Multirate Algorithm Research Articles

A conservative discontinuous-Galerkin-in-time (DGiT) multirate time integration framework for interface-coupled problems with applications to solid–solid interaction and air–sea models

In this paper we extend the DGiT multirate framework, developed in Connors and Sockwell (2022) for scalar transmission problems, to a solid–solid interaction (SSI) problem involving two coupled elastic solids and a coupled air–sea model with the rotating, thermal shallow water equations. In so doing we aim to demonstrate the broad applicability of the mathematical theory and governing principles established in Connors and Sockwell (2022) to coupled problems characterized by subproblems evolving at different temporal scales. Multirate time integration algorithms employing different time steps, optimized for the dynamics of each subproblem, can significantly improve simulation efficiency for such coupled problems. However, development of multirate algorithms is a highly non-trivial task due to the coupling, which can impact accuracy, stability or other desired properties such as preservation of system invariants. DGiT provides a general template for multirate time integration that can achieve these properties. To elucidate the manner in which DGiT accomplishes this task, we fully detail each step in the application of the framework to the SSI and air–sea coupled problems. Numerical examples illustrate key properties of the resulting multirate schemes for both problems.

A Multirate Discontinuous-Galerkin-in-Time Framework for Interface-Coupled Problems

A framework is presented to design multirate time stepping algorithms for two dissipative models with coupling across a physical interface. The coupling takes the form of boundary conditions imposed on the interface, relating the solution variables for both models to each other. The multirate aspect arises when numerical time integration is performed with different time step sizes for the component models. In this paper, we seek to identify a unified approach to develop multirate algorithms for these coupled problems. This effort is pursued though the use of discontinuous-Galerkin time stepping methods, acting as a general unified framework, with different time step sizes. The subproblems are coupled across user-defined intervals of time, called coupling windows, using polynomials that are continuous on the window. The coupling method is shown to reproduce the correct interfacial energy dissipation, discrete conservation of fluxes, and asymptotic accuracy. In principle, methods of arbitrary order are possible. As a first step, herein we focus on the presentation and analysis of monolithic methods for advection-diffusion models coupled via generalized Robin-type conditions. The monolithic methods could be computed using a Schur-complement approach. The framework and analysis herein accommodate multirate coupling for a wide variety of single-step, multistep, and multistage methods. A concrete example of Crank--Nicolson integrators coupled in a multirate fashion is also included to illustrate the ideas. We conclude with some discussion of future developments, such as different interface conditions and partitioned methods.

Multi‐rate co‐simulation of power system transients using dynamic phasor and EMT solvers

This study presents a novel multi-rate algorithm for the co-simulation of power system transients using base-frequency dynamic phasor solver for frequency adaptive simulation of transient (BFAST) and electromagnetic transient (EMT) solvers. The BFAST solver alters its solution technique from dynamic phasors to EMT based upon the frequency contents of the waveforms being simulated. A changeover algorithm between the two solvers is also presented. The BFAST solver is then integrated with an industrial-grade EMT solver to develop a BFAST–EMT multi-rate co-simulator. The interface between the two solvers is established using transmission lines at the partitioning locations. The co-simulator combines the benefits of dynamic phasors, frequency adaptive simulation of transients, parallel processing, and multi-rate simulation. The study describes the several solution modes of the proposed co-simulator. Illustrative examples are included to demonstrate the accuracy and computational benefits of the proposed co-simulator.

Implementation of Ultrahigh-Speed Decimators

Traditionally, the data rate of a digital signal processing system is bound by the processing speed. In this article, a formal approach for overcoming this limitation and allowing high-data-rate applications to run on relatively slow processors is presented. This approach allows the time sampling period to be much shorter than the time required to process an input sample; in effect, an ultrahigh-speed system is obtained where the sample rate exceeds the processing rate by a factor controlled by the system designer. The proposed approach is applied to the multirate decimation algorithm and its associated dependence graph. A directed acyclic graph (DAG) is then obtained from it using a scheduling policy. The DAG is then partitioned using an interlaced partitioning scheme. Multiphase/multirate clocking is used to synchronize the different components of the system. The number of partitions required depends on the I/O rate and processor speed. The proposed approach speeds up the system at the expense of extra latency and hardware resources.

