Approximate Solver Research Articles

Bayesian control variates for optimal covariance estimation with pairs of simulations and surrogates

ABSTRACT Predictions of the mean and covariance matrix of summary statistics are critical for confronting cosmological theories with observations, not least for likelihood approximations and parameter inference. Accurate estimates require running costly N-body and hydrodynamics simulations. Approximate solvers, or surrogates, greatly reduce the computational cost but introduce biases, especially in the non-linear regime of structure growth. We propose ‘CARPool Bayes’ to solve the inference problem for both the means and covariances using a combination of simulations and surrogates. Our approach allows incorporating prior information for the mean and covariance. We derive closed-form solutions for maximum a posteriori covariance estimates that are efficient Bayesian shrinkage estimators, guarantee positive semidefiniteness, and can optionally leverage analytical covariance approximations. We discuss choices of the prior and propose a procedure for obtaining optimal prior hyperparameter values with a small set of test simulations. We test our method by estimating the covariances of clustering statistics of gadget-iiiN-body simulations at redshift z = 0.5 using surrogates from a 100–1000× faster particle-mesh code. Taking the sample covariance from 15 000 simulations as the truth, and using an empirical Bayes prior with diagonal blocks, our estimator produces nearly identical Fisher matrix contours for ΛCDM parameters using only 15 simulations of the non-linear dark matter power spectrum. In this case, the number of simulations is so small that the sample covariance is degenerate. We show cases where even with a naïve prior our method improves the estimate. Our framework is applicable to a wide range of cosmological problems where fast surrogates are available.

Inverse design enables large-scale high-performance meta-optics reshaping virtual reality

Meta-optics has achieved major breakthroughs in the past decade; however, conventional forward design faces challenges as functionality complexity and device size scale up. Inverse design aims at optimizing meta-optics design but has been currently limited by expensive brute-force numerical solvers to small devices, which are also difficult to realize experimentally. Here, we present a general inverse-design framework for aperiodic large-scale (20k × 20k λ2) complex meta-optics in three dimensions, which alleviates computational cost for both simulation and optimization via a fast approximate solver and an adjoint method, respectively. Our framework naturally accounts for fabrication constraints via a surrogate model. In experiments, we demonstrate aberration-corrected metalenses working in the visible with high numerical aperture, poly-chromatic focusing, and large diameter up to the centimeter scale. Such large-scale meta-optics opens a new paradigm for applications, and we demonstrate its potential for future virtual-reality platforms by using a meta-eyepiece and a laser back-illuminated micro-Liquid Crystal Display.

Many-body approach to superfluid nuclei in axial geometry

Starting from a general many-body fermionic Hamiltonian, we derive the equations of motion (EOM) for nucleonic propagators in a superfluid system. The resulting EOM is of the Dyson type formulated in the basis of Bogoliubov's quasiparticles. As the leading contributions to the dynamical kernel of this EOM in strongly-coupled regimes contain phonon degrees of freedom in various channels, an efficient method of calculating phonon's characteristics is required to successfully model these kernels. The traditional quasiparticle random phase approximation (QRPA) solvers are typically used for this purpose in nuclear structure calculations, however, they become very prohibitive in nonspherical geometries. In this paper, by linking the notion of the quasiparticle-phonon vertex to the variation of the Bogoliubov's Hamiltonian, we show that the recently developed finite-amplitude method (FAM) can be efficiently employed to compute the vertices within the FAM-QRPA. To illustrate the validity of the method, calculations based on the relativistic density-dependent point-coupling Lagrangian are performed for the single-nucleon states in heavy and medium-mass nuclei with axial deformations. The cases of $^{38}\mathrm{Si}$ and $^{250}\mathrm{Cf}$ are presented and discussed.

