High-quality Approximation Research Articles

Fast regression of the tritium breeding ratio in fusion reactors

The tritium breeding ratio (TBR) is an essential quantity for the design of modern and next-generation D-T fueled nuclear fusion reactors. Representing the ratio between tritium fuel generated in breeding blankets and fuel consumed during reactor runtime, the TBR depends on reactor geometry and material properties in a complex manner. In this work, we explored the training of surrogate models to produce a cheap but high-quality approximation for a Monte Carlo (MC) TBR model in use at the UK Atomic Energy Authority. We investigated possibilities for dimensional reduction of its feature space, reviewed 9 families of surrogate models for potential applicability, and performed hyperparameter optimization. Here we present the performance and scaling properties of these models, the fastest of which, an artificial neural network, demonstrated and a mean prediction time of , representing a relative speedup of with respect to the expensive MC model. We further present a novel adaptive sampling algorithm, Quality-Adaptive Surrogate Sampling, capable of interfacing with any of the individually studied surrogates. Our preliminary testing on a toy TBR theory has demonstrated the efficacy of this algorithm for accelerating the surrogate modelling process.

Three-way approximation of decision granules based on the rough set approach

We discuss the three-way rough set based approach for approximation of decision granules in Intelligent Systems (IS's). The novelty of the approach is in presenting a new concept of approximation space which is based on advanced reasoning tools. Many generalisations of the rough set approaches developed over the years are mainly concentrated around reasoning concerning (partial) inclusion of sets. However, such approximation spaces are not satisfactory to deal with important aspects of approximate reasoning by IS's aiming to construct of the high quality approximations of compound decision granules. We demonstrate a number of examples supporting this claim. In particular, in solving the considered in the paper problems are involved complex algorithmic optimization processes directed by reasoning tools supporting searching for (semi-)optimal approximations of decision granules in huge spaces. This paper is a step toward developing tools for derivation of granules supporting IS's in perceiving situations to a degree satisfactory for making the right decisions.

High-quality approximation of log-aesthetic curves based on the fourth-order derivative

Abstract We propose a new method for approximating log-aesthetic curves ${\boldsymbol C}_{\mathrm{LA}}$ using high-degree Bézier curves. By leveraging the property that higher order derivatives are more sensitive to the quality of approximation, the method minimizes an objective function based on the fourth-order derivative; consequently, ${\boldsymbol C}_{\mathrm{LA}}$ is approximated with high accuracy. In addition, the proposed method is composed of two steps to ensure stable optimization so as not to be negatively affected because of a local minimum and to evaluate the fourth-order derivative. Furthermore, we reveal the difficulty in sufficiently approximating ${\boldsymbol C}_{\mathrm{LA}}$ with Bézier curves from two aspects. One aspect entails the uncertainty of how accurately the low-degree Bézier curves can approximate ${\boldsymbol C}_{\mathrm{LA}}$. The other aspect is the existence of a subset of ${\boldsymbol C}_{\mathrm{LA}}$ that is inherently difficult to approximate with such conventional parametric curves. The experimental results and comparisons demonstrated the validity of the proposed method.

State Preparation Boosters for Early Fault-Tolerant Quantum Computation

Quantum computing is believed to be particularly useful for the simulation of chemistry and materials, among the various applications. In recent years, there have been significant advancements in the development of near-term quantum algorithms for quantum simulation, including VQE and many of its variants. However, for such algorithms to be useful, they need to overcome several critical barriers including the inability to prepare high-quality approximations of the ground state. Current challenges to state preparation, including barren plateaus and the high-dimensionality of the optimization landscape, make state preparation through ansatz optimization unreliable. In this work, we introduce the method of ground state boosting, which uses a limited-depth quantum circuit to reliably increase the overlap with the ground state. This circuit, which we call a booster, can be used to augment an ansatz from VQE or be used as a stand-alone state preparation method. The booster converts circuit depth into ground state overlap in a controllable manner. We numerically demonstrate the capabilities of boosters by simulating the performance of a particular type of booster, namely the Gaussian booster, for preparing the ground state of N2 molecular system. Beyond ground state preparation as a direct objective, many quantum algorithms, such as quantum phase estimation, rely on high-quality state preparation as a subroutine. Therefore, we foresee ground state boosting and similar methods as becoming essential algorithmic components as the field transitions into using early fault-tolerant quantum computers.