A New Class of High-Order Methods for Multirate Differential Equations

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. The proposed methods are based on a specific subset of explicit one-step exponential integrators. More precisely, starting from an explicit exponential Runge--Kutta method of the appropriate form, we derive a multirate algorithm to approximate the action of the matrix exponential through the definition of modified “fast” initial-value problems. These fast problems may be solved using any viable solver, enabling multirate simulations through use of a subcycled method. Due to this structure, we name these multirate exponential Runge--Kutta (MERK) methods. In addition to showing how MERK methods may be derived, we provide rigorous convergence analysis, showing that for an overall method of order $p$, the fast problems corresponding to internal stages may be solved using a method of order $p-1$, while the final fast problem corresponding to the time-evolved solution must use a method of order $p$. Numerical simulations are then provided to demonstrate the convergence and efficiency of MERK methods with orders three through five on a series of multirate test problems.

A self adjusting multirate algorithm for robust time discretization of partial differential equations

We show the distinctive potential advantages of a self adjusting multirate method based on diagonally implicit solvers for the robust time discretization of partial differential equations. The properties of the specific ODE methods considered are reviewed, with special focus on the TR-BDF2 solver. A general expression for the stability function of a generic one stage multirate method is derived, which allows to study numerically the stability properties of the proposed algorithm in a number of examples relevant for applications. Several numerical experiments, aimed at the time discretization of hyperbolic partial differential equations, demonstrate the efficiency and accuracy of the resulting approach.

Extendable multirate real-time simulation of active distribution networks based on field programmable gate arrays

Real-time simulation of large-scale active distribution networks exhibiting a wide range of time-scales puts forward higher requirements for simulation accuracy and efficiency. This paper presents an extendable method and design for the real-time simulation of active distribution networks utilising high-performance hardware field programmable gate arrays. In the aspect of numerical algorithm, a high-accuracy and stable multirate simulation algorithm is proposed. The entire active distribution network is decoupled into different subsystems by their inherent time-scales and distinct time steps are used to solve the subsystems. Then root-matching method is adopted to form the exponential difference equations that represent the behaviours of the electric distribution network being modelled, which eliminates the truncation errors and thus provides a highly accurate time-domain solution. To handle the interface between the subsystems, a multirate interfacing method is proposed. The hardware design of the multirate interfaces is presented as well. With the multirate algorithm, a fully functional extendable real-time simulator based on field programmable gate arrays is designed and implemented. Two modified IEEE 33-node systems with photovoltaics and energy storage are simulated on the 4-field-programmable-gate-arrays-based real-time simulator. Simulation results are compared with the commercial simulation tool PSCAD/EMTDC to validate the correctness and effectiveness of the proposed method and design.

Adaptive Stabilization of Discrete-Time Nonminimum Phase Systems

In this paper, we present a direct multirate adaptive control algorithm that ensures global stabilization of a class of (potentially unstable and invertible) linear time-invariant discrete-time plants of known order and relative degree one. An essential feature of our scheme is that no projections are needed, no appeal is made to the persistence of excitation arguments for the stability proof and it has a better transient performance than existing schemes. The implementation of the controller requires some prior information about its Markov parameters, namely, upper and lower bounds on the systems impulse response. It is directly applicable to plants with interlacing real poles and zeros, i.e., with Cauchy index equal to the plant order, provided the adaptation gain is restricted to be smaller than some value determined by the measure of relative primeness of the pole and zero polynomial. This class contains some practically interesting systems, for instance, resistor–inductor or resistance–capacitance circuits.

Using Signal Subtraction Operation to Suppress Electro-Acoustic Leakage in Ultrasonic Low-Frequency Dual Element Transducers

It was shown that the use of complex modulated signals with subsequent multi-rate radio engineering processing of ultrasound echo-signals allows the creation of new methods for ultrasonic non-destructive inspection that are inaccessible for conventional devices. An example of using an ultrasonic multifunctional adaptive measuring complex demonstrates the possibility of creating a method for suppressing the parasitic signal of electro-acoustic leakage in low-frequency dual-element transducers by means of a multi-rate signal processing algorithm including the simplest signal subtraction operation.