Multi-Species Compressible Solver for Non-Continuum Flow Through a Micro-Channel

Compressible multi species flow solver, developed by the authors Kumar and Maheshwari [May 15, 2020. “Viscous Multi-Species Lattice Boltzmann Solver for Simulating Shock-Wave Structure.” Computers and Fluids 203. Article ID 104539], has been extended further to non-continuum flows. The approach uses the Onsager-BGK model, which ensures Onsager's maximum entropy production principle is followed, thus avoids unphysical solutions. Equations for velocity slip and temperature jump have been derived. The viscous slip, thermal slip and diffusion slip coefficients have been successfully validated with 13-moment solver's and experimental data. The departure from equilibrium in such flows has been approximated by a collision probability function (), instead of using the complicated second order perturbation over the Knudsen number (that leads to Burnett equations). This provided the necessary reduction in (viscous, thermal, mass) dissipation and the developed approximate solver is able to predict the thermal and viscous profiles in a non-continuum regime and validates against DSMC simulations. Test cases consisting of flow through micro-channels demonstrated the ability of solver to handle monoatomic, diatomic and poly atomic gases. The developed non-continuum solver is in a finite volume framework and is easily adaptable to legacy in-house Navier–Stokes solvers.

Adversarial decisions on complex dynamical systems using game theory

We apply computational Game Theory to a unification of physics-based models that represent decision-making across a number of agents within both cooperative and competitive processes. Here the competitors try to both positively influence their own returns, while negatively affecting those of their competitors. Modelling these interactions with the so-called Boyd–Kuramoto–Lanchester (BKL) complex dynamical system model yields results that can be applied to business, gaming and security contexts. This paper studies a class of decision problems on the BKL model, where a large set of coupled, switching dynamical systems are analysed using game-theoretic methods.Due to their size, the computational cost of solving these BKL games becomes the dominant factor in the solution process. To resolve this, we introduce a novel Nash Dominant solver, which is both numerically efficient and exact. The performance of this new solution technique is compared to traditional exact solvers, which traverse the entire game tree, as well as to approximate solvers such as Myopic and Monte Carlo Tree Search (MCTS). These techniques are assessed, and used to gain insights into both nonlinear dynamical systems and strategic decision making in adversarial environments.

A Hydro-particle-mesh Code for Efficient and Rapid Simulations of the Intracluster Medium

Abstract We introduce the cosmological HYPER code based on an innovative hydro-particle-mesh (HPM) algorithm for efficient and rapid simulations of gas and dark matter. For the HPM algorithm, we update the approach of Gnedin & Hui to expand the scope of its application from the lower-density intergalactic medium (IGM) to the higher-density intracluster medium (ICM). While the original algorithm tracks only one effective particle species, the updated version separately tracks the gas and dark matter particles, as they do not exactly trace each other on small scales. For the approximate hydrodynamics solver, the pressure term in the gas equations of motion is calculated using robust physical models. In particular, we use a dark matter halo model, ICM pressure profile, and IGM temperature–density relation, all of which can be systematically varied for parameter-space studies. We show that the HYPER simulation results are in good agreement with the halo model expectations for the density, temperature, and pressure radial profiles. Simulated galaxy cluster scaling relations for Sunyaev–Zel’dovich (SZ) and X-ray observables are also in good agreement with mean predictions, with scatter comparable to that found in hydrodynamic simulations. HYPER also produces lightcone catalogs of dark matter halos and full-sky tomographic maps of the lensing convergence, SZ effect, and X-ray emission. These simulation products are useful for testing data analysis pipelines, generating training data for machine learning, understanding selection and systematic effects, and for interpreting astrophysical and cosmological constraints.