Random Features for Kernel Approximation: A Survey on Algorithms, Theory, and Beyond.

The class of random features is one of the most popular techniques to speed up kernel methods in large-scale problems. Related works have been recognized by the NeurIPS Test-of-Time award in 2017 and the ICML Best Paper Finalist in 2019. The body of work on random features has grown rapidly, and hence it is desirable to have a comprehensive overview on this topic explaining the connections among various algorithms and theoretical results. In this survey, we systematically review the work on random features from the past ten years. First, the motivations, characteristics and contributions of representative random features based algorithms are summarized according to their sampling schemes, learning procedures, variance reduction properties and how they exploit training data. Second, we review theoretical results that center around the following key question: how many random features are needed to ensure a high approximation quality or no loss in the empirical/expected risks of the learned estimator. Third, we provide a comprehensive evaluation of popular random features based algorithms on several large-scale benchmark datasets and discuss their approximation quality and prediction performance for classification. Last, we discuss the relationship between random features and modern over-parameterized deep neural networks (DNNs), including the use of high dimensional random features in the analysis of DNNs as well as the gaps between current theoretical and empirical results. This survey may serve as a gentle introduction to this topic, and as a users' guide for practitioners interested in applying the representative algorithms and understanding theoretical results under various technical assumptions. We hope that this survey will facilitate discussion on the open problems in this topic, and more importantly, shed light on future research directions. Due to the page limit, we suggest the readers refer to the full version of this survey https://arxiv.org/abs/2004.11154.

GRosSo: mining statistically robust patterns from a sequence of datasets

Pattern mining is a fundamental data mining task with applications in several domains. In this work, we consider the scenario in which we have a sequence of datasets generated by potentially different underlying generative processes, and we study the problem of mining statistically robust patterns, which are patterns whose probabilities of appearing in transactions drawn from such generative processes respect well-defined conditions. Such conditions define the patterns of interest, describing the evolution of their probabilities through the datasets in the sequence, which may, for example, increase, decrease, or stay stable, through the sequence. Due to the stochastic nature of the data, one cannot identify the exact set of the statistically robust patterns by analyzing a sequence of samples, i.e., the datasets, taken from the generative processes, and has to resort to approximations. We then propose gRosSo, an algorithm to find rigorous approximations of the statistically robust patterns that do not contain false positives or false negatives with high probability. We apply our framework to the mining of statistically robust sequential patterns and statistically robust itemsets. Our extensive evaluation on pseudo-artificial and real data shows that gRosSo provides high-quality approximations for the problem of mining statistically robust sequential patterns and statistically robust itemsets.

OPTIMIZATION OF RESOURCE DISTRIBUTION UNDER THE CONDITIONS OF FUZZY INITIAL DATA

The problem of resource distribution in several directions of its spending is considered for the case when the parameters of the distribution efficiency criterion are fuzzy numbers with given membership functions. The purpose of the study is the development of mathematical models and methods for solving the problem of resource allocation for practically the most important criteria, taking into account the fuzziness of the numerical values – of their parameters. An analysis of the well-known approach to solving the problem is carried out and its main shortcomings are identified, which motivate the continuation of research. A method for solving the stated problem is proposed, the computational implementation of which contains three stages. At the first stage, using the membership functions of the fuzzy parameters of the problem, the membership function of the criterion is formed. The function obtained in this case is approximated at the second stage using a four-parameter distribution. An important advantage of this distribution is the possibility, by varying the numerical values – of its parameters over a wide range, to change the mathematical expectation, variance, and asymmetry of the values – specified by this distribution, providing a high quality of approximation. Thus, the criterion for the effectiveness of the task is determined. At the third stage, a mathematical model of the optimization problem of the distribution of a limited resource is formulated. The following three options for constructing an optimality criterion are considered: maximizing the criterion with the maximum possible value of its membership function; maximization of the criterion, provided that the value of its membership function is not lower than the specified one; maximization of the criterion, provided that the value of the membership function of each of its terms is not lower than the specified one. Each of the resulting problems is a standard problem of mathematical programming and is solved by known methods. A possible direction for further research is discussed in order to improve the adequacy of the used analytical descriptions of the membership functions of the fuzzy parameters of the problem.

