Deterministic Sampling Method Research Articles

Efficient Blind Hyperspectral Unmixing Framework Based on CUR Decomposition (CUR-HU)

Hyperspectral imaging captures detailed spectral data for remote sensing. However, due to the limited spatial resolution of hyperspectral sensors, each pixel of a hyperspectral image (HSI) may contain information from multiple materials. Although the hyperspectral unmixing (HU) process involves estimating endmembers, identifying pure spectral components, and estimating pixel abundances, existing algorithms mostly focus on just one or two tasks. Blind source separation (BSS) based on nonnegative matrix factorization (NMF) algorithms identify endmembers and their abundances at each pixel of HSI simultaneously. Although they perform well, the factorization results are unstable, require high computational costs, and are difficult to interpret from the original HSI. CUR matrix decomposition selects specific columns and rows from a dataset to represent it as a product of three small submatrices, resulting in interpretable low-rank factorization. In this paper, we propose a new blind HU framework based on CUR factorization called CUR-HU that performs the entire HU process by exploiting the low-rank structure of given HSIs. CUR-HU incorporates several techniques to perform the HU process with a performance comparable to state-of-the-art methods but with higher computational efficiency. We adopt a deterministic sampling method to select the most informative pixels and spectrum components in HSIs. We use an incremental QR decomposition method to reduce computation complexity and estimate the number of endmembers. Various experiments on synthetic and real HSIs are conducted to evaluate the performance of CUR-HU. CUR-HU performs comparably to state-of-the-art methods for estimating the number of endmembers and abundance maps, but it outperforms other methods for estimating the endmembers and the computational efficiency. It has a 9.4 to 249.5 times speedup over different methods for different real HSIs.

Are dividends impacted by the timing of macro data releases? Case of WIG, DAX and S&P500

Purpose: This paper attempts to reveal, on the one hand, whether dividend paid by dividend-paying companies from the WIG, DAX and S&P500 indices for the period 2017-2022 were characterized by positive dynamics of change in real terms, and whether decisions on the level of dividends recommended for payout are more determined by macroeconomic factors from the end of the year, in which the company generated financial results, or from the period, in which dividend decisions were made. Design/methodology/approach: The research objective of this paper is accomplished by means of a thorough literature analysis. In the area of statistical methods, the authors refer to classical methods of correlational analysis, with a focus on non-parametric methods (Spearman’s rho and Kendall’s tau). Findings: The analyses carried out made it possible to indicate that, for companies listed on the FSE and WSE, the only quantity correlated with changes in dividends paid is the PMI index, with a negative correlation for Polish companies and a positive one for German companies. This correlation occurs only if one considers December macroeconomic data readings. On the other hand, for NYSE-listed entities, statistically significant relationships were obtained for the PMI index (negative), the inflation rate (negative) and interest rates (positive). With the exception of interest rates, the correlations apply to both December readings and those from May of the dividend payout year. At the same time, the results indicate different decisions made by the boards of US, German and Polish companies in the face of the SARS-CoV-2 pandemic. Research limitations/implications: The study was conducted on a limited number of analyzed companies and for a limited time range. Therefore, it could be biased, due to the deterministic stock sampling method and the research period. Practical implications: Expanded knowledge of the impact of the timing of publication of macroeconomic parameters on decisions on dividend payouts and their amount. This knowledge is important for both investors and investment funds’ boards. Consequently, one can make better investment decisions. Social implications: Among the paper’s social implications, the most important appears to be a possible change in the investors’ attitude towards dividend-paying companies that not only pay dividends systematically, but also have positive dynamics of change, and the realization that dividend decisions are affected not only by a number of determinants, but also by the timing of their occurrence. Ultimately, investors’ needs could be better addressed. Originality/value: The paper evaluates the impact of macroeconomic determinants on changes in dividend payouts by companies for the period 2017-2022. What is new in the paper is the analysis of whether the timing of the publication of macroeconomic parameters significantly affects the level of dividends paid, thereby filling our knowledge gap. Keywords: dividend-paying companies, determinants of dividend policy, macroeconomic indicators, inflation. Category of the paper: Research paper

Deterministic sampling method using simplex ensemble and scaling method for efficient and robust uncertainty quantification

ABSTRACT Uncertainty quantification (UQ) of the neutron multiplication factor is important to investigate the appropriate safety margin for a target system. Although the random sampling method is a practical and useful UQ method, a large computational cost is required to reduce the statistical error of the estimated uncertainty. Furthermore, if an input variable follows a normal distribution with a large standard deviation, the perturbed input variable by the random sampling method may become a physically inappropriate or negative value. To address these issues for the efficient and robust UQ, a modified deterministic sampling method using the simplex ensemble and the scaling method is proposed. The features of the proposed method are summarized as follows: The sample size is r + 2 , where r corresponds to the effective rank of the covariance matrix between the input variables; depending on a situation of target UQ, the amounts of perturbations for the input parameters can be arbitrarily given by the scaling factor method; the scaling factor can be updated to avoid physically inappropriate in the perturbed input variables. The effectiveness of the proposed method is demonstrated through the UQ of the neutron multiplication factor due to fuel manufacturing uncertainties for a typical PWR pin-cell burnup calculation.

