Accurate Solver Research Articles

Black-Box Acceleration of Monotone Convex Program Solvers

When and where was the study conducted: This work was done in 2018, 2019 and 2020 when Palma London was a PhD student at Caltech and Shai Vardi was a postdoc at Caltech. This work was also done in part while Palma London was visiting Purdue University, and while Reza Eghbali was a postdoctoral fellow the Simons Institute for the Theory of Computing. Adam Wierman is a professor at Caltech. Article Summary and Talking Points: Please describe the primary purpose/findings of your article in 3 sentences or less. This paper presents a framework for accelerating (speeding up) existing convex program solvers. Across engineering disciplines, a fundamental bottleneck is the availability of fast, efficient, accurate solvers. We present an acceleration method that speeds up linear programing solvers such as Gurobi and convex program solvers such as the Splitting Conic Solver by two orders of magnitude. Please include 3-5 short bullet points of “Need to Know” items regarding this research and your findings. - Optimizations problems arise in many engineering and science disciplines, and developing efficient optimization solvers is key to future innovation. - We speed up linear programing solver Gurobi by two orders of magnitude. - This work applies to optimization problems with monotone objective functions and packing constraints, which is a common problem formulation across many disciplines. Please identify 2 pull quotes from your article that best capture the novelty and impact of your research. “We propose a framework for accelerating exact and approximate convex programming solvers for packing linear programming problems and a family of convex programming problems with linear constraints. Analytically, we provide worst-case guarantees on the run time and the quality of the solution produced. Numerically, we demonstrate that our framework speeds up Gurobi and the Splitting Conic Solver by two orders of magnitude, while maintaining a near-optimal solution.” “Our focus in this paper is on a class of packing problems for which data is either very costly or hard to obtain. In these situations, the number of data points available is much smaller than the number of variables. In a machine-learning setting, this regime is increasingly prevalent because it is often advantageous to consider larger and larger feature spaces, while not necessarily obtaining proportionally more data.” Article Implications - Please describe in 5 sentences or less the innovative takeaway(s) of your research. This framework applies to optimization problems with monotone objective functions and packing constraints, which is a common problem formulation across many disciplines, including machine learning, inference, and resource allocation. Providing fast solvers for these problems is crucial. We exploit characteristics of the problem structure and leverage statistical properties of the problem constraints to allow us to speed up optimization solvers. We present worst-case guarantees on run-time, and empirically demonstrate speedups of two orders of magnitude. - Please describe in 5 sentences or less why your findings would be of interest to the general public. Many problems in engineering, science, math, and machine learning involve solving an optimization problem. Fast, efficient optimization solvers are key to future innovation in science and engineering. This work presents a tool to accelerate existing convex solvers, and thus can also be applied to future solvers. As the size of datasets grow it is even more crucial to have fast solvers. - Who would be the most impacted by your research (i.e. by industry, job title, consumer category). Our work impacts machine-learning researchers and optimization researchers, in industry or academia.

An efficient IMEX-DG solver for the compressible Navier-Stokes equations for non-ideal gases

We propose an efficient, accurate and robust IMEX solver for the compressible Navier-Stokes equations describing non-ideal gases with general cubic equation of state and Stiffened-Gas EOS. The method is based on an h-adaptive Discontinuous Galerkin spatial discretization and on an Additive Runge Kutta IMEX method for time discretization. It is specifically tailored for low Mach number applications and allows to simulate low Mach regimes at a significantly reduced computational cost, while maintaining full second order accuracy also for higher Mach number regimes. The method has been implemented in the framework of the deal.II numerical library, whose adaptive mesh refinement capabilities are employed to enhance efficiency. Refinement indicators appropriate for real gas phenomena have been introduced. A number of numerical experiments on classical benchmarks for compressible flows and their extension to real gases demonstrate the properties of the proposed method.

