Separable Approximation Research Articles

Relationship between drift kinetics, gyrokinetics and magnetohydrodynamics in the long-wavelength limit

Gyrokinetic theories, even ‘global’ models, typically rely on a separation in scale (in perpendicular wavelength) between the fluctuations and the system scale. In such models direct simulation of system-scale dynamics like magnetohydrodynamic (MHD) motion is formally not consistent. Drift-kinetic theory, on the other hand, may be directly applied to model system-scale MHD-ordered behaviour. I review the long-wavelength limit of standard gyrokinetics and drift kinetics to present the relationships between these theories in an elementary fashion. This provides a pathway to global gyrokinetic modelling, resulting in an approach that is structurally similar to kinetic MHD, and I present dynamical equations for solving global field evolution in this framework. Departures from certain earlier global gyrokinetic theories include the appearance of magnetosonic (fast) modes, and the cross-coupling of the parallel and perpendicular currents with perpendicular and parallel magnetic field components. A periodic two-dimensional testcase is outlined as a benchmarking and implementation target, to help clarify practical aspects of these theories, with minimal complexity in terms of boundary conditions, and a proof-of-principle implementation of a field-solver is exhibited. To motivate this work, I first illustrate certain limitations of existing global gyrokinetic frameworks and directly identify how scale separation approximations lead to certain ‘missing’ system-scale field terms in global gyrokinetics, largely as a result of simplifications associated with the field representation in terms of $A_{\|}$ and $B_{\|}$ . As a result, the currents in the gyrokinetic Ampère's law resulting from a gyrokinetic equilibrium distribution do not match the currents implied by Ampère's law in a force-balance MHD equilibrium. I present a simple choice of equilibrium distribution function whose drift-kinetic currents are consistent with MHD currents; a specific Grad–Shafranov equilibrium is used to illustrate the size of the components of the parallel currents.

Optimal decision-making of mutual fund temporary borrowing problem via approximate dynamic programming

Temporary borrowing is a liquidity risk management tool for mutual fund managers to meet investor redemption demands. We develop a new Markov decision process model to describe the temporary borrowing process, considering the multiple lending channels, the validity period of loans, and the uncertainties of cost, demand, and maximum loan amount simultaneously. An approximate dynamic programming (ADP) algorithm is initiated to solve the cost-minimizing temporary borrowing problem. We construct the value function as a separable approximation and prove the convexity of its components with respect to the available funds in different channels. Moreover, a new value function updating formula, DMAX, is designed to overcome value function overestimation. The proved convexity and the proposed formula contribute to fast and reliable value function estimation. Numerical experiments based on actual business data show that the proposed algorithm can obtain near-optimal decisions in deterministic cases and maintain high robustness in stochastic cases.

Interpolative Separable Density Fitting for Accelerating Two-Electron Integrals: A Theoretical Perspective.

Low-rank approximations have long been considered an efficient way to accelerate electronic structure calculations associated with the evaluation of electron repulsion integrals (ERIs). As an accurate and efficient algorithm for compressing the ERI tensor, the interpolative separable density fitting (ISDF) decomposition has recently attracted great attention in this context. In this perspective, we introduce the ISDF decomposition from the theoretical aspects and technique details. The ISDF decomposition can construct a fully separable low-rank approximation (tensor hypercontraction factorization) of ERIs in real space with a cubic cost, offering great flexibility for accelerating high-scaling electronic structure calculations. We review the typical applications of ISDF in hybrid functionals, time-dependent density functional theory, and GW approximation. Finally, we discuss the promising directions for future development of ISDF.

Reductions of non-separable approximate linear programs for network revenue management

We suggest a novel choice of non-separable basis functions for an approximate linear programming approach to the well-known network revenue management problem. Considering non-separability is particularly important when interdependencies between resources are large. Such a situation can be illustrated for example by a bus line, where different origin-destination pairs have many overlapping segments. Traditional separable approximation approaches tend to ignore the resulting interactions.We suggest to group resources into non-separable subnetworks. For each chosen subnetwork, basis functions either span the whole function space or consist of linear functions. Given this more general choice of basis functions, we extend existing reductions of approximate linear programs. If there is only one subnetwork, for which the basis functions span the whole function space, we prove the equivalence to a compact linear program of polynomial size. For the general case, we suggest an approximate reduction. Numerical examples illustrate our novel upper bounds for the maximum expected revenue and the corresponding competitive policies. In particular, we find that the added benefit of non-separability heavily depends on the network structure and the capacity.Our work helps to better understand the impact of assuming separability in network revenue management. The polynomial sized reductions make it possible to estimate the added average revenue resulting from incorporating interactions between resources. The theory we develop demonstrates how the interpretation of dual variables as state-action probabilities can be applied to reduce exponentially large approximate linear programs via variable aggregation.

