Conventional Iterative Scheme Research Articles

AI‐enhanced iterative solvers for accelerating the solution of large‐scale parametrized systems

SummaryRecent advances in the field of machine learning open a new era in high performance computing for challenging computational science and engineering applications. In this framework, the use of advanced machine learning algorithms for the development of accurate and cost‐efficient surrogate models of complex physical processes has already attracted major attention from scientists. However, despite their powerful approximation capabilities, surrogate model predictions are still far from being near to the ‘exact’ solution of the problem. To address this issue, the present work proposes the use of up‐to‐date machine learning tools in order to equip a new generation of iterative solvers of linear equation systems, capable of very efficiently solving large‐scale parametrized problems at any desired level of accuracy. The proposed approach consists of the following two steps. At first, a reduced set of model evaluations is performed using a standard finite element methodology and the corresponding solutions are used to establish an approximate mapping from the problem's parametric space to its solution space using a combination of deep feedforward neural networks and convolutional autoencoders. This mapping serves as a means of obtaining very accurate initial predictions of the system's response to new query points at negligible computational cost. Subsequently, an iterative solver inspired by the Algebraic Multigrid method in combination with Proper Orthogonal Decomposition, termed POD‐2G, is developed that successively refines the initial predictions of the surrogate model towards the exact solution. The application of POD‐2G as a standalone solver or as preconditioner in the context of preconditioned conjugate gradient methods is demonstrated on several numerical examples of large scale systems, with the results indicating its strong superiority over conventional iterative solution schemes.

A fast-converging scheme for the phonon Boltzmann equation with dual relaxation times

The phonon Boltzmann transport equation with dual relaxation times is often used to describe the heat conduction in semiconductor materials when the classical Fourier's law is no longer valid. For practical engineering designs, accurate and efficient numerical methods are highly demanded to solve the equation. At a large Knudsen number (i.e., the ratio of the phonon mean free path to a characteristic system length), steady-state solutions can be obtained via the conventional iterative scheme (CIS) within a few iterations. However, when the Knudsen number becomes small, i.e., when the phonon transport occurs in the diffusive or hydrodynamic regime, thousands of iterations are required to obtain converged results. In this work, a general synthetic iterative scheme (GSIS) is proposed to tackle the inefficiency of CIS. The key ingredient of the GSIS is that a set of macroscopic synthetic equations, which is exactly derived from the Boltzmann transport equation, is simultaneously solved with the kinetic equation to obtain the temperature and heat flux. During the iteration, the macroscopic quantities are used to evaluate the equilibria in the scattering terms of the kinetic equation, thus guiding the evolution of the phonon distribution function, while the distribution function, in turn, provides closures to the synthetic equations. The Fourier stability analysis is conducted to reveal the superiority of the GSIS over the CIS in terms of fast convergence in periodic systems. It is shown that the convergence rate of the GSIS can always be maintained under 0.2 so that only two iterations are required to reduce the iterative error by one order of magnitude. Numerical results in wall-bounded systems are presented to demonstrate further the efficiency of GSIS, where the CPU time is reduced by up to three orders of magnitude, especially in both the diffusive and hydrodynamic regimes where the Knudsen number is small.

Single-FFT Receiver With Pairwise Maximum Likelihood for Layered ACO-OFDM

A single Fourier transform (FFT) receiver with pairwise maximum likelihood (ML) is proposed for layered asymmetrically clipped optical orthogonal frequency division multiplexing (LACO-OFDM). To improve the single-FFT receiver, the pairwise ML detection is made in time domain after a layer diversion in LACO-OFDM. With only one FFT block, the proposed receiver has a low complexity. Simulation results show that the proposed receiver has the better bit error ratio (BER) performance than the conventional iterative scheme for the LACO-OFDM system with 2 layers in additive white Gaussian noise (AWGN) channel. Furthermore, it is proved that compared with the conventional iterative receiver, the proposed receiver is less sensitive to the clipping distortion. As the nonlinearity increased, the proposed scheme can outperform the conventional iterative solution in nonlinearity channel for LACO-OFDM in visible light communications.