Partially Observable Markov Decision Processes and Robotics

Planning under uncertainty is critical to robotics. The partially observable Markov decision process (POMDP) is a mathematical framework for such planning problems. POMDPs are powerful because of their careful quantification of the nondeterministic effects of actions and the partial observability of the states. But for the same reason, they are notorious for their high computational complexity and have been deemed impractical for robotics. However, over the past two decades, the development of sampling-based approximate solvers has led to tremendous advances in POMDP-solving capabilities. Although these solvers do not generate the optimal solution, they can compute good POMDP solutions that significantly improve the robustness of robotics systems within reasonable computational resources, thereby making POMDPs practical for many realistic robotics problems. This article presents a review of POMDPs, emphasizing computational issues that have hindered their practicality in robotics and ideas in sampling-based solvers that have alleviated such difficulties, together with lessons learned from applying POMDPs to physical robots.

Full waveform inversion through double-sweeping solver

An efficient method is proposed to accurately approximate the gradient and the Hessian operator for the full-waveform inversion (FWI) in large-scale problems. The key idea is an approximate solver called double-sweeping solver, which divides the domain into smaller slabs and sequentially solves the wavefields through a downward and an upward sweeping. The sequential solution is facilitated by approximating the continuity conditions that suppress the multiples, thus relaxing long-range coupling between the subdomains. The double-sweeping solver is incorporated into an inexact Gauss-Newton approach to perform FWI, where the gradient and the Hessian vector multiplication are computed more efficiently. Through numerical experiments, we show that the convergence of FWI with respect to the number of iterations does not degrade when the double-sweeping approximation is used. Given that the double-sweeping solver is computationally cheaper than full-wave simulation, the proposed method is more efficient than the standard FWI. This paper contains the complete formulation of the proposed methodology as well as an illustration of its effectiveness to problems of varying complexity including the inversion of the Marmousi model from the Geophysics community.

GeoTangle : Interactive Design of Geodesic Tangle Patterns on Surfaces

Tangles are complex patterns, which are often used to decorate the surface of real-world artisanal objects. They consist of arrangements of simple shapes organized into nested hierarchies, obtained by recursively splitting regions to add progressively finer details. In this article, we show that 3D digital shapes can be decorated with tangles by working interactively in the intrinsic metric of the surface. Our tangles are generated by the recursive application of only four operators, which are derived from tracing the isolines or the integral curves of geodesics fields generated from selected seeds on the surface. Based on this formulation, we present an interactive application that lets designers model complex recursive patterns directly on the object surface without relying on parametrization. We reach interactive speed on meshes of a few million triangles by relying on an efficient approximate graph-based geodesic solver.

Block low‐rank single precision coarse grid solvers for extreme scale multigrid methods

AbstractExtreme scale simulation requires fast and scalable algorithms, such as multigrid methods. To achieve asymptotically optimal complexity, it is essential to employ a hierarchy of grids. The cost to solve the coarsest grid system can often be neglected in sequential computings, but cannot be ignored in massively parallel executions. In this case, the coarsest grid can be large and its efficient solution becomes a challenging task. We propose solving the coarse grid system using modern, approximate sparse direct methods and investigate the expected gains compared with traditional iterative methods. Since the coarse grid system only requires an approximate solution, we show that we can leverage block low‐rank techniques, combined with the use of single precision arithmetic, to significantly reduce the computational requirements of the direct solver. In the case of extreme scale computing, the coarse grid system is too large for a sequential solution, but too small to permit massively parallel efficiency. We show that the agglomeration of the coarse grid system to a subset of processors is necessary for the sparse direct solver to achieve performance. We demonstrate the efficiency of the proposed method on a Stokes‐type saddle point system solved with a monolithic Uzawa multigrid method. In particular, we show that the use of an approximate sparse direct solver for the coarse grid system can outperform that of a preconditioned minimal residual iterative method. This is demonstrated for the multigrid solution of systems of order up to degrees of freedom on a petascale supercomputer using 43,200 processes.