Internal Boundary Between Entanglement and Separability Within a Quantum State

Quantum states are the key mathematical objects in quantum mechanics, and entanglement lies at the heart of the nascent fields of quantum information processing and computation. However, there has not been a general, necessary and sufficient, and operational separability condition to determine whether an arbitrary quantum state is entangled or separable. In this paper, we show that whether a quantum state is entangled or not is determined by a threshold within the quantum state. We first introduce the concept of finer and optimal separable states based on the properties of separable states in the role of higher-level witnesses. Then we show that any bipartite quantum state can be decomposed into a convex mixture of its optimal entangled state and its optimal separable state. Furthermore, we show that whether an arbitrary quantum state is entangled or separable, as well as positive partial transposition (PPT) or not, is determined by the robustness of its optimal entangled state to its optimal separable state with reference to a crucial threshold. Moreover, for an arbitrary quantum state, we provide operational algorithms to obtain its optimal entangled state, its optimal separable state, its best separable approximation (BSA) decomposition, and its best PPT approximation decomposition. And the open question of how to calculate the BSA in high-dimension systems is partially done.

A Non-Cooperative Resource Utilization Game Between Two Competing Malware

In this letter, we consider a population of digital nodes (such as phones, computers, etc.) that are under the attack of two competing malware. These malware infect the nodes in order to exploit their computational resources for specific purposes such as mining crypto-currency, cloud computing, etc. We suppose that each virus spreads following the susceptible-infected-susceptible (SIS) compartmental model. Additionally, we assume that the malware designers can tune the percentage of resource utilization from their host nodes. A higher resource utilization implies a higher instantaneous profit but will also lead to faster detection and elimination (node recovery) of the malware. Once the malware is detected, complete protection of the infected node by means of anti-malware software is also possible at a smaller rate. The proposed setup results in a non-cooperative game between the two players (the malware designers) trying to maximize their profit i.e., the resources utilized from the infected nodes. We characterize and analyze the Nash equilibrium for such a game using a time-scale separation approximation. Finally, we numerically validate the approximation and we compute the price of anarchy.

Compact Approximate Linear Programs for Network Revenue Management: Reduction via Constraint Decomposition and Adaptation for Locally Separable Approximation

Compact Approximate Linear Programs for Network Revenue Management: Reduction via Constraint Decomposition and Adaptation for Locally Separable Approximation

Gauge-Invariant Perturbations at a Quantum Gravity Bounce

We study the dynamics of gauge-invariant scalar perturbations in cosmological scenarios with a modified Friedmann equation, such as quantum gravity bouncing cosmologies. We work within a separate universe approximation which captures wavelengths larger than the cosmological horizon; this approximation has been successfully applied to loop quantum cosmology and group field theory. We consider two variables commonly used to characterise scalar perturbations: the curvature perturbation on uniform-density hypersurfaces ζ and the comoving curvature perturbation R. For standard cosmological models in general relativity as well as in loop quantum cosmology, these quantities are conserved and equal on super-horizon scales for adiabatic perturbations. Here we show that while these statements can be extended to a more general form of modified Friedmann equations similar to that of loop quantum cosmology, in other cases, such as the simplest group field theory bounce scenario, ζ is conserved across the bounce whereas R is not. We relate our results to approaches based on a second-order equation for a single perturbation variable, such as the Mukhanov–Sasaki equation.