Low-Latency Unfolded-KES Architecture for Emerging Storage Class Memories

This paper presents an advanced key-equation solver (KES) algorithm that can reduce the computing latency of BCH decoding for the high-speed storage class memory (SCM). Adopting the unfolding algorithm as a factor of two, compared to the conventional iterative KES scheme, the proposed work potentially halves the number of processing cycles for KES module, which is normally dominates the overall BCH decoding latency. In contrast that the straight-forward unfolding method increases the critical delay, we accelerate the major computing path that is activated at the most of SCM lifetime, preserving the critical delay of the proposed KES module as similar to that of the original one. When the minor cases are detected, the recovery processing is added at the end of the corresponding iteration. In order to reduce the additional energy consumption due to the unfolded architecture, we carefully deactivate the internal modules during the accelerated processing, which only necessitate for the recovery cycle. Implementation results show that the proposed KES architecture greatly reduces the decoding latency of arbitrary BCH decoder, leading to the high-speed and reliable emerging storages.

Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks

We study the training process of Deep Neural Networks (DNNs) from the Fourier analysis perspective. We demonstrate a very universal Frequency Principle (F-Principle) --- DNNs often fit target functions from low to high frequencies --- on high-dimensional benchmark datasets such as MNIST/CIFAR10 and deep neural networks such as VGG16. This F-Principle of DNNs is opposite to the behavior of most conventional iterative numerical schemes (e.g., Jacobi method), which exhibit faster convergence for higher frequencies for various scientific computing problems. With a simple theory, we illustrate that this F-Principle results from the regularity of the commonly used activation functions. The F-Principle implies an implicit bias that DNNs tend to fit training data by a low-frequency function. This understanding provides an explanation of good generalization of DNNs on most real datasets and bad generalization of DNNs on parity function or randomized dataset.

Can we find steady-state solutions to multiscale rarefied gas flows within dozens of iterations?

One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iterative scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the “converged” solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the number of iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iterative scheme (GSIS) to find the steady-state solutions of rarefied gas flows within dozens of iterations at any Knudsen number. The key ingredient of our scheme is that the macroscopic equations, which are solved together with the Boltzmann equation and help to adjust the velocity distribution function, not only asymptotically preserve the Navier-Stokes limit in the framework of Chapman-Enskog expansion, but also contain the Newton's law for stress and the Fourier's law for heat conduction explicitly. For this reason, like the implicit UGKS, the constraint that the spatial cell size should be smaller than the mean free path of gas molecules is removed, but we do not need the complex evaluation of numerical flux at cell interfaces. What's more, as the GSIS does not rely on the specific collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are expected to accelerate the slow convergence in the simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.

A multi-group extended linear discontinuous method for fixed-source discrete ordinates problems in slab geometry

At present, neutron density calculation in non-multiplying media is relevant in many areas of engineering and science. In this paper, we propose the Extended Linear Discontinuous (ELD) method in multi-group discrete ordinates formulation, originally formulated for one-energy group fixed-source problems with isotropic scattering source in slab geometry. The proposed auxiliary equations are uncoupled on angular directions and combine the linear discontinuous approximation of the finite element method and the quasi-analytical general solution of the spectral nodal method. Thus, we can implement an efficient and simple algorithm using the conventional source iteration scheme for the sweeping equations. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of the ELD method. The work shows that the main advantages of the proposed method are that the numerical scheme is stable for coarse-meshes, and its numerical results are more accurate than those generated by the Diamond Difference (DD) and Linear Discontinuous (LD) methods.

A multi-group extended linear discontinuous method for fixed-source discrete ordinates problems in slab geometry

At present, neutron density calculation in non-multiplying media is relevant in many areas of engineering and science. In this paper, we propose the Extended Linear Discontinuous (ELD) method in multi-group discrete ordinates formulation, originally formulated for one-energy group fixed-source problems with isotropic scattering source in slab geometry. The proposed auxiliary equations are uncoupled on angular directions and combine the linear discontinuous approximation of the finite element method and the quasi-analytical general solution of the spectral nodal method. Thus, we can implement an efficient and simple algorithm using the conventional source iteration scheme for the sweeping equations. Numerical results for benchmark problems are presented to illustrate the accuracy and computational performance of the ELD method. The work shows that the main advantages of the proposed method are that the numerical scheme is stable for coarse-meshes, and its numerical results are more accurate than those generated by the Diamond Difference (DD) and Linear Discontinuous (LD) methods.