Fast Multipole Methods for N-body Simulations of Collisional Star Systems

Abstract Direct N-body simulations of star clusters are accurate but expensive, largely due to the numerous  ( N 2 ) pairwise force calculations. To solve the post-million-body problem, it will be necessary to use approximate force solvers, such as tree codes. In this work, we adapt a tree-based, optimized Fast Multipole Method (FMM) to the collisional N-body problem. The use of a rotation-accelerated translation operator and an error-controlled cell opening criterion leads to a code that can be tuned to arbitrary accuracy. We demonstrate that our code, Taichi, can be as accurate as direct summation when N > 104. This opens up the possibility of performing large-N, star-by-star simulations of massive stellar clusters and would permit large parameter space studies that would require years with the current generation of direct summation codes. Using a series of tests and idealized models, we show that Taichi can accurately model collisional effects, such as dynamical friction and the core-collapse time of idealized clusters, producing results in strong agreement with benchmarks from other collisional codes such as NBODY6++GPU or PeTar. Parallelized using OpenMP and AVX, Taichi is demonstrated to be more efficient than other CPU-based direct N-body codes for simulating large systems. With future improvements to the handling of close encounters and binary evolution, we clearly demonstrate the potential of an optimized FMM for the modeling of collisional stellar systems, opening the door to accurate simulations of massive globular clusters, super star clusters, and even galactic nuclei.

Approximating inverse FEM matrices on non-uniform meshes with {mathcal{H}}-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse {mathcal{H}}-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the {mathcal{H}}-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

FUZZOLIC: Mixing fuzzing and concolic execution

In the last few years, a large variety of approaches and methodologies have been explored in the context of software testing, ranging from black-box techniques, such as fuzzing, to white-box techniques, such as concolic execution, with a full spectrum of instances in between. Using these techniques, developers and security researchers have been able to identify in the last decade a large number of critical vulnerabilities in thousands of software projects.In this article, we investigate how to improve the performance and effectiveness of concolic execution, proposing two main enhancements to the original approach. On one side, we devise a novel concolic executor that can analyze complex binary programs while running under QEMU and efficiently produce symbolic queries, which could generate valuable program inputs when solved. On the other side, we investigate whether techniques borrowed from the fuzzing domain can be applied to solve the symbolic queries generated by concolic execution, providing a viable alternative to accurate but expensive SMT solving techniques. We show that the combination of our concolic engine, Fuzzolic, and our approximate solver, Fuzzy-Sat, can perform better in terms of code coverage than popular state-of-the-art fuzzers on a variety of complex programs and can identify different unknown bugs in several real-world applications.

Revisiting 2D Convolutional Neural Networks for Graph-Based Applications.

Graph convolutional networks (GCNs) are widely used in graph-based applications such as graph classification and segmentation. However, current GCNs have limitations on implementation such as network architectures due to their irregular inputs. In contrast, convolutional neural networks (CNNs) are capable of extracting rich features from large-scale input data, but they do not support general graph inputs. To bridge the gap between GCNs and CNNs, in this paper we study the problem of how to effectively and efficiently map general graphs to 2D grids that CNNs can be directly applied to, while preserving graph topology as much as possible. We therefore propose two novel graph-to-grid mapping schemes, namely, graph-preserving grid layout (GPGL) and its extension Hierarchical GPGL (H-GPGL) for computational efficiency. We formulate the GPGL problem as integer programming and further propose an approximate yet efficient solver based on a penalized Kamada-Kawai method, a well-known optimization algorithm in 2D graph drawing. We propose a novel vertex separation penalty that encourages graph vertices to lay on the grid without any overlap. Along with this image representation, even extra 2D maxpooling layers contribute to the PointNet, a widely applied point-based neural network. We demonstrate the empirical success of GPGL on general graph classification with small graphs and H-GPGL on 3D point cloud segmentation with large graphs, based on 2D CNNs including VGG16, ResNet50 and multi-scale maxout (MSM) CNN.