Quantifying deviations from separability in space-time functional processes

The estimation of covariance operators of spatio-temporal data is in many applications only computationally feasible under simplifying assumptions, such as separability of the covariance into strictly temporal and spatial factors. Powerful tests for this assumption have been proposed in the literature. However, as real world systems, such as climate data are notoriously inseparable, validating this assumption by statistical tests, seems inherently questionable. In this paper we present an alternative approach: By virtue of separability measures, we quantify how strongly the data’s covariance operator diverges from a separable approximation. Confidence intervals localize these measures with statistical guarantees. This method provides users with a flexible tool, to weigh the computational gains of a separable model against the associated increase in bias. As separable approximations we consider the established methods of partial traces and partial products, and develop weak convergence principles for the corresponding estimators. Moreover, we also prove such results for estimators of optimal, separable approximations, which are arguably of most interest in applications. In particular we present for the first time statistical inference for this object, which has been confined to estimation previously. Besides confidence intervals, our results encompass tests for approximate separability. All methods proposed in this paper are free of nuisance parameters and do not require computationally expensive resampling procedures. A simulation study underlines the advantages of our approach and its applicability is demonstrated by the investigation of German annual temperature data.

Efficient compact linear programs for network revenue management

We are concerned with computing bid prices in network revenue management using approximate linear programming. It is well-known that affine value function approximations yield bid prices which are not sensitive to remaining capacity. The analytic reduction to compact linear programs allows the efficient computation of such bid prices. On the other hand, capacity-dependent bid prices can be obtained using separable piecewise linear value function approximations. Even though compact linear programs have been derived for this case also, they are still computationally much more expensive compared to using affine functions. We propose compact linear programs requiring substantially smaller computing times while, simultaneously, significantly improving the performance of capacity-independent bid prices. This simplification is achieved by taking into account remaining capacity only if it becomes scarce. Although our proposed linear programs are relaxations of the unreduced approximate linear programs, we conjecture equivalence and provide according numerical support. We measure the quality of an approximation by the difference between the expected performance of an induced policy and the corresponding theoretical upper bound. Using this paradigm in numerical experiments, we demonstrate the competitiveness of our proposed linear programs.

Electron-Affinity Time-Dependent Density Functional Theory: Formalism and Applications to Core-Excited States.

The lack of particle-hole attraction and orbital relaxation within time-dependent density functional theory (TDDFT) lead to extreme errors in the prediction of K-edge X-ray absorption spectra (XAS). We derive a linear-response formalism that uses optimized orbitals of the n - 1-electron system as the reference, building orbital relaxation and a proper hole into the initial density. Our approach is an exact generalization of the static-exchange approximation that ameliorates the particle-hole interaction error associated with the adiabatic approximation and reduces errors in TDDFT XAS by orders of magnitude. With a statistical performance of just 0.5 eV root-mean-square error and the same computational scaling as TDDFT under the core-valence separation approximation, we anticipate that this approach will be of great utility in XAS calculations of large systems.

TTDFT: A GPU accelerated Tucker tensor DFT code for large-scale Kohn-Sham DFT calculations

We present the Tucker tensor DFT (TTDFT) code which uses a tensor-structured algorithm with graphic processing unit (GPU) acceleration for conducting ground-state DFT calculations on large-scale systems. The Tucker tensor DFT algorithm uses a localized Tucker tensor basis computed from an additive separable approximation to the Kohn-Sham Hamiltonian. The discrete Kohn-Sham problem is solved using Chebyshev filtered subspace iteration method that relies on matrix-matrix multiplications of a sparse symmetric Hamiltonian matrix and a dense wavefunction matrix, expressed in the localized Tucker tensor basis. These matrix-matrix multiplication operations, which constitute the most computationally intensive step of the solution procedure, are GPU accelerated providing ∼8-fold GPU-CPU speedup for these operations on the largest systems studied. The computational performance of the TTDFT code is presented using benchmark studies on aluminum nano-particles and silicon quantum dots with system sizes ranging up to ∼7,000 atoms. Program summaryProgram Title: TTDFT: Tucker tensor density functional theory codeCPC Library link to program files:https://doi.org/10.17632/8dgmcs8ys2.1Licensing provisions: LGPLProgramming language: C/C++External routines/libraries: TuckerMPI (https://gitlab.com/tensors/TuckerMPI), cuBLAS (https://docs.nvidia.com/cuda/cublas/index.html), cuSparse (https://docs.nvidia.com/cuda/cusparse/index.html), ALGLIB (http://www.alglib.net/), Boost (https://www.boost.org/), BLAS (http://www.netlib.org/blas/), LAPACK (http://www.netlib.org/lapack/), PETSc (https://www.mcs.anl.gov/petsc), SLEPc (http://slepc.upv.es)Nature of problem: Real-space Kohn-Sham density functional theory calculations using localized Tucker tensor basis.Solution method: We present a real-space Kohn-Sham density functional theory code based on tensor-structured techniques with GPU acceleration. Tensor-structured techniques are adopted for computing a Tucker tensor basis, representing the eigenfunctions of an additive separable approximation to the Kohn-Sham Hamiltonian. The Tucker tensor basis is further localized using L1 regularization to improve the sparsity of the Kohn-Sham Hamiltonian matrix, and improve the computational efficiency and parallel scalability of the proposed algorithm. The solution to the Kohn-Sham problem in the localized Tucker tensor basis is computed using the Chebyshev filtered subspace iteration (ChFSI) method.Additional comments including restrictions and unusual features: The code works with Troullier-Martin (TM) pseudopotentials in Kleinman-Bylander form. The current release supports only non-periodic DFT calculations with the local density approximation (LDA) for exchange-correlation functional.This TTDFT project uses GitHub via Git, a free distributed version control software. The archived version at the time of submission of this work can be found on the CPC program library through program files DOI provided above. The GitHub repository of this project can be found on https://github.com/ttdftdev/ttdft_public.

