Design Of Numerical Algorithms Research Articles

Fully discrete Spectral-Galerkin scheme for a ternary Allen–Cahn type mass-conserved Nakazawa–Ohta phase-field model for triblock copolymers

In this paper, we first use the Allen–Cahn relaxation dynamics to simulate the phase-field model of triblock copolymers, where the mass-conserving property is achieved by adding two additional nonlocal Lagrange multipliers. The numerical approximations of the obtained model are further considered, and an efficient fully-discrete numerical scheme based on the Spectral-Galerkin approach for spatial discretization and the so-called explicit-IEQ (invariant energy quadratization) method for time marching is developed. Two auxiliary variables are introduced to reformulate the governing system into an equivalent form, which facilitates the design of numerical algorithms by simply discretizing nonlinear terms explicitly. The resulting scheme is not only second-order accurate in time and spectrally accurate in space, but also easy to implement, i.e., the scheme can be carried out by only solving multiple independently decoupled, linear, and constant-coefficient elliptic equations at each time step. We also prove the unconditional energy stability of the developed scheme and demonstrate its effectiveness by implementing several benchmarking numerical examples in 2D and 3D.

A Novel Hybrid Algorithm for the Forward Kinematics Problem of 6 DOF Based on Neural Networks.

The closed kinematic structure of Gough–Stewart platforms causes the kinematic control problem, particularly forward kinematics. In the traditional hybrid algorithm (backpropagation neural network and Newton–Raphson), it is difficult for the neural network part to train different datasets, causing training errors. Moreover, the Newton–Raphson method is unable to operate on a singular Jacobian matrix. In this study, in order to solve the forward kinematics problem of Gough–Stewart platforms, a new hybrid algorithm is proposed based on the combination of an artificial bee colony (ABC)–optimized BP neural network (ABC–BPNN) and a numerical algorithm. ABC greatly improves the prediction ability of neural networks and can provide a superb initial value to numerical algorithms. In the design of numerical algorithms, a modification of Newton’s method (QMn-M) is introduced to solve the problem that the traditional algorithm model cannot be solved when it is trapped in singular matrix. Results show that the maximal improvement in ABC–BPNN error optimization was 46.3%, while the RMSE index decreased by 42.1%. Experiments showed the feasibility of QMn-M in solving singular matrix data, while the percentage improvement in performance for the average number of iterations and required time was 14.4% and 13.9%, respectively.

Resiliency in numerical algorithm design for extreme scale simulations

This work is based on the seminar titled ‘Resiliency in Numerical Algorithm Design for Extreme Scale Simulations’ held March 1–6, 2020, at Schloss Dagstuhl, that was attended by all the authors. Advanced supercomputing is characterized by very high computation speeds at the cost of involving an enormous amount of resources and costs. A typical large-scale computation running for 48 h on a system consuming 20 MW, as predicted for exascale systems, would consume a million kWh, corresponding to about 100k Euro in energy cost for executing 1023 floating-point operations. It is clearly unacceptable to lose the whole computation if any of the several million parallel processes fails during the execution. Moreover, if a single operation suffers from a bit-flip error, should the whole computation be declared invalid? What about the notion of reproducibility itself: should this core paradigm of science be revised and refined for results that are obtained by large-scale simulation? Naive versions of conventional resilience techniques will not scale to the exascale regime: with a main memory footprint of tens of Petabytes, synchronously writing checkpoint data all the way to background storage at frequent intervals will create intolerable overheads in runtime and energy consumption. Forecasts show that the mean time between failures could be lower than the time to recover from such a checkpoint, so that large calculations at scale might not make any progress if robust alternatives are not investigated. More advanced resilience techniques must be devised. The key may lie in exploiting both advanced system features as well as specific application knowledge. Research will face two essential questions: (1) what are the reliability requirements for a particular computation and (2) how do we best design the algorithms and software to meet these requirements? While the analysis of use cases can help understand the particular reliability requirements, the construction of remedies is currently wide open. One avenue would be to refine and improve on system- or application-level checkpointing and rollback strategies in the case an error is detected. Developers might use fault notification interfaces and flexible runtime systems to respond to node failures in an application-dependent fashion. Novel numerical algorithms or more stochastic computational approaches may be required to meet accuracy requirements in the face of undetectable soft errors. These ideas constituted an essential topic of the seminar. The goal of this Dagstuhl Seminar was to bring together a diverse group of scientists with expertise in exascale computing to discuss novel ways to make applications resilient against detected and undetected faults. In particular, participants explored the role that algorithms and applications play in the holistic approach needed to tackle this challenge. This article gathers a broad range of perspectives on the role of algorithms, applications and systems in achieving resilience for extreme scale simulations. The ultimate goal is to spark novel ideas and encourage the development of concrete solutions for achieving such resilience holistically.