Systematic evaluation for multi-rate simulation of DC grids

With wide applications of power electronic devices in modern power systems, simulation using traditional electromechanical and electromagnetic tools suffers low speed and imprecision. Multi-rate methods can enhance efficiency of simulation by decreasing the scale of systems in small time-steps. However, the existing traditional methods for multi-rate simulation suffer the problems of instability and simulation errors. These have hindered the application of multi-rate simulation in power industry. Therefore theoretical evaluation on different multi-rate simulation methods is crucial to understand the feasibility and limitation of the methods, and to contribute to overcome the drawbacks of the traditional methods. In this paper, the multi-rate simulation performance based on two traditional technologies and a Modified Thevenin Interface are evaluated to provide an overall feasibility of multi-rate algorithms in the power simulation. The Modified Thevenin Interface is proposed to overcome the drawbacks in synchronization. Three theorems are proposed and proved for theoretically analyzing the stability of the simulation methods. Error analyses of the multi-rate methods are performed to identify the relationships between errors and simulation conditions. Besides, the accuracy and efficiency performance in a practical project of VSC-MTDC shows the feasibility and necessity by using multi-rate simulation. Through the theoretical analysis, the issues of stability and accuracy of multi-rate simulation for the DC grids have been better understood, based on which an improved simulation algorithm has been proposed to overcome these issues. Long-term system dynamics of large-scale systems containing DC grids and fast transients of HVDC converters can be investigated simultaneously with high speed and sufficient accuracy.

Adaptive filtering for intelligent sensing speech based on multi-rate LMS algorithm

Adaptive filtering algorithm is one of the most active research topics in signal processing at present. Speech signal processing is the hot issue in the field of digital signal processing. Based on theoretical research of adaptive filter, the work simulated Least mean square error (LMS) adaptive filtering algorithm under different signal-to-noise ratios, spreading to speech signal sequences of different sampling rates. By changing sampling rate factors and filter orders, the work discussed adaptive filtering problems of LMS algorithm under narrowband and broadband interference. Simulation result is consistent with theoretical analysis. With good noise reduction effect, this algorithm is simple and easy to implement.

Multirate method for co-simulation of electrical-chemical systems in multiscale modeling

Multiscale modeling by means of co-simulation is a powerful tool to address many vital questions in neuroscience. It can for example be applied in the study of the process of learning and memory formation in the brain. At the same time the co-simulation technique makes it possible to take advantage of interoperability between existing tools and multi-physics models as well as distributed computing. However, the theoretical basis for multiscale modeling is not sufficiently understood. There is, for example, a need of efficient and accurate numerical methods for time integration. When time constants of model components are different by several orders of magnitude, individual dynamics and mathematical definitions of each component all together impose stability, accuracy and efficiency challenges for the time integrator. Following our numerical investigations in Brocke et al. (Frontiers in Computational Neuroscience, 10, 97, 2016), we present a new multirate algorithm that allows us to handle each component of a large system with a step size appropriate to its time scale. We take care of error estimates in a recursive manner allowing individual components to follow their discretization time course while keeping numerical error within acceptable bounds. The method is developed with an ultimate goal of minimizing the communication between the components. Thus it is especially suitable for co-simulations. Our preliminary results support our confidence that the multirate approach can be used in the class of problems we are interested in. We show that the dynamics ofa communication signal as well as an appropriate choice of the discretization order between system components may have a significant impact on the accuracy of the coupled simulation. Although, the ideas presented in the paper have only been tested on a single model, it is likely that they can be applied to other problems without loss of generality. We believe that this work may significantly contribute to the establishment of a firm theoretical basis and to the development of an efficient computational framework for multiscale modeling and simulations.