Numerical simulation method of a circulating fluidized bed reactor using a modified MP-PIC solver of OpenFOAM

There are some previous works for simulating a bubbling fluidized bed (BFB) reactor using OpenFOAM's MPPICFoam solver, but those for circulating fluidized bed (CFB) reactors are few. In this study, the MPPICFoam solver was verified with a BFB reactor by comparing the solver's simulation results with the experiment and simulation results from the literature. By introducing the inter-boundary particle circulation function and a new module for setting sand particle size distribution, simulations were performed for a 0.1 MW th pilot-scale CFB combustor under isothermal conditions. The modified solver for the CFB reactor gives similar results as the experiment and commercial computational fluid dynamics (CFD) tool, well-predicting the pressure distribution and solid volume fraction in the riser. • MPPICFoam solver reliability for lab-scale bubbling fluidized-bed reactor verified. • Accurate solver prediction of solid volume fraction (horizontal) distribution. • Model used for numerical simulation to study riser hydrodynamics of a CFB combustor. • Particle size distribution module and new circulating boundary conditions added. • Improved solver shows superior predicted experimental results in dense bed region.

Fast and accurate solvers for weakly singular integral equations

Fast and accurate solvers for weakly singular integral equations

Method of Moment Analysis of Carbon Nanotubes Embedded in a Lossy Dielectric Slab Using a Multilayer Dyadic Green’s Function

Modeling the electromagnetic response of carbon nanotube (CNT) reinforced composites is inherently a 3-D multiscale problem that is challenging to solve in real time for nondestructive evaluation (NDE) applications. This article presents a fast and accurate full-wave electromagnetic solver based on a multilayer dyadic Green’s function approach. In this approach, we account for the effects of the dielectric slab, where the CNTs are embedded, without explicitly discretizing its interfaces. Due to their large aspect ratios, the CNTs are modeled as arbitrary thin wires (ATWs), and the method-of-moment (MoM) formulation with distributed line impedance is used to solve for their coupled currents. The accuracy of the in-house solver is validated against the commercial MoM and the finite element method (FEM) solvers over a broad range of frequencies (from 1 GHz to 10 THz) and for a wide range of dielectric slab properties. Examples of 100 nm-long vertical and horizontal CNTs embedded in a 1 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu \text{m}$ </tex-math></inline-formula> -thick lossy dielectric substrate are presented. The in-house solver provides more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$50\times $ </tex-math></inline-formula> speed up while solving the vertical CNT and more than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$570\times $ </tex-math></inline-formula> speed up while solving the horizontal CNT than a commercial MoM solver over the GHz-to-THz frequency range.

An accelerated, high-order accurate direct solver for the Lippmann–Schwinger equation for acoustic scattering in the plane

An efficient direct solver for solving the Lippmann–Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem with N degrees of freedom, the solver constructs an approximate inverse in $\mathcal {O}(N^{3/2})$ operations and then, given an incident field, can compute the scattered field in $\mathcal {O}(N \log N)$ operations. The solver is based on a previously published direct solver for integral equations that relies on rank-deficiencies in the off-diagonal blocks; specifically, the so-called Hierarchically Block Separable format is used. The particular solver described here has been reformulated in a way that improves numerical stability and robustness, and exploits the particular structure of the kernel in the Lippmann–Schwinger equation to accelerate the computation of an approximate inverse. The solver is coupled with a Nyström discretization on a regular square grid, using a quadrature method developed by Ran Duan and Vladimir Rokhlin that attains high-order accuracy despite the singularity in the kernel of the integral equation. A particularly efficient solver is obtained when the direct solver is run at four digits of accuracy, and is used as a preconditioner to GMRES, with each forwards application of the integral operators accelerated by the FFT. Extensive numerical experiments are presented that illustrate the high performance of the method in challenging environments. Using the 10th-order accurate version of the Duan–Rokhlin quadrature rule, the scheme is capable of solving problems on domains that are over 500 wavelengths wide to relative error below 10− 10 in a couple of hours on a workstation, using 26M degrees of freedom.