Mathematical Modeling of Hydrodynamic Cavitation as Low Energy Extraction Technique for Lipid Removal from Nannochloropsis sp.

Lipid extraction assisted by hydrodynamic cavitation (HCLE) is one of the promising processes with low energy requirements. This study aims to reduce the energy requirement using a discrete flow system and evaluate two models to calculate the volumetric mass transfer coefficient. The variations of the number of repetitions, cavitation number, microalgae concentration, and temperature have affected the energy requirement value. The first model uses total lipid mass transfer approximation (Model 1) and the second uses separated lipid mass transfer approximation (Model 2). Based on Model 1 the value of total volumetric mass transfer coefficient ( ) f were 1.166 x 10-2, 3.113 x 10-3 and 1.285 x 10-3 min-1 with coefficient of determination (R2) is 0.9797. Whereas, based on Model 2 the value of volumetric mass transfer coefficient from disrupted microalgae ( ) were 1.131 x 10-2, 2.925 x 10-3 and 1.260 x 10-3 min-1 and from the intact microalgae ( ) was 0.051, 0.030 and 0.011 1/min with R2 of 0.9766. Both models gave a similar result. It was shown that lipid release from disrupted microalgae was dominant compared to the intact microalgae. Therefore, the discrete flow system of HCLE is a promising technique for extracting lipids from microalgae

Building separable approximations for quantum states via neural networks

Finding the closest separable state to a given target state is a notoriously difficult task, even more difficult than deciding whether a state is entangled or separable. To tackle this task, we parametrize separable states with a neural network and train it to minimize the distance to a given target state with respect to a differentiable distance, such as the trace distance or Hilbert--Schmidt distance. By examining the output of the algorithm, we obtain an upper bound on the entanglement of the target state, and construct an approximation for its closest separable state. We benchmark the method on a variety of well-known classes of bipartite states and find excellent agreement, even up to local dimension of $d=10$, while providing conjectures and analytic insight for isotropic and Werner states. Moreover, we show our method to be efficient in the multipartite case, considering different notions of separability. Examining three- and four-party GHZ and W states, we recover known bounds and obtain additional ones, for instance, for triseparability.

Dynamical approximations for composite quantum systems: assessment of error estimates for a separable ansatz

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work Burghardt et al (2021 J. Phys. A: Math. Theor. 54 414002). Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The time-dependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate.

On the Asymptotic Evaluation of the Physical Optics Approximation for Plane Wave Scattering by Circular Conducting Cylinders

In this paper, the scattering far-field from a circular electric conducting cylinder has been analyzed by physical optics (PO) approximation for both H and E polarizations. The evaluation of radiation integrations due to the PO current is conducted numerically and analytically. While non-uniform and uniform asymptotic solutions have been derived by the saddle point method, a separate approximation has been made for forward scattering direction. Comparisons among our approximation, direct numerical integration and exact solution results yield a good agreement for electrically large cylinders.