Efficient simulation of a slow-fast dynamical system using multirate finite difference schemes

We consider a system of ordinary differential equations describing a slow-fast dynamical system, in particular, a predator-prey system that is highly susceptible to local time variations. This model exhibits coexistence of predatorprey dynamics in the case when the prey population grows much faster than that of the predators with a quite diversified time response. For particular parametric values their interactions show a stable relaxation oscillation in the positive octant. Such characteristics are difficult to mimic using conventional time integrators that are used to solve systems of ordinary differential equations. To resolve this, we design and analyze multirate time integration methods to solve a mathematical model for a slow-fast dynamical system. Proposed methods are based on using extrapolation multirate discretisation algorithms. Through these methods, we reduce the integration time by integrating the slow sub-system with a larger step length than the fast sub-system. This allows us to efficiently solve multiscale ordinary differential equations. Besides theoretical results, we provide thorough numerical experiments which confirm that these multirate schemes outperform corresponding single-rate schemes substantially both in terms of computational work and CPU times.

A MULTIRATE VARIABLE-TIMESTEP ALGORITHM FOR N-BODY SOLAR SYSTEM SIMULATIONS WITH COLLISIONS

ABSTRACT We present and analyze the performance of a new algorithm for performing accurate simulations of the solar system when collisions between massive bodies and test particles are permitted. The orbital motion of all bodies at all times is integrated using a high-order variable-timestep explicit Runge–Kutta Nyström (ERKN) method. The variation in the timestep ensures that the orbital motion of test particles on eccentric orbits or close to the Sun is calculated accurately. The test particles are divided into groups and each group is integrated using a different sequence of timesteps, giving a multirate algorithm. The ERKN method uses a high-order continuous approximation to the position and velocity when checking for collisions across a step. We give a summary of the extensive testing of our algorithm. In our largest simulation—that of the Sun, the planets Earth to Neptune and 100,000 test particles over 100 million years—the relative error in the energy after 100 million years was of the order of 10−11.

11.25-ms-Group-Delay and Low-Complexity Algorithm Design of 18-Band Quasi-ANSI S1.11 1/3 Octave Digital Filterbank for Hearing Aids

This paper presents a novel algorithm and architecture design for 18-band quasi-class-2 ANSI S1.11 1/3 octave filterbank. The proposed design has several advantages such as lower group delay, lower computational complexity, and lower matching error. The technique we developed in this paper can be summarized as follows: 1) a simple low-pass filter (LPF) and discrete cosine transform (DCT) modulation are utilized to generate a uniform 9-band filterbank first, and then all elements of z <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> are replaced by all-pass filters to obtain a non-uniform filterbank; 2) a fast recursive structure and variable-length algorithm is further developed to efficiently accomplish DCT modulation. Thus, the spectrum of LPF can be easily spanned and flexibly extended to the location of the desired central frequency; 3) after employing the multi-rate algorithm, an 18-band non-uniform filterbank is generated from two 9-band sub filterbanks by following the proposed design steps and parameter determinations. Compared with the latest Liu 's quasi-class-2 ANSI S1.11 design, the proposed method-I (Proposed-I) totally has 72.8% reduction for multiplications per sample, 11.25-ms group delay, and 59 additions decreased per sample. Moreover, the maximum matching error of the proposed method-II (Proposed-II) is averagely equal to 1.79 dB much smaller than that of the latest Wei 's design. For the proposed variable-length DCT modulation, only 2 adders, 2 multipliers, 2 multiplexers, and 5 registers are required for hardware implementation after applying VLSI retiming scheme. Overall, the proposed filterbank design would be a new solution for future applications in the area of hearing aids.

The Design and Implementation of Revise System on Frequency Response about Power Amplifier of Shortwave Transmitter

A revise system on frequency response based on power amplifier of shortwave transmitter is designed to solve the problem of nonlinear response at different frequencies on power amplifier. This system is able to guarantee the output of power amplifier to fluctuate slightly around the rated power though the two processes named pre-revise on frequency response of power amplifier and current revise on frequency response of power amplifier. A modified variable step and multi-rate updated LMS algorithm is put into use to reduce the calculated amount and boost the rate of the revise. It is demonstrated by experiment that the revise system of this paper is able to guarantee the output of power amplifier around the rated power with the deviation of dB at a high convergence rate.