Modified Variational Iteration Method with Chebyshev Polynomials for Solving 12th order Boundary Value problems

We consider in this paper an illustration of the modified variational iteration method (MVIM) as an effective and accurate solver of 12th order boundary value problem (BVP). For this reason, the Chebyshev polynomials of the principal kind was utilized as a premise capabilities in the guess of the logical capability of the given issue. The strategy is applied in an immediate manner without utilizing linearization or irritation. The subsequent mathematical confirmations recommend that the strategy is without a doubt successful and exact as applied to a few direct and nonlinear issues as mathematical trial and error. Maple 18 was used for all computational simulations carried out in this research.©2022 JNSMR UIN Walisongo. All rights reserved.

Approximate solution to the generalized radial Boussinesq equation

This paper presents a method for constructing approximate semi-analytical solutions for the radially symmetric generalized Boussinesq equation, also known as the porous medium equation (PME). This work is a further extension of analysis done by Telyakovskiy et al. (2016), that studied water injection governed by the Boussinesq equation into an unconfined initially dry aquifer. The newly derived approximate solution also has two parts; one to approximate singular behavior near the well and a polynomial part to model far-field behavior. With the prescribed initial and boundary conditions, the analyzed problem can be rewritten in terms of similarity variables resulting in a boundary value problem for a nonlinear ordinary differential equation. Using higher-order physically significant moments of the solution, a closed-form expression for the position of the wetting front and the shape of the phreatic surface is obtained. The introduced solution is valid over a wide range of injection regimes: the time-independent, the power-law injection function with positive exponents, and exponential-law of injection. A highly accurate numerical solver similar to Telyakovskiy et al. (2016) is used to validate the presented approximate solutions. Plus, comparison is made with the experimental data on prediction of the wetting front position for the special case of the Boussinesq equation.

Ions transport in electromembrane desalination: A numerical modeling for the return flow ion-concentration-polarization desalination system

In this work, via numerical simulation, we conducted a detailed study on the manner of ion transport, the factors controlling the ion transport, and the effect of ion transport on the desalinating efficiency of the return-flow ion concentration polarization desalination system. An accurate numerical solver was developed to model the ions' transport in the system. Various configurations of applied voltage and fluid flow were enforced to investigate the system response in Ohmic and limiting. Via examining the system in the different operating regimes and various conditions, we report a detailed theoretical understanding of system operation. With the increasing feeding flow rate, the simulation results show that the current and current utilization increases, the salt removal ratio changes inversely, and the energy per ion removed decreases. The system reaches its maximum value of current utilization at a value of porous membrane flow (PM-flow) of 0.04 mm/s in our investigation into the effects of this condition on the performance of the system. The findings are useful in optimizing the design and operation of the return flow - ion concentration polarization (RF-ICP) desalination system, giving firm evidence for the advantage of the ICP-desalination technology.

Development and validation of a combustion large-eddy-simulation solver based on fully compressible formulation and tabulated chemistry

This study concerns the development and validation of a high-fidelity CFD solver for large-eddy simulations (LES) of combustion and reacting flows. The solver is built upon a high-resolution numerical scheme and leverages a compressible flamelet formulation, so that it can capture both the turbulent combustion and thermoacoustic effects simultaneously. The validation study is first performed by considering a scalar-mixing case in homogeneous turbulence, and then extended to LES of a non-premixed jet flame. The solver accuracy is assessed by comparing the numerical solutions with the available experimental data. It is found that the newly developed solver is able to accurately predict the time-averaged combustion fields as well as the fluctuation quantities. The predictive accuracy is comparable to that of the state-of-the-art low-Mach solvers in literature.

An efficient and accurate implicit DG solver for the incompressible Navier–Stokes equations

AbstractWe propose an efficient, accurate, and robust implicit solver for the incompressible Navier–Stokes equations, based on a DG spatial discretization and on the TR‐BDF2 method for time discretization. The effectiveness of the method is demonstrated in a number of classical benchmarks, which highlight its superior efficiency with respect to other widely used implicit approaches. The parallel implementation of the proposed method in the framework of the deal.II software package allows for accurate and efficient adaptive simulations in complex geometries, which makes the proposed solver attractive for large scale industrial applications.