Learning-based high-quality image recovery from 1D signals obtained by single-pixel imaging

As an innovative and computational imaging technique, it is critical for single-pixel imaging (SPI) to achieve a high reconstruction quality. However, the reconstruction image quality in conventional SPI is heavily dependent on the sampling rate. In this work, we present a learning-based SPI approach for high-quality image reconstruction under a low sampling rate. A bucket detector included in our optical setup is used to collect the one-dimensional (1D) signals reflected from a two-dimensional (2D) object and an end-to-end generative adversarial network (EGAN) which is pre-trained with simulated data is utilized to implement the reconstruction of the object. The results show that the proposed approach is able to produce high quality approximations of 2D images from optically collected 1D bucket signals at a very low sampling ratio. It can also be shown a better performance can be achieved by compared with previous studies on the same dataset, such as conventional SPI, compressive-sensing ghost imaging (CSGI) and U-net-based SPI approaches.

Parallel incremental update of semi-monolayer covering rough set based on object set changes

The semi-monolayer covering rough set has the characteristics of high approximation quality and efficient computation. In the face of massive data changes, the study of rough sets will face the challenge of high complexity computation. To address the large-scale changes of object sets in set-valued decision information systems, this paper proposes a dynamic update method based on Spark framework to solve the approximation set problem of semi-monolayer covering rough set. This method is mainly embodied in four aspects: firstly, for the change of information grains in the proposed single-layer covering rough set, this paper proposes an update strategy for the change of information units; then for the update of reliable elements, an update strategy is proposed for 〖▁C_"GC0" 〗^' (X) and 〖¯C_"GC0" 〗^' (X); secondly, for the change of reliable elements and disputed elements, an RSM matrix update strategy; and finally the update strategy of the approximation set is proposed for the changes of information units. Since the incremental methods using other models will yield different approximation sets and the prerequisite of the comparison experiments requires the approximation sets to be consistent, in order to ensure the correctness of the experimental results, this paper designs the comparison experiments of static and dynamic algorithms based on the semi-monolayer covering rough set model. The experimental results with several data sets show that the incremental algorithm speeds up in 3.67~7.72 times than the static algorithm when there is an object set update.

Security Constrained Unit Commitment Based on Modified Line Outage Distribution Factors

The N-1 criterion is universally taken as the security constrains in the security-constrained unit commitment (SCUC) problem. The line outage distribution factors (LODF) is widely used in the N-1 continency for its state independence and computational efficiency. However, the LODF based model fails to consider reactive power and voltage magnitude. This paper closes this gap by deriving modified line outage distribution factors (MLODF) based on a linear power flow model. The proposed model, though state-independent, provides high-quality approximation of voltage magnitudes. Then, we apply this model to formulate the SCUC problem and propose an iterative solution process based on MLODF post-contingency filter to set up the contingency constraints. The simulation results of the benchmark test systems verify the accuracy of proposed MLODF. The six-bus and IEEE 118-bus systems show that the MLODF based model can provide a more accurate and secure generation schedule comparing with the traditional LODF based model while ensuring the AC feasibility for most N-1 continency.