Multi-Group Multicast Beamforming: Optimal Structure and Efficient Algorithms

This paper considers the multi-group multicast beamforming optimization problem, for which the optimal solution has been unknown due to the non-convex and NP-hard nature of the problem. By utilizing the successive convex approximation numerical method and Lagrangian duality, we obtain the optimal multicast beamforming solution structure for both the quality-of-service (QoS) problem and the max-min fair (MMF) problem. The optimal structure brings valuable insights into multicast beamforming: We show that the notion of uplink-downlink duality can be generalized to the multicast beamforming problem. The optimal multicast beamformer is a weighted MMSE filter based on a group-channel direction: a generalized version of the optimal downlink multi-user unicast beamformer. We also show that there is an inherent low-dimensional structure in the optimal multicast beamforming solution independent of the number of transmit antennas, leading to efficient numerical algorithm design, especially for systems with large antenna arrays. We propose efficient algorithms to compute the multicast beamformer based on the optimal beamforming structure. Through asymptotic analysis, we characterize the asymptotic behavior of the multicast beamformers as the number of antennas grows, and in turn, provide simple closed-form approximate multicast beamformers for both the QoS and MMF problems. This approximation offers practical multicast beamforming solutions with a near-optimal performance at very low computational complexity for large-scale antenna systems.

Model predictive control for linear differential-algebraic equations

We are concerned with Model Predictive Control (MPC) of constrained linear systems governed by regular differential-algebraic equations. The contribution is twofold: First, we provide a characterization of the set of admissible control functions to clarify what the actual input is. This is essential for the design of numerical algorithms. Secondly, we present a blueprint for the construction of suitable terminal costs and terminal constraints such that asymptotic stability of the origin w.r.t. the MPC closed-loop is guaranteed provided initial feasibility. To this end, we exploit recent results on the unconstrained linear quadratic regulator problem using recently introduced concepts of input index and an augmented system.

Structure-preserving Analysis on Folding and Unfolding Process of Undercarriage

The main idea of the structure-preserving method is to preserve the intrinsic geometric properties of the continuous system as much as possible in numerical algorithm design. The geometric constraint in the multi-body systems, one of the difficulties in the numerical methods that are proposed for the multi-body systems, can also be regarded as a geometric property of the multi-body systems. Based on this idea, the symplectic precise integration method is applied in this paper to analyze the kinematics problem of folding and unfolding process of nose undercarriage. The Lagrange governing equation is established for the folding and unfolding process of nose undercarriage with the generalized defined displacements firstly. And then, the constrained Hamiltonian canonical form is derived from the Lagrange governing equation based on the Hamiltonian variational principle. Finally, the symplectic precise integration scheme is used to simulate the kinematics process of nose undercarriage during folding and unfolding described by the constrained Hamiltonian canonical formulation. From the numerical results, it can be concluded that the geometric constraint of the undercarriage system can be preserved well during the numerical simulation on the folding and unfolding process of undercarriage using the symplectic precise integration method.

Thermodynamic formalization of the fluid dynamics equations for a charged dielectric in an electromagnetic field

Minkowski’s classical work underlying modern electrodynamics is described. Primary attention is given to the mathematical refinements that are required if the parameters ɛ and μ depend on the properties of the dielectric fluid, i.e., the medium carrying charges in the field under study. It is shown that the motion of the medium and the accompanying evolution of the electromagnetic field are described by differential equations that are symmetric and hyperbolic in the sense of Friedrichs. This property guarantees their well-posedness. Note that this class of equations was not known in Minkowski’s time. At present, it plays an important role in the mathematical simulation of nonstationary processes and in the design of numerical algorithms. The author’s view of the mathematical foundations of Minkowski’s work is presented, which relates the latter to present-day insights into the theory of differential equations. This paper can possibly be of interest to physicists.