Power performance comparison analysis of FPGA using Riemann Definite Integral Equation Solver

Definite integral equation solvers were hitherto developed using software which take long time to generate results. The complexity of designs and verifications, due to advancement in calculus and linear algebra, data dependency in designs, and the need for parallelism in models, necessitated the need for faster, accurate, and reliable hardware-based solvers. This will reduce latency and excessive memory requirement in generating values of definite integrals. The Field Programmable Gate Array (FPGA)-Based Definite Integral Solver developed in this research is able to solve definite integral equations with less resource utilization (1.08% look up tables (LUT), 0.07% flip flops (FF), 10.81% digital signal processing (DSP) and 23.86% input-output (IO)) and hence minimum switching activities at lower power (Signals: 0.421W, Logic: 0.345W, Digital Signal Processing: 1.904W and Input/Output: 0.015W) that translates to fast and accurate results generation. The power and resource utilization evaluation on the solver and the performance comparison with similar works showed that excessive memory consumption and resource utilization has been reduced significantly by the design.

Solving 2D Poisson's equation based on conditional generative adversarial network

AbstractThe numerical simulation of electromagnetic problems is usually complex and time‐consuming, so a fast and accurate solver is urgently needed. In this paper, a deep learning method based on conditional generative adversarial network (CGAN) is used to solve the 2D Poisson's equation in electrostatic field. The relative permittivity distribution and source charge distribution in the two dimensional region are regarded as label of CGAN, and the potential distribution is the real image or generated image corresponding to the label. This deep learning network can be trained by reasonably designing the loss functions of generator and discriminator, and based on a large amount of samples obtained by the finite difference method. The average relative prediction error of the trained CGAN for all samples is less than 0.5%, with a significant reduction in prediction time, while that of convolution neural network (CNN) is less than 1.5%. Furthermore, the average relative prediction error of CGAN is less than 1% for the scenes different from those of training and testing samples, including triangle, square, and pentagon. The prediction results indicate that CGAN has stronger learning ability and fast prediction speed, and can be used as a fast solver for electromagnetic numerical problems.

Quantitative ultrasound brain imaging with multiscale deconvolutional waveform inversion

High-resolution images of human brain are critical for monitoring the neurological conditions in a portable and safe manner. Sound speed mapping of brain tissues provides unique information for such a purpose. In addition, it is particularly important for building digital human acoustic models, which form a reference for future ultrasound research. Conventional ultrasound modalities can hardly image the human brain at high spatial resolution inside the skull due to the strong impedance contrast between hard tissue and soft tissue. We carry out numerical experiments to demonstrate that the time-domain waveform inversion technique, originating from the geophysics community, is promising to deliver quantitative images of human brains within the skull at a sub-millimeter level by using ultra-sound signals. The successful implementation of such an approach to brain imaging requires the following items: signals of sub-megahertz frequencies transmitting across the inside of skull, an accurate numerical wave equation solver simulating the wave propagation, and well-designed inversion schemes to reconstruct the physical parameters of targeted model based on the optimization theory. Here we propose an innovative modality of multiscale deconvolutional waveform inversion that improves ultrasound imaging resolution, by evaluating the similarity between synthetic data and observed data through using limited length Wiener filter. We implement the proposed approach to iteratively update the parametric models of the human brain. The quantitative imaging method paves the way for building the accurate acoustic brain model to diagnose associated diseases, in a potentially more portable, more dynamic and safer way than magnetic resonance imaging and x-ray computed tomography.

A fourth-order accurate adaptive solver for incompressible flow problems

We present a numerical solver for the incompressible Navier–Stokes equations that combines fourth-order-accurate discrete approximations and an adaptive tree grid (i.e. h-refinement). The scheme employs a novel compact-upwind advection scheme and a 4th-order accurate projection algorithm whereby the numerical solution exactly satisfies the incompressibility constraint. Further, we introduce a new refinement indicator that is tailored to this solver. We show tests and examples to illustrate the consistency, convergence rate and the application for the adaptive solver. The combination of the solver scheme and the proposed grid adaptation algorithm result in fourth-order convergence rates whilst only tuning a single grid-refinement parameter. The speed performance is benchmarked against a well-established second-order accurate adaptive solver alternative. We conclude that the present 4th order solver is an efficient design for problems with strong localization in the spatial-temporal domain and where a high degree of convergence for the solution statistics is desired.