Accurate and efficient computation of the Boltzmann equation for Couette flow: Influence of intermolecular potentials on Knudsen layer function and viscous slip coefficient

The Couette flow is one of the fundamental problems of rarefied gas dynamics, which has been investigated extensively based on the linearized Boltzmann equation (LBE) of hard-sphere molecules and simplified kinetic model equations. However, how the different intermolecular potentials affect the viscous slip coefficient and the structure of Knudsen layer remains unclear. Here, a novel synthetic iteration scheme (SIS) is developed for the LBE to find solutions of Couette flow accurately and efficiently: the velocity distribution function is first solved by the conventional iterative scheme, then it is modified such that in each iteration i) the flow velocity is guided by an ordinary differential equation that is asymptotic-preserving at the Navier–Stokes limit and ii) the shear stress is equal to the average shear stress. Based on the Bhatnagar–Gross–Krook model, the SIS is assessed to be efficient and accurate. Then we investigate the Knudsen layer function for gases interacting through the inverse power-law, shielded Coulomb, and Lennard-Jones potentials, subject to diffuse-specular and Cercignani–Lampis gas-surface boundary conditions. When the tangential momentum accommodation coefficient (TMAC) is not larger than one, the Knudsen layer function is strongly affected by the potential, where its value and width increase with the effective viscosity index of gas molecules. Moreover, the Knudsen layer function exhibits similarities among different values of TMAC when the intermolecular potential is fixed. For Cercignani–Lampis boundary condition with TMAC larger than one, both the viscous slip coefficient and Knudsen layer function are affected by the intermolecular potential, especially when the “backward” scattering limit is approached. With the asymptotic theory by Jiang and Luo (2016) [14] for the singular behavior of the velocity gradient in the vicinity of solid surfaces, we find that the whole Knudsen layer function can be well fitted by the power series ∑n=02∑m=02cn,mxn(xln⁡x)m, where x is the distance to the solid surface. Finally, the experimental data of the Knudsen layer profile are explained by the LBE solution with proper values of the viscosity index and TMAC.

A high-order hybridizable discontinuous Galerkin method with fast convergence to steady-state solutions of the gas kinetic equation

The mass flow rate of Poiseuille flow of rarefied gas through long ducts of two-dimensional cross-sections with arbitrary shape is critical in the pore-network modeling of gas transport in porous media. Here, for the first time, the high-order hybridizable discontinuous Galerkin (HDG) method is used to find the steady-state solution of the linearized Bhatnagar–Gross–Krook equation on two-dimensional triangular meshes. The velocity distribution function and its traces are approximated in piecewise polynomial spaces (of degree up to 4) on the triangular meshes and mesh skeletons, respectively. By employing a numerical flux that is derived from the first-order upwind scheme and imposing its continuity weakly on the mesh skeletons, global systems for unknown traces are obtained with fewer coupled degrees of freedom when compared to the original discontinuous Galerkin formulation. To achieve fast convergence to the steady-state solution, a diffusion-like equation for flow velocity, which is asymptotic-preserving into the fluid dynamic limit, is solved by the HDG simultaneously on the same meshes. The proposed HDG-synthetic iterative scheme is proved to be accurate and efficient. Specifically, for flows in the near-continuum regime, numerical simulations have shown that, to achieve the same level of accuracy, our scheme could be faster than the conventional iterative scheme by two orders of magnitude, also it is faster than the synthetic iterative scheme based on the finite difference discretization in the spatial space by one order of magnitude. In addition, the implicit HDG method is more efficient than an explicit discontinuous Galerkin gas kinetic solver, as well as the implicit discontinuous Galerkin scheme when the degree of approximating polynomial is larger than 2. The HDG-synthetic iterative scheme is ready to be extended to simulate rarefied gas mixtures and the Boltzmann collision operator.

A fast iterative scheme for the linearized Boltzmann equation

Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L=(L+Nδh)−Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier–Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne–Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard–Jones intermolecular potential for the first time, and the difference between these results and those using the hard-sphere potential is discussed.