Approximating multi-purpose AC Optimal Power Flow with reinforcement trained Artificial Neural Network

Solving AC-Optimal Power Flow (OPF) problems is an essential task for grid operators to keep the power system safe for the use cases such as minimization of total generation cost or minimization of infeed curtailment from renewable DERs (Distributed Energy Resource). Mathematical solvers are often able to solve the AC-OPF problem but need significant computation time. Artificial neural networks (ANN) have a good application in function approximation with outstanding computational performance. In this paper, we employ ANN to approximate the solution of AC-OPF for multiple purposes. The novelty of our work is a new training method based on the reinforcement learning concept. A high-performance batched power flow solver is used as the physical environment for training, which evaluates an augmented loss function and the numerical action gradient. The augmented loss function consists of the objective term for each use case and the penalty term for constraints violation. This training method enables training without a reference OPF and the integration of discrete decision variable such as discrete transformer tap changer position in the constrained optimization. To improve the optimality of the approximation, we further combine the reinforcement training approach with supervised training labeled by reference OPF. Various benchmark results show the high approximation quality of our proposed approach while achieving high computational efficiency on multiple use cases.

Bi-objective facility location under uncertainty with an application in last-mile disaster relief

Multiple and usually conflicting objectives subject to data uncertainty are main features in many real-world problems. Consequently, in practice, decision-makers need to understand the trade-off between the objectives, considering different levels of uncertainty in order to choose a suitable solution. In this paper, we consider a two-stage bi-objective single source capacitated model as a base formulation for designing a last-mile network in disaster relief where one of the objectives is subject to demand uncertainty. We analyze scenario-based two-stage risk-neutral stochastic programming, adaptive (two-stage) robust optimization, and a two-stage risk-averse stochastic approach using conditional value-at-risk (CVaR). To cope with the bi-objective nature of the problem, we embed these concepts into two criterion space search frameworks, the epsilon -constraint method and the balanced box method, to determine the Pareto frontier. Additionally, a matheuristic technique is developed to obtain high-quality approximations of the Pareto frontier for large-size instances. In an extensive computational experiment, we evaluate and compare the performance of the applied approaches based on real-world data from a Thies drought case, Senegal.

Estimation of the growth rate of wild ungulate populations in the territory of the Saratov region by the dynamic series of their numbers

The present paper is devoted to the study of the possibility of estimating the reproductive potentials of wild ungulate populations, and possibly other large mammals, by the time series of their numbers. We have found out that this is possible, which is confirmed by the high quality of approximation of the time series of abundance by logistic curves, and the corresponding coefficients of their determination for different species ranged from 75 to 96%. For such calculations, one circumstance is necessary, which is that the population of the studied species has been briefly exposed to some unfavorable factor causing a significant reduction in its numbers with subsequent restoration to the previous level, or the time series should contain a well-expressed and extended section of the transition of the population from some lower level to the upper level of the population, passing into a stationary state. The values of the maximum exponential growth rates of ungulate populations that we obtained do not fundamentally differ from the data available in other researchers’ works. In addition, it should be borne in mind that our method for assessing the reproductive potentials of ungulates is statistical, with features accompanying all such methods, for example, in the presence of statistical errors in all determined parameters. However, the evaluation of the magnitude of these errors is a topic for a separate study.

Procedural modeling applied to the 3D city model of Bogota: A case study

BackgroundComputer Generated Animations (CGA), when applied to three-dimensional (3D) city models (3DCM), can be used as powerful tools to support urban decision-making. This leads to a new paradigm, based on procedural modeling, that allows the integration of known urban structures. MethodsThis paper introduces a new workflow for the development of high-quality approximations of urban models in a short time and enables facilities to be imported from other cities into a given city model, following specific generation rules. ResultsThus, this workflow provides a very simple approach to observe, study, and simulate the implementation of models already developed in other cities, in a city where they are not yet adopted. Examples of these models include all types of mobility systems and urban infrastructure. ConclusionsThis allows us to perceive the environmental impact of certain decisions in the real world, as well as to carry out simple simulations to determine the changes that can occur in the flows of people, traffic, and other city activities.