A new 3D OpenFoam solver with improved resolution for hyperbolic systems on hybrid unstructured grids

Although the accuracy and the robustness of discretized schemes solving hyperbolic systems have been greatly improved over the last two decades, devising a low-dissipative and non-oscillatory method remains a challenge. Especially, it is algorithmically complicated and computationally expensive to extend and apply high-resolution non-oscillatory schemes to 3D hybrid unstructured grids. Therefore, in this work we develop accurate, practical, and robust 3D hybrid unstructured OpenFoam solvers which can be applied in wide industrial applications. Different from existing 3D high order numerical solvers of which reconstruction schemes are based on polynomials of high degree, the proposed 3D scheme employs a polynomial function and a hyperbolic tangent basis function as two candidates for reconstruction functions. The MUSCL (Monotone Upstream-centered Schemes for Conservation law) scheme with the MLP (Multi-dimensional Limiting Process) slope limiter is adopted as the polynomial function. The multi-dimensional THINC (Tangent of Hyperbola for INterface Capturing) function with quadratic surface representation and Gaussian quadrature, so-called THINC/QQ, is used as the non-polynomial reconstruction candidate. With these reconstruction candidates, a multi-stage boundary variation diminishing (BVD) algorithm which significantly minimizes numerical dissipation is designed on 3D unstructured grids to select the final reconstruction function. With this accurate and robust reconstruction scheme, then we developed new solvers for simulating inviscid high speed compressible single-phase flow and two-phase flow via the implementation of the proposed scheme and algorithm into the open-source computational fluid dynamics (CFD) software, OpenFOAM. The solver with the proposed low-dissipative scheme, named as EulerEquationFoam-M-TQ-BVD, is constructed with the density-based solver for inviscid high speed compressible flow simulations. To simulate compressible multi-material flow with moving material interfaces, a solver named as FiveEquationFoam-M-TQ-BVD is developed based on mechanical-equilibrium diffusive interface model and MieGrüneisen equation of state. The performance of new solvers is evaluated through wide range benchmark tests and are compared with existing high order schemes such as the third-order WENO (Weighted Essentially Non-Oscillatory) scheme. The new solvers produce superior solution across critical points, contact discontinuities and material interfaces. Moreover, in comparison with WENO schemes, the proposed solver enjoys algorithmic simplicity and greater flexibility in 3D hybrid unstructured grids. Thus, this work provides accurate, practical and robust numerical solvers for high speed compressible single- and multi-material flow simulations.

An Efficient Jet Marcher for Computing the Quasipotential for 2D SDEs

We present a new algorithm, the efficient jet marching method (EJM), for computing the quasipotential and its gradient for two-dimensional SDEs. The quasipotential is a potential-like function for nongradient SDEs that gives asymptotic estimates for the invariant probability measure, expected escape times from basins of attractors, and maximum likelihood escape paths. The quasipotential is a solution to an optimal control problem with an anisotropic cost function which can be solved for numerically via Dijkstra-like label-setting methods. Previous Dijkstra-like quasipotential solvers have displayed in general 1st order accuracy in the mesh spacing. However, by utilizing higher order interpolations of the quasipotential as well as more accurate approximations of the minimum action paths, EJM achieves second-order accuracy for the quasipotential and nearly second-order for its gradient. Moreover, by using targeted search neighborhoods for the fastest characteristics following the ideas of Mirebeau, EJM also enjoys a reduction in computation time. This highly accurate solver enables us to compute the prefactor for the WKB approximation for the invariant probability measure and the Bouchet–Reygner sharp estimate for the expected escape time for the Maier–Stein SDE. Our codes are available on GitHub.