Repeated anticipatory network traffic control using iterative optimization accounting for model bias correction

Anticipatory signal control in traffic networks adapts the signal timings with the aim of controlling the resulting (equilibrium) flows and route choice patterns in the network. This study investigates a method to support control decisions for successful applications in real traffic systems that operate repeatedly, for instance from day to day, month to month, etc. The route choice response to signal control is usually predicted through models; however this leads to suboptimality because of unavoidable prediction errors between model and reality. This paper proposes an iterative optimizing control method to drive the traffic network towards the real optimal performance by observing modeling errors and correcting for them. Theoretical analysis of this Iterative Optimizing Control with Model Bias Correction (IOCMBC) on matching properties between the modeled optimal solution and the real optimum is presented, and the advantages over conventional iterative schemes are demonstrated. A local convergence analysis is also elaborated to investigate conditions required for a convergent scheme. The main innovation is the calculation of the sensitivity (Jacobian) information of the real route choice behavior with respect to signal control variables. To avoid performing additional perturbations, we introduce a measurement-based implementation method for estimating the operational Jacobian that is associated with the reality. Numerical tests confirm the effectiveness of the proposed IOCMBC method in tackling modeling errors, as well as the influence of the optimization step size on the reality-tracking convergence.

Scheme for Utilizing the Soft Feedback Information in Bit-Patterned Media Recording System

This paper presents a scheme that utilizes soft feedback information obtained from a decoder to eliminate detrimental interference in bit-patterned media recording (BPMR) systems. BPMR is a promising technology that can increase the storage area density of magnetic storage systems, but one of the main challenges is to overcome 2-D spatial interference. The performance of the proposed scheme is compared with that of a conventional iterative detection scheme, and the results indicate that the proposed model is better by approximately 1 dB at 20% track misregistration. In particular, the performance of the system is quite superior, allowing the design of systems with higher area densities.

Improved resonance calculation of fluoride salt-cooled high-temperature reactor based on subgroup method

The subgroup method is improved in several aspects to address challenges brought on by design features of the Fluoride salt-cooled High-temperature Reactor (FHR). Firstly, the Dancoff correction is applied to resolve the double heterogeneity arising from embedding TRISO fuel particles in the matrix of pebbles. Secondly, a fast Resonance Interference Factor (RIF) scheme is proposed to treat the resonance interference effect in the FHR. In this scheme, the heterogeneous system is converted into a homogeneous one according to self-shielded cross section conservation of the dominant resonant nuclide. The resonance interference effect is considered in the equivalent homogenous system by correcting the non-interfered self-shielded cross sections with RIFs which are obtained by solving the slowing down equation in hyper-fine energy group (∼1M number of energy groups). Finally, the resonance elastic scattering effect becomes considerable due to high temperatures in the FHR. This effect is considered by substitution of the conventional Resonance Integral (RI) table with that generated by the Monte Carlo method. The Monte Carlo method is modified via the Doppler Broadening Correction Rejection (DBRC) method to implement the Doppler broadened scattering kernel. The numerical results show that the Dancoff correction can significantly reduce errors brought about by the double heterogeneity. The fast RIF scheme provides more accurate effective self-shielded cross sections than the conventional iteration scheme. In addition, the speedup ratio of the fast RIF scheme is ∼3.3 compared with the conventional on-the-fly RIF schemes for TRU TRISO. The scheme to generate RI table can resolve the resonance elastic scattering effect encountered by the conventional scheme.

LDPC 부호화된 MIMO 시스템을 위한 패리티 검사 기반 반복 간섭 제거 기법

In this letter, a parity check based iterative IC scheme is proposed for LDPC coded MIMO systems. After the decoding procedures in each iteration of the proposed scheme, each decoded codeword is utilized for the IC procedures only when the ratio of the check nodes satisfying the parity check equations to the total number of check nodes is not smaller than the pre-defined threshold value. Simulation results verify that the proposed scheme can achieve an improved BLER at the high SNR region compared to the conventional iterative IC scheme.