Tiered Sampling

We introduce Tiered Sampling , a novel technique for estimating the count of sparse motifs in massive graphs whose edges are observed in a stream. Our technique requires only a single pass on the data and uses a memory of fixed size M , which can be magnitudes smaller than the number of edges. Our methods address the challenging task of counting sparse motifs—sub-graph patterns—that have a low probability of appearing in a sample of M edges in the graph, which is the maximum amount of data available to the algorithms in each step. To obtain an unbiased and low variance estimate of the count, we partition the available memory into tiers (layers) of reservoir samples. While the base layer is a standard reservoir sample of edges, other layers are reservoir samples of sub-structures of the desired motif. By storing more frequent sub-structures of the motif, we increase the probability of detecting an occurrence of the sparse motif we are counting, thus decreasing the variance and error of the estimate. While we focus on the designing and analysis of algorithms for counting 4-cliques, we present a method which allows generalizing Tiered Sampling to obtain high-quality estimates for the number of occurrence of any sub-graph of interest, while reducing the analysis effort due to specific properties of the pattern of interest. We present a complete analytical analysis and extensive experimental evaluation of our proposed method using both synthetic and real-world data. Our results demonstrate the advantage of our method in obtaining high-quality approximations for the number of 4 and 5-cliques for large graphs using a very limited amount of memory, significantly outperforming the single edge sample approach for counting sparse motifs in large scale graphs.

Two-stage stochastic program optimizing the cost of electric vehicles in commercial fleets

The possibility of electric vehicles to technically replace internal combustion engine vehicles and to deliver economic benefits still mainly depends on the battery size and the charging infrastructure costs as well as on annual mileage (utilizing the lower variable costs of electric vehicles). Current studies on electric vehicles’ total cost of ownership often neglect two important factors that influence the investment decision and operational costs: firstly, the trade-off between battery and charging capacity; secondly the uncertainty in energy consumption. This paper proposes a two-stage stochastic program that minimizes the total cost of ownership of a commercial electric vehicle under uncertain energy consumption and available charging times induced by mobility patterns and outside temperature. The optimization program is solved by sample average approximation based on mobility and temperature scenarios. A hidden Markov model is introduced to predict mobility demand scenarios. Three scenario reduction heuristics are applied to reduce computational effort while keeping a high-quality approximation. The proposed framework is tested in a case study of the home nursing service. The results show the large influence of the uncertain mobility patterns on the optimal solution. In the case study, the total cost of ownership can be reduced by up to 3.9% by including the trade-off between battery and charging capacity. The introduction of variable energy prices can lower energy costs by 31.6% but does not influence the investment decision in this case study. Overall, this study provides valuable insights for real applications to determine the techno-economic optimal electric vehicle and charging infrastructure configuration.

TipTap: Approximate Mining of Frequent k -Subgraph Patterns in Evolving Graphs

“Perhaps he could dance first and think afterwards, if it isn’t too much to ask him.” S. Beckett, Waiting for Godot Given a labeled graph, the collection of k -vertex induced connected subgraph patterns that appear in the graph more frequently than a user-specified minimum threshold provides a compact summary of the characteristics of the graph, and finds applications ranging from biology to network science. However, finding these patterns is challenging, even more so for dynamic graphs that evolve over time, due to the streaming nature of the input and the exponential time complexity of the problem. We study this task in both incremental and fully-dynamic streaming settings, where arbitrary edges can be added or removed from the graph. We present TipTap , a suite of algorithms to compute high-quality approximations of the frequent k -vertex subgraphs w.r.t. a given threshold, at any time (i.e., point of the stream), with high probability. In contrast to existing state-of-the-art solutions that require iterating over the entire set of subgraphs in the vicinity of the updated edge, TipTap operates by efficiently maintaining a uniform sample of connected k -vertex subgraphs, thanks to an optimized neighborhood-exploration procedure. We provide a theoretical analysis of the proposed algorithms in terms of their unbiasedness and of the sample size needed to obtain a desired approximation quality. Our analysis relies on sample-complexity bounds that use Vapnik–Chervonenkis dimension, a key concept from statistical learning theory, which allows us to derive a sufficient sample size that is independent from the size of the graph. The results of our empirical evaluation demonstrates that TipTap returns high-quality results more efficiently and accurately than existing baselines.