Design of numerical algorithms for the problem of charge transport in a 2D silicon MOSFET transistor with a silicon oxide nanochannel

We are concerned with the problem of charge transport in a 2D silicon MOSFET transistor occupying a domain Ω with a silicon oxide nanochannel occupying a domain ΩG. After proposing an additional boundary condition for the electric potential on the common boundary of the domains Ω and ΩG we design two numerical algorithms for funding approximate solutions of this problem. The first algorithm is a new one and uses interpolation polynomials of spline-collocation and the sweep method. The second algorithm is based on the well-known longitudinal-transverse sweep (l.t.s.) method. By using these algorithms we obtain graphs of stationary solutions of our problem. We also compare the workability and efficiency of the proposed algorithms for various values of parameters.

Stochastic models in kinetic theory

The paper is concerned with some aspects of stochastic modeling in kinetic theory. First, an overview of the role of particle models with random interactions is given. These models are important both in the context of foundations of kinetic theory and for the design of numerical algorithms in various engineering applications. Then, the class of jump processes with a finite number of states is considered. Two types of such processes are studied, where particles change their states either independently of each other (monomolecular processes) or via binary interactions (bimolecular processes). The relationship of these processes with corresponding kinetic equations is discussed. Equations are derived both for the average relative numbers of particles in a given state and for the fluctuations of these numbers around their averages. The simplicity of the models makes several aspects of the theory more transparent.

Belief propagation as a dynamical system: the linear case and open problems

Systems and control theory have found wide application in the analysis and design of numerical algorithms. The authors present an equivalent discrete-time dynamical system interpretation of an algorithm commonly used in information theory called belief propagation (BP). BP is one instance of the so-called sum–product algorithm and arises, for example, in the context of iterative decoding of low-density parity-check codes. The authors review a few known results from information theory in the language of dynamical systems and show that the typically very high-dimensional, non-linear dynamical system corresponding to BP has interesting structural properties. For the linear case, they completely characterise the behaviour of this dynamical system in terms of its asymptotic input–output map. Finally, the authors state some of the open problems concerning BP in terms of the dynamical system presented.

Design of numerical algorithms for the ballistic diode problem

Numerical algorithms for finding stationary solutions to a hydrodynamic model of charge transport in semiconductors are proposed and described in detail.

Stellar Core Collapse: A Case Study in the Design of Numerical Algorithms for Scalable Radiation Hydrodynamics

The fields of radiation and hydrodynamics are time-honored disciplines in physics. However, it is only since the nuclear age that the importance of radiation hydrodynamics - a single field studying the interplay of radiation and material hydrodynamics together - has come into prominence. Similarly, it is only since the computer age that physically interesting problems in this field have become tractable. In this article, the authors discuss the process of building a model to study one of the fundamental problems of astrophysics: the supernova explosion of a massive star caused by the collapse of its core. An explanation for the mechanism of this explosion involves the interaction of material hydrodynamics and neutrino radiation.

A local minimax characterization for computing multiple nonsmooth saddle critical points

This paper is concerned with characterizations of nonsmooth saddle critical points for numerical algorithm design. Most characterizations for nonsmooth saddle critical points in the literature focus on existence issue and are converted to solve global minimax problems. Thus they are not helpful for numerical algorithm design. Inspired by the results on computational theory and methods for finding multiple smooth saddle critical points in [14, 15, 19, 21, 23], a local minimax characterization for multiple nonsmooth saddle critical points in either a Hilbert space or a reflexive Banach space is established in this paper to provide a mathematical justification for numerical algorithm design. A local minimax algorithm for computing multiple nonsmooth saddle critical points is presented by its flow chart.

Existence and Stability for Mathematical Models of Sedimentation–Consolidation Processes in Several Space Dimensions

In this paper, we study several spatially multidimensional initial-boundary value problems modelling sedimentation–consolidation processes of a flocculated suspension. This solid–fluid mixture is considered as two superimposed continuous media differing in density and viscosity. The phenomenological foundation and derivation of the mathematical model are based on the previous work by R. Bürger et al. (2000, Z. Angew. Math. Mech. 80, 79–92). We study the full coupling of the conservation of mass equation and the conservation of linear momentum equation. For different types of regularization, we establish energy estimates. The dissipative nature of the whole system assures the existence as well as the stability (long time asymptotics) of a solution of the system (provided that the viscosities of the fluid are large enough). Moreover, the energy estimates might serve as the foundation for the design of numerical algorithms to simulate the system.