Belief propagation-based joint iterative algorithm for detection and decoding in asynchronous CDMA satellite systems

As the geostationary orbit (GEO) is congested in the commercial Ku band, the adjacent satellites will receive the interference from small aperture terminals which have a wide field of view. To reduce the interference toward the adjacent satellites, code division multiple access (CDMA) with spread spectrum needs to be adopted in the satellite communications on-the-move system. However, the mutual interference between CDMA users leads to the performance loss, which is not acceptable for the power-sensitive small aperture terminals, and the computation complexity is still an intractable problem when the conventional iterative minimum mean square error (MMSE) scheme is employed to suppress the multiple access interference (MAI) in satellite systems especially that is coded by low-density parity-check (LDPC) code. Moreover, the long transmission delays of different users caused by the long transmission distance require the robustness of the interference suppression for the asynchronous case. In this paper, a joint chip-level algorithm for CDMA multiuser detection and LDPC decoding with a new and simple iterative structure based on belief propagation (BP) for satellite systems is developed, which integrates the iterations of multiuser detection and the iterations of LDPC decoding with factor graphs. Through analysis and simulations, it is verified that the proposed algorithm is robust to the asynchronous CDMA satellite system, and with the similar iterative complexity, the performance of the proposed algorithm outperforms that of the conventional iterative MMSE scheme in the asynchronous CDMA satellite system.

A factored variant of the Newton iteration for the solution of algebraic Riccati equations via the matrix sign function

In this paper we introduce a variant of the Newton iteration for the matrix sign function that results in an efficient numerical solver for a certain class of algebraic Riccati equations (AREs). In particular, when the Hamiltonian matrix associated with the ARE can be composed as A B B T C T C ? A T $\left [\begin {array}{llll}{A}&{BB^{T}}\\{C^{T}C}&{-A^{T}}\end {array}\right ]$ , with B and C T $C^{T}$ having a much larger number of rows than columns, the new algorithm exploits the special structure of the off-diagonal blocks to yield an alternative factored Newton iteration which reduces the cost per iteration by a factor of up to 8 (16 in case A is symmetric negative definite) w.r.t. the conventional iterative scheme. Experiments with a large collection of benchmark examples show that the factored iteration attains numerical accuracy similar to that of the conventional Newton iteration as well as the structure-preserving doubling algorithm. High-performance implementations of these methods, making heavy use of LAPACK linked to a multi-threaded implementation of BLAS, demonstrate the clear advantage of the new iteration on a 48-core AMD-based platform.

Analysis of vertically loaded piles in multilayered transversely isotropic soils by BEM

This paper presents a boundary element method (BEM) for the static analysis of vertically loaded single piles embedded in multilayered transversely isotropic soils by using an analytical layer-element solution. The solution which is especially suitable for numerical calculation is deduced through the combination of the order reduction approach and the technology of integral transform. A boundary element formula for axially loaded piles with both enlarged tip and head cap is established by taking the solution as the kernel function, and the compressibility of piles is further considered based on a direct algorithm which can avoid the occurrence of non-convergence results from the conventional iterative scheme while the stiffness of piles approaches to that of the soil foundation. Numerical examples including piles in a finite depth soil and multilayered soils of transverse isotropy are carried out to study the influence of soil properties on single pile behaviors.

Coordinated Spatial Multiplexing with Orthogonalized Channels for Multiuser MIMO Downlink Systems

In this paper, we propose a new coordinated spatial multiplexing algorithm for multiuser multiple-input multiple-output (MIMO) downlink systems. Unlike other conventional coordination methods, our scheme exploits orthogonalized effective channels to yield the transmit precoding and receive combining matrices without requiring an iterative operation. In order to determine each row of the effective channels, a greedy type successive process is utilized. Simulation results show that the performance of our proposed scheme is very close to the conventional iterative scheme with significantly reduced complexity. Also, our method achieves a 3 dB performance gain over the conventional non-iterative scheme.

Using relations between stress and fluid pressure for improved compaction modelling in flow simulation and increased efficiency in coupled rock mechanics simulation

ABSTRACT The conventional compaction model used in reservoir simulators defines compaction as a function of fluid pressure, whereas, in reality, it is a function of effective stress. The interrelationship between fluid pressure, effective stress and reservoir parameters (materials distribution, geometry, production scheme) is investigated. By modifying the conventional concept of flow simulator compaction a predictor is constructed for the rock mechanics computations in a coupled flow–rock mechanics simulation. This predictor reduces the time to converge the stress computations by reducing or eliminating the number of pore volume iterations in the coupling scheme. Overall computing time is thereby reduced considerably, while maintaining accuracy in the stress computations. Additionally, the compaction state in the flow simulator will be more accurate than in a conventional iterative coupling scheme.