High‐order isogeometric modified method of characteristics for two‐dimensional coupled Burgers' equations

AbstractThis paper presents a novel isogeometric modified method of characteristics for the numerical solution of the two‐dimensional nonlinear coupled Burgers' equations. The method combines the modified method of characteristics and the high‐order NURBS () elements to discretize the governing equations. The Lagrangian interpretation in this isogeometric analysis greatly reduces the time truncation errors in the Eulerian methods. A third‐order explicit Runge–Kutta scheme is used for the discretization in time. We present a detailed description of the algorithm used for the calculation of departure points and the interpolation stage. Our focus is on constructing highly accurate and stable solvers for the two‐dimensional nonlinear coupled Burgers' equations at high Reynolds numbers. A variety of benchmark tests and numerical examples are provided to show the effectiveness, accuracy, and performance of the proposed modified method of characteristics by virtue of potential advantages of isogeometric analysis. The method developed is anticipated to provide new research directions to the practical calculation of incompressible flows and to studies of their physical behavior.

Power Allocation for Reliable and Energy-Efficient Optical LEO-to-Ground Downlinks with Hybrid ARQ Schemes

Satellites in low earth orbit (LEO) are currently being deployed for numerous communication, positioning, space and Earth-imaging missions. To provide higher data rates in direct-to-user links and earth observation downlinks, the free-space optics technology can be employed for LEO-to-ground downlinks. Moreover, the hybrid automatic repeat request (HARQ) can be adopted since the propagation latency is low for LEO satellites. In this work, a power allocation methodology is proposed for optical LEO-to-ground downlinks under weak turbulence employing HARQ retransmission schemes. Specifically, the average power consumption is minimized given a maximum transmitted power constraint and a target outage probability threshold to ensure energy efficiency and reliability, respectively. The optimization problem is formulated as a constrained nonlinear programming problem and solved for Type I HARQ, chase combining (CC) and incremental redundancy (IR) schemes. The solutions are derived numerically via iterative algorithms, namely interior-point (IP) and sequential quadratic programming (SQP), and validated through an exhaustive (brute-force) search. The numerical simulations provide insight into the performance of the retransmission schemes regarding average power. More specifically, Type I HARQ has the worst output, CC has a moderate one, and IR exhibits the best performance. Finally, the IP algorithm is a slower but more accurate solver, and SQP is faster but slightly less accurate.

Evaluation of trans-critical transition of single- and multi-component sprays under diesel engine-like conditions

• The discrepancies in trans-critical transition of different sprays are unraveled. • Droplets in dilute sprays cannot represent the thermodynamic state of dense sprays. • Diesel sprays are more prone to transition to supercritical than n-dodecane sprays. The trans-critical transition of multi-component sprays under the ECN Spray-A conditions has attracted extensive interest. However, the drastic change of critical points of multi-components makes this controversial. In this study, the trans-critical transition of sprays before and after the end of injection is evaluated by numerical methods, and new findings have been obtained. First, the robust and accurate thermodynamic equilibrium solver and multi-component droplet evaporation model were developed and implemented into the OpenFOAM. These models have been validated against the measured phase change diagram and isolated droplet evaporation rate. Then, the mixture critical temperatures of n-dodecane/nitrogen and multi-component diesel/nitrogen were calculated under a wide range of pressures based on the thermodynamic equilibrium solver. The mixture critical temperature decreases almost linearly with the increase of the pressure. Following that, extensive sprays of n-dodecane and multi-component diesel under high temperature and pressure conditions were simulated, and the predicted spray liquid penetrations were validated against the experimental data. The results found that the dilute sprays after the end of injection are more prone to transition to the supercritical mixing regime, while the trans-critical possibility of dense sprays before the end of spray decreases due to the cooling effect of extensive evaporation. Finally, the trans-critical transitions of multi-component diesel sprays before and after the end of injection were evaluated and compared with those of n-dodecane sprays. Due to the low mixture critical temperature and evaporation rate of diesel, its sprays are more likely to transition to the supercritical regime. Therefore, it can conclude that the dilute sprays after the end of injection cannot represent the thermodynamic state of dense sprays. Moreover, the multi-component diesel and n-dodecane sprays follow a different trans-critical transition pathway.