An efficient parallel implementation of explicit multirate Runge–Kutta schemes for discontinuous Galerkin computations

Although explicit time integration schemes require small computational efforts per time step, their efficiency is severely restricted by their stability limits. Indeed, the multi-scale nature of some physical processes combined with highly unstructured meshes can lead some elements to impose a severely small stable time step for a global problem. Multirate methods offer a way to increase the global efficiency by gathering grid cells in appropriate groups under local stability conditions. These methods are well suited to the discontinuous Galerkin framework. The parallelization of the multirate strategy is challenging because grid cells have different workloads. The computational cost is different for each sub-time step depending on the elements involved and a classical partitioning strategy is not adequate any more. In this paper, we propose a solution that makes use of multi-constraint mesh partitioning. It tends to minimize the inter-processor communications, while ensuring that the workload is almost equally shared by every computer core at every stage of the algorithm. Particular attention is given to the simplicity of the parallel multirate algorithm while minimizing computational and communication overheads. Our implementation makes use of the MeTiS library for mesh partitioning and the Message Passing Interface for inter-processor communication. Performance analyses for two and three-dimensional practical applications confirm that multirate methods preserve important computational advantages of explicit methods up to a significant number of processors.

An efficient explicit algorithm for multi-rate predictive control systems

In this paper, explicit model predictive control (MPC) schemes for discrete-time linear-invariant multi-rate systems with constraints on inputs and states are studied. An efficient multi-rate predictive control algorithm is proposed by solving a multiparametric linear programming (mp-LP), and the explicit state feedback control laws are developed. Further, based on the closed-loop multi-rate piecewise affine (PWA) structure, stability analysis is presented by linear matrix inequality (LMI). The simulation results show the effectiveness of the explicit MPC for multi-rate system.

Advanced vibration suppression algorithms in adaptive optics systems

Vibration suppression in astronomical adaptive optics (AO) systems has gathered great attention in the context of next-generation instrumentation for current telescopes and future Extremely Large Telescopes. Laser tomographic AO systems require natural guide stars to measure the low-order modes such as tip-tilt (TT) and TT-anisoplanatism. To increase the sky coverage, the guide stars are often faint, thus requiring lower temporal sampling frequencies to work on a more favorable signal-to-noise regime. Such sampling frequencies can be of the order of, or even lower than, the range of frequencies where vibrations are likely to appear. Ideally, vibrations affecting the low-order modes could be corrected at the higher laser loop frame rate using an upsampling procedure. This paper compares the most relevant solutions proposed hitherto to a novel multirate algorithm using the linear-quadratic-Gaussian (LQG) approach capable of upsampling the correction to further reduce the impact of vibrations. Results from numerical Monte Carlo simulations span a large range of parameters from pure sinusoids to relatively broad peak vibrations, covering the likely-to-be signals in a realistic AO system. The improvement is shown at sampling frequencies from 20 to 800 Hz, including below the vibration itself, in the example of 29.5 Hz on a Thirty Meter Telescope-like scenario. The multirate LQG ensures the least residual for both faint and bright stars for all the peak widths considered based on telemetry from the Keck Observatory.

Multirate Energy-Bounding Algorithm for High-Fidelity Stable Haptic Interaction Control

This brief describes a multirate energy-bounding algorithm that significantly increases the displayable stiffness while guaranteeing the overall haptic interaction stability. Unlike haptic interaction control algorithms that are based on the multirate simulation of virtual environments, the proposed multirate approach separates the low-level control thread from the haptic thread. The haptic thread updates the haptic interaction computation, or the collision detection and response and dynamic or deformation simulation, of a virtual environment at slow rate while the low-level control thread updates the actuator command force determined by the energy-bounding algorithm to guarantee the interaction stability at a high update rate. An analytical derivation study shows that the displayable stiffness of virtual environments can be significantly increased by an order of N, which is the ratio of the fast actuator command signal update to the slow haptic simulation update. Experiments using a commercial haptic device demonstrate that the displayed stiffness significantly increases while guaranteeing stability even when the haptic rendering rate is slow.