A multi-resolution method for two-phase fluids with complex equations of state by binomial solvers in three space dimensions

In this paper, we propose approximate solvers for two-phase fluids with general equations of state (EOS) in high dimension. The standard finite volume scheme is used for each fluid away from material interface. The two-phase interfaces are captured by the level set method coupled with a multi-resolution algorithm. For the Riemann solver of two-phase fluids with general equations of state, we construct an iterative approximation method by the solver for binomial equations of state. The velocity of the interface and the interface exchange fluxes are obtained precisely. With the help of the adaptive multi-resolution algorithms, we extend the method to three space dimensions conveniently. Numerical examples are carried out to demonstrate the strength and robustness of this method.

New Features in Feko and WinProp 2019

This paper describes some of the latest features in the commercial electromagnetic software Feko (including WinProp). These include the modeling of non-ideal cable shield connections, the parallel direct adaptive cross approximation (ACA) solver, edge and wedge diffraction for the ray launching geometrical optics (RL-GO) solver, and several new features related to automotive radar.

Reciprocal-Log Approximation and Planar PDE Solvers

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 25 September 2020Accepted: 13 July 2021Published online: 01 November 2021Keywordsreciprocal-log approximation, rational approximation, lightning solverAMS Subject Headings65N35, 41A30, 65E05Publication DataISSN (print): 0036-1429ISSN (online): 1095-7170Publisher: Society for Industrial and Applied MathematicsCODEN: sjnaam

An all speed second order well-balanced IMEX relaxation scheme for the Euler equations with gravity

We present an implicit-explicit well-balanced finite volume scheme for the Euler equations with a gravitational source term which is able to deal also with low Mach flows. To visualize the different scales we use the non-dimensionalized equations on which we apply a pressure splitting and a Suliciu relaxation. On the resulting model, we apply a splitting of the flux into a linear implicit and an non-linear explicit part that leads to a scale independent time-step. The explicit step consists of a Godunov type method based on an approximative Riemann solver where the source term is included in the flux formulation. We develop the method for a first order scheme and give an extension to second order. Both schemes are designed to be well-balanced, preserve the positivity of density and internal energy and have a scale independent diffusion. We give the low Mach limit equations for well-prepared data and show that the scheme is asymptotic preserving. These properties are numerically validated by various test cases.

Simple non-extensive sparsification of the hierarchical matrices

In this paper, we consider the matrices approximated in ${\mathscr{H}}^{2}$ format. The direct solution, as well as the preconditioning of systems with such matrices, is a challenging problem. We propose a non-extensive sparse factorization of the ${\mathscr{H}}^{2}$ matrix that allows to substitute direct ${\mathscr{H}}^{2}$ solution with the solution of the system with an equivalent sparse matrix of the same size. The sparse factorization is constructed of parameters of the ${\mathscr{H}}^{2}$ matrix. In the numerical experiments, we show the consistency of this approach in comparison with the other approximate block low-rank hierarchical solvers, such as HODLR [3], H2Lib [5], and IFMM [11].

A finite volume WENO scheme for immiscible inviscid two-phase flows

This article presents a two-phase finite volume (FV) method based on the concept of equilibrium volume fraction for simulating inviscid two-phase flows. A reduced three-equation hyperbolic system is adopted as the governing equations, which is closed with a barotropic equation of state for both phases. A modified weighted essentially non-oscillatory (WENO) scheme is proposed for reconstructing the fluid state at cell-interfaces, which can avoid strictly pressure or velocity oscillations near material interfaces in the test-case of the advection of an isolated interface, while retaining the convergence order of the WENO scheme in 1D and 2D cases. Besides, with the generalized Riemann invariants (GRI), the eigenstructure of the hyperbolic system is analyzed and an approximate Riemann flux solver is proposed based on the linearization of the GRI. The semi-discrete system is explicitly integrated in time by means of the classical 4th-order Runge-Kutta (RK4) scheme. The proposed scheme is validated in a series of two-phase test-cases, such as advection of a two-phase vortex, linear sloshing, long-time wave propagation and dam-break flows, for which good agreements are obtained with the references.