On the pollution effect of quasi‐compressibility methods in magneto‐hydrodynamics and reactive flows

The study of the asymptotics of quasi-compressibility methods is an essential tool to construct and improve numerical schemes in hydrodynamics, like e.g., stabilized finite element methods or time-splitting projection methods. The goal of this work is to illustrate different asymptotical solution behaviour for equations in magneto-hydrodynamics and those that describe reactive flows for standard quasi-compressibility methods that influences the design of numerical algorithms. Copyright © 1999 John Wiley & Sons, Ltd.

On the pollution effect of quasi-compressibility methods in magneto-hydrodynamics and reactive flows

The study of the asymptotics of quasi-compressibility methods is an essential tool to construct and improve numerical schemes in hydrodynamics, like e.g., stabilized finite element methods or time-splitting projection methods. The goal of this work is to illustrate different asymptotical solution behaviour for equations in magneto-hydrodynamics and those that describe reactive flows for standard quasi-compressibility methods that influences the design of numerical algorithms.

Rank Reducibility of a Covariance Matrix in the Frisch Scheme

The Frisch scheme for identification of mathematical models from data corrupted by additive noise contains many unsolved aspects. One of the principal problems, of particular interest for factor analysis and structural regression methodologies, concerns rank reducibility of a covariance matrix simply by changing its diagonal entries. With reference to this topic, the paper shows that the mathematical models compatible with the data are the solutions of a set of polynomial equations which satisfy some well-defined constraints. The approach is based on the rank reducibility criteria suggested in a well-known paper by Ledermann, generalized to take into account the definiteness conditions on the noise-free covariance matrix. The results obtained give a deeper insight on the theoretical properties of the Frisch scheme and can represent a starting point for the design of numerical algorithms to solve the problem.

Displacement Structures of Covariance Matrices, Lossless Systems, and Numerical Algorithm Design

Low displacement rank theory underlies many fast algorithms designed for structured covariance matrices. Some of these have gained notoriety for their numerical instability problems, particularly fast least-squares algorithms. Recent studies have shown that instability is not inherent to fast algorithms, but rather comes from the violation of backward consistency constraints. This paper thus details the connection between covariance matrices of a given displacement inertia and lossless rational matrices, as well as the role of this connection in numerically consistent algorithms. This basic connection allows displacement structures to be parametrized via a sequence of rotation angles obtained from a lossless system. The utility of this approach is that, irrespective of errors in the rotation parameter set, they remain consistent with a positive definite matrix of a prescribed displacement inertia. This property in turn may be rephrased as meaningful forms of backward consistency in numerical algorithms. The rotation parameters then take the form of Givens or Jacobi angles applied to data, in contrast to classical approaches which directly manipulate dyadic decompositions of the displacement structure. The concepts are illustrated in popular signal processing applications. In particular, these connections lend clear insight into the stable computation of reflection coefficients of Toeplitz systems, and also serve to resolve the numerical instability problem of fast least-squares algorithms.

Implicit matrix multiplication with maximum accuracy on various transputer networks

The majority of numerical algorithms employs floating-point vector and matrix operations. On a parallel computer these algorithms should be solved fastand reliably in order to avoid a time-consuming error analysis. The XSC-languages (high-level language extensions for eXtended Scientific Computation) are well-suited for this purpose since they support the design of numerical algorithms delivering correct and automatically verified results. This goal is attained by an arithmetic with maximum accuracy (especially for vector and matrix operations), highly accurate standard functions, and exact evaluation of dot product expressions. Within theESPRIT Parallel Computing Action, one XSC-language, PASCAL-XSC, was implemented on a Supercluster Transputer System under the operating system HELIOS. Parallel algorithms for computationally intensive and maximally accurate matrix operations were implemented and tested on various transputer architectures. We will sketch some features of these architectures and present some benchmarks for the algorithms used. These algorithms form a parallel C runtime library of PASCAL-XSC (or any other XSC-language that uses a C runtime library) and are called automatically. This can be considered a basis for implicit parallelization in an XSC-language.

Matrix Visualization in the Design of Numerical Algorithms

At the heart of much scientific computing are the algorithmic kernels often found in numerical software libraries. Numerical analysts and algorithm designers can be aided by various software tools in the design of their algorithms. We present a tool for matrix visualization and its application in the design and development of numerical algorithms for supercomputers. We discuss the development of the tool as an object-oriented distributed system and show examples of its use, including applications in linear algebra and performance monitoring. By using color computer graphics, one can gain insights into algorithm behavior, which can then be used to design more efficient numerical algorithms. In particular, we present a case study of the application of data structure visualization in the development of hybrid parallel algorithms for the singular value decomposition. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.