Stochastic alternating direction method of multipliers for Byzantine-robust distributed learning

This paper aims to solve a distributed learning problem under Byzantine attacks. In the underlying distributed master-worker architecture, there exist a number of unknown but malicious workers that can send arbitrary messages to the master to deviate the learning process, called Byzantine workers. In the literature, a total variation (TV) norm-penalized approximation formulation has been investigated to alleviate the effect of Byzantine attacks. To be specific, the TV norm penalty not only forces the local variables at the regular workers to be close, but is robust to the outliers sent by the Byzantine workers as well. For handling the separable TV norm-penalized approximation formulation, we propose a Byzantine-robust stochastic alternating direction method of multipliers (ADMM). Theoretically, we prove that the proposed method converges to a bounded neighborhood of the optimal solution at a rate of O(1/k) under mild assumptions, where k is the number of iterations and the size of neighborhood is determined by the number of Byzantine workers. Numerical experiments on the MNIST and COVERTYPE datasets further demonstrate the effectiveness of the proposed method to various Byzantine attacks.

How separable median filters can get better results than full 2D versions

The well-known method of median filtering is used both in a wide range of application frameworks and as a standalone filter. Small-window median filters can highly reduce the power of salt and pepper or additive Gaussian noise and minimize edge blurring. Large-window filters are also used, for example to estimate the background of images. Currently, the window size should not be an issue as a constant time algorithm and several implementations, including GPU-based codes, have been proposed in recent years. Unfortunately, none of these constant time implementations manage to fully exploit the capabilities of modern GPUs, and thus the throughputs of large-window median filters remain far below the peak throughputs allowed by recent GPU models. This paper aims at showing that a separable approximation of a 2D median filtering is often stronger than its full 2D implementation. Statistical and theoretical analysis are conducted to show and explain this fact which had so far remained unobserved. It is confirmed by experimentations on a dataset composed of 10,000 images, corrupted by different levels of salt and pepper noise. Separable and full 2D median filter algorithms are compared with several metrics, notably PSNR and MSSIM. In addition, a GPU implementation of 2D separable median filters is also proposed. This implementation is able to output up to 125 billion pixels per second on a recent Volta V100, which significantly outperforms existing implementations like Nvidia’s NPP library or Green’s code, resulting in the fastest median filtering solution to date.

Training very large scale nonlinear SVMs using Alternating Direction Method of Multipliers coupled with the Hierarchically Semi-Separable kernel approximations

• Large scale nonlinear Support Vector Machines can be trained using Alternating Direction Method of Multipliers and Hierarchically-Semi- Separable kernel approximations. • Reduced computational times can be obtained without worsening the classification accuracy. • Grid search for the misclassification parameter selection is extremely efficient in this framework. • Our efficient C++ implementation shows clear advantages over LIBSVM. Typically, nonlinear Support Vector Machines (SVMs) produce significantly higher classification quality when compared to linear ones but, at the same time, their computational complexity is prohibitive for large-scale datasets: this drawback is essentially related to the necessity to store and manipulate large, dense and unstructured kernel matrices. Despite the fact that at the core of training an SVM there is a simple convex optimization problem, the presence of kernel matrices is responsible for dramatic performance reduction, making SVMs unworkably slow for large problems. Aiming [at] an efficient solution of large-scale nonlinear SVM problems, we propose the use of the Alternating Direction Method of Multipliers coupled with Hierarchically Semi-Separable (HSS) kernel approximations. As shown in this work, the detailed analysis of the interaction among their algorithmic components unveils a particularly efficient framework and indeed, the presented experimental results demonstrate[, in the case of Radial Basis Kernels,] a significant speed-up when compared to the state-of-the-art nonlinear SVM libraries (without significantly affecting the classification accuracy).

Symplectic integration of learned Hamiltonian systems.

Hamiltonian systems are differential equations that describe systems in classical mechanics, plasma physics, and sampling problems. They exhibit many structural properties, such as a lack of attractors and the presence of conservation laws. To predict Hamiltonian dynamics based on discrete trajectory observations, the incorporation of prior knowledge about Hamiltonian structure greatly improves predictions. This is typically done by learning the system's Hamiltonian and then integrating the Hamiltonian vector field with a symplectic integrator. For this, however, Hamiltonian data need to be approximated based on trajectory observations. Moreover, the numerical integrator introduces an additional discretization error. In this article, we show that an inverse modified Hamiltonian structure adapted to the geometric integrator can be learned directly from observations. A separate approximation step for the Hamiltonian data is avoided. The inverse modified data compensate for the discretization error such that the discretization error is eliminated. The technique is developed for Gaussian processes.