social influence have focused on settings where individuals are nested in predefined groups and do not interact across groups. Such study designs permit using standard estimation methods such as multilevel models for estimating treatment effects but restrict social influence to originate only from individuals within the same group. In more general settings, such as social networks where an individual is free to interact with any other individual, the absence of discernible clusters or scientifically meaningful groups precludes existing estimation methods. In this article, we introduce a new class of methods for assessing social influence in social networks in the context of randomized experiments in psychology. Our proposal builds on the potential outcomes framework from the causal inference literature. In particular, we exploit the concept of (treatment) interference, which occurs between individuals when one individual's outcome is affected by other individuals' treatments. Estimation proceeds using randomization-based approaches that are established in other disciplines and guarantee valid inference by construction. We compared the proposed methods with standard methods empirically using Monte Carlo simulation studies. We illustrated the method using publicly available data from an experiment assessing the effects of an anticonflict intervention among students' peer networks. The R scripts used to implement the proposed methods in the simulation studies and the applied example are freely available online. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

An Instrumental Variable Approach to Dynamic Models

Abstract We present a new class of methods for identification and inference in dynamic models with serially correlated unobservables, which typically imply that state variables are econometrically endogenous. In the context of Industrial Organization, these state variables often reflect econometrically endogenous market structure. We propose the use of Generalized Instrument Variables methods to identify those dynamic policy functions that are consistent with instrumental variable (IV) restrictions. Extending popular “two-step” methods, these policy functions then identify a set of structural parameters that are consistent with the dynamic model, the IV restrictions and the data. We provide computed illustrations to both single-agent and oligopoly examples. We also present a simple empirical analysis that, among other things, supports the counterfactual study of an environmental policy entailing an increase in sunk costs.

The W4 method: A new multi-dimensional root-finding scheme for nonlinear systems of equations

We propose a new class of method for solving nonlinear systems of equations, which, among other things, has four nice features: (i) it is inspired by the mathematical property of damped oscillators, (ii) it can be regarded as a simple extension to the Newton-Raphson (NR) method, (iii) it has the same local convergence as the NR method does, (iv) it has a significantly wider convergence region or the global convergence than that of the NR method. In this article, we present the evidence of these properties, applying our new method to some examples and comparing it with the NR method.

Voyager: Revisiting Available Bandwidth Estimation With a New Class of Methods—Decreasing- Chirp-Train Methods

The available bandwidth (ABW) of a network path is a crucial metric for various applications, such as traffic engineering, congestion control, multimedia streaming, and path selection in software-defined wide-area networks (SDWAN). In recent years, a new class of measurement methods have been proposed to estimate the available bandwidth, decreasing-chirp-train methods. However, the performance and limitations of this new class of methods are neither well studied nor fairly compared beyond simulation studies. In this work, we implement Voyager, a modular framework that allows us to conduct a fair and thorough comparison of how a variety of modern bandwidth estimation methods perform under different network paths and traffic conditions. We shed light on the characteristics and limitations of the internal algorithms of various methods, and propose two new methods. We investigate the impact of various bottleneck types and traffic types, and we explore the performance of these methods on high speed links where interrupt coalescence can cause measurement noise. We finally test Voyager on long-distance Internet links and with live traffic and report our findings.

Ultra-specific fluorescence detection of DNA modifying enzymes by dissipation system

Abnormal expression of DNA modifying enzymes (DMEs) is linked to a variety of diseases including cancers. It is desirable to develop accurate methods for DME detection. However, the substrate-based probe for target DMEs is disturbed by various non-target DMEs that have similar activity resulting in a loss of specificity. Here we utilized dissipative DNA networks to develop an ultra-specific fluorescence assay for DME, absolutely distinguishing between target and non-target enzymes. Unlike the conventional sensors in which the discrimination of target and non-target relies on signal intensity, in our system, target DMEs exhibit featured fluorescence oscillatory signals, while non-target DMEs show irreversible ‘one-way’ fluorescence increase. These dissipation-enabled probes (DEPs) exhibit excellent generality for various types of DMEs including DNA repair enzyme apurinic/apyrimidinic endonuclease 1 (APE1), polynucleotide kinase (T4 PNK), and methyltransferase (Dam). DEPs provide a novel quantification mode based on area under curve which is more robust than those intensity-based quantifications. The detection limits of APE1, T4 PNK, and Dam reach 0.025 U/mL, 0.44 U/mL, and 0.113 U/mL, respectively. DEPs can accurately identify their corresponding DMEs with excellent specificity in cell extracts. Fluorescence sensors based on DEPs herein represent a conceptually new class of methods for enzyme detection, which can be easily adapted to other sensing platforms such as electrochemical sensors.

The analysis of the fractional-order system of third-order KdV equation within different operators

This paper studies the fractional system of the third-order Korteweg-De Vries equation under the fractional derivatives of Caputo, Atangana-Baleanu and Yang-Srivastava-Machado. The Homotopy perturbation transform technique defined with the Caputo, Atangana-Baleanu and Yang-Srivastava-Machado operators is applied to achieve the non-linear model’s actual results. This non-linear scheme generally arises as a description of waves in traffic flow, electric circuits, elastic media and electrodynamics, multi-component plasmas, shallow water waves etc. This research aims to develop a new class of methods that do not require small parameters to find approximate solutions of fractional coupled systems and eliminate unrealistically and linearization factors. Analytical simulation demonstrates that the suggested method is reliable, efficient, and straightforward to various physical systems. This study demonstrates that Homotopy perturbation transform method numerical solutions are very accurate and effective for analyzing the non-linear behavior of a system. According to the research, the Homotopy perturbation transform method is much easier, more convenient, and more efficient than other analytical techniques.