MO-NFSA for solving unconstrained multi-objective optimization problems

Normative Fish Swarm Algorithm (NFSA) is a novel variant of the Artificial Fish Swarm Algorithm (AFSA) proposed in 2019, and has been proven effective in solving single-objective optimization problems. Inspired by the potential of NFSA, this paper proposes an evolutionary multi-objective optimization algorithm, namely “Multi-Objective Normative Fish Swarm Algorithm (MO-NFSA)”. However, due to the fact that NFSA was originally modeled for single-objective optimization, a certain degree of transformation, modification, and additional strategies (related to multi-objective optimization features) must be integrated into the original NFSA as part of MO-NFSA modeling for further extension. This article adopts a total of 15 multi-objective optimization test cases in any category of fixed-dimensional, non-fixed-dimensional (ZDT set) or scalable multi-objective (DTLZ set) optimization types. These multi-objective optimization test cases are used to compare the performance of MO-NFSA with other comparative algorithms. However, performance evaluation of multi-objective optimization is a daunting task. Their performance results can only be digitized through quality indicators (i.e., performance metrics), which are mainly tested from three different aspects of high-quality approximation: convergence, uniformity, and spread. Here, quality indicators including generation distance (GD), spacing (S), and spread-delta (∆/∆*) are used to check the quality performance of each algorithm based on the corresponding approximation. In the work, 20 simulations are run, and hence, 20 sets of data are collected. The collected results prove that MO-NFSA is superior to other comparison algorithms in all aspects of high-quality approximation. MO-NFSA can solve different types of multi-objective optimization problems.

Semi-monolayer covering rough set on set-valued information systems and its efficient computation

For set-valued information systems, there are many original dot-based approximation models based on tolerance relations and other developed tolerance relations. Because of the lack of efficient algorithms, they are not to accommodate the bigger and bigger set-valued information table. Therefore, it is a real challenge on how to efficiently calculate a high-quality approximation set in set-valued information systems. To address the challenge, we propose reliable approximation operators based on semi-monolayer covering for set-valued information systems. Benefiting from considerable research about tolerance rough set models and covering rough set models, the proposed approximation operators used a piecewise design to effectively reduce the negative effects of the set-valued records and provided high-quality approximation sets for set-valued information systems. Furthermore, the reliable semi-monolayer covering approximation sets are more easily granulated and efficiently calculated than before. Based on the equivalent granule-based forms, the corresponding granular algorithms are designed for the improved approximation sets. The experiments on some UCI data sets show the improved approximation sets are high quality and efficient computational in set-valued information systems.

High-Order Resolution of Multidimensional Compressible Reactive Flow Using Implicit Shock Tracking

A recently developed high-order implicit shock tracking method is novelly applied to a benchmark problem in two-dimensional compressible reactive flow, and results of remarkably high accuracy are achieved relative to competing shock capturing schemes. High-order implicit shock tracking is a discontinuous Galerkin discretization of conservation laws that simultaneously computes an approximate flow solution and aligns faces of the computational mesh with discontinuities in the flow to provide nonlinear stabilization and an improved approximation to the solution. The model problem is chosen such that its exact solution is available in analytic form to facilitate a detailed study of the truncation error of the tracking method relative to a nominally fifth-order weighted essentially nonoscillatory method. We show the implicit tracking method is able to robustly align the mesh with the shock and, particularly for polynomial bases of degree greater than 1, provide a high-quality approximation to the exact solution on meshes far coarser than required by standard methods. Finally, we demonstrate the tracking method obtains near optimal convergence rates in several error metrics for the problem under consideration.