Deterministic sampling methods coupled with dynamic coverage factors for uncertainty analyses in MELCOR simulation

This study proposed a modified uncertainty quantification (UQ) approach which couples deterministic sampling (DS) methods with dynamic coverage factors, to overcome the disadvantage of the previous UQ approach (Wang and Ma, 2023b) in obtaining numerically equivalent estimates of 95/95 tolerance limits for MELCOR simulations of postulated severe accidents in a Swedish pressurized water reactor (PWR). In the modified UQ approach two deterministic sampling (DS) methods were used to obtain the first two statistical moments, and dynamic coverage factors calculated from fitted beta distributions are employed to extend the moment information to the 95th percentile. The modified UQ approach was compared with the previous UQ approach (coupling DS methods with a fixed coverage factor) and conventional UQ approach of the first order Wilks’ method. The comparative results showed that the modified UQ approach not only enabled eliminating unrealistic estimates in the previous UQ approach, but also offered estimates that are covered by boxplots from the first order Wilks’ method with a significant reduction of computational cost. In the modified UQ approach 10–18 samples were sufficient, while at least 59 samples were required for the first order Wilks’ method. In the scope of the present study, DS-Standard method was recommended in the modified UQ approach in view of conservatism, while DS-Simplex method was preferred in view of accuracy and computational cost. Since the behavior of a DS method is dependent of time and outputs, in practice different DS methods should be integrated to the modified UQ approach for numerically equivalent estimates of 95/95 tolerance limits in MELCOR severe accident simulations.

Application of deterministic sampling methods to uncertainty quantification in MELCOR severe accident simulation

This study is concerned with uncertainty analysis of MELCOR simulation of a hypothetical severe accident initiated by station blackout (SBO) in a Swedish pressurized water reactor (PWR). 9 input parameters are chosen, and 12 safety-related output parameters are selected as the figures of merit (FOMs). In random sampling (RS) method, 800 MELCOR cases are run to produce empirical cumulative distribution functions (CDFs) and empirical 95th percentiles of FOMs. Given this sufficient sample size, uncertainty analyses through statistical analysis, can be performed to obtain the first two statistical moments and 95/95 estimates from the first order Wilks’ method. However, RS method including Wilks’ method turns out to be time consuming and computationally expensive since many MELCOR cases require iterative tuning of MELCOR input to restart and finish calculations. To overcome this issue encountered in RS method, in the present study three deterministic sampling (DS) methods are applied to uncertainty analyses, and a coupled approach by combining DS methods with a coverage factor 1.65 is proposed. Comparable results show that DS methods can generally capture the first two statistical moments quickly, and acceptably accurate 95th percentiles can be calculated by the coupled approach when the output parameters can be described as normal distributions. Besides, estimated 95th percentiles from the coupled approach are covered by estimates (boxplots) from the first order Wilks’ method. Hence, it can be concluded that the coupled approach has its potential to work as an alternative in engineering to RS method including Wilks’ method for numerically equivalent estimates of 95/95 tolerance limit with an acceptable accuracy and a significantly less computational cost in MELCOR severe accident simulation.

Nuclear data adjustment using a deterministic sampling method with unscented transformation

ABSTRACT A nuclear data adjustment method using a deterministic sampling method based on an unscented transform was developed, and its validity was confirmed through a twin experiment using a critical benchmark problem. Conventional nuclear data adjustment methods require sensitivity analysis using generalized perturbation theory or many forward calculations using stochastically sampled nuclear data. To address this issue, this study focused on unscented transform-based sampling (UTS), which is used in uncertainty quantification. Based on the UTS, perturbed nuclear data can be deterministically sampled to reproduce the population covariance matrix with minimum sample size. Therefore, UTS can significantly reduce the computational cost compared to conventional nuclear data adjustment using random sampling (RS). Furthermore, the UTS was improved to prevent the sampling of negative nuclear data while accurately reproducing the population covariance matrix. The proposed method was applied to the numerical experiment of Godiva, and the adjusted nuclear data were compared with those obtained using conventional methods. Consequently, it was demonstrated that UTS can adjust nuclear data at a lower computational cost than RS.

Constrained minimum energy designs

Space-filling designs are important in computer experiments, which are critical for building a cheap surrogate model that adequately approximates an expensive computer code. Many design construction techniques in the existing literature are only applicable for rectangular bounded space, but in real world applications, the input space can often be non-rectangular because of constraints on the input variables. One solution to generate designs in a constrained space is to first generate uniformly distributed samples in the feasible region, and then use them as the candidate set to construct the designs. Sequentially Constrained Monte Carlo (SCMC) is the state-of-the-art technique for candidate generation, but it still requires large number of constraint evaluations, which is problematic especially when the constraints are expensive to evaluate. Thus, to reduce constraint evaluations and improve efficiency, we propose the Constrained Minimum Energy Design (CoMinED) that utilizes recent advances in deterministic sampling methods. Extensive simulation results on 15 benchmark problems with dimensions ranging from 2 to 13 are provided for demonstrating the improved performance of CoMinED over the existing methods.

Quantification of uncertainties due to manufacturing tolerances using deterministic sampling methods

The objective of this paper is to assess the applicability of three Deterministic Sampling (DS) methods (namely DS-Standard, DS-Simplex and DS-Hadamard) for uncertainty quantification of manufacturing tolerances for global safety parameters, such as the multiplicative factor (keff), and local quantities, such as the fission rate. More specifically, the IPEN/MB-01 research reactor (from LWR-RESR benchmark) has been modeled in the Monte Carlo code SERPENT to which the sampling methods have been applied. Twenty different input parameters (related to geometry and materials) were considered as sources of uncertainties. Uncertainty estimates for keff by DS were all compatible among the three methodologies (with a maximum nominal difference of 5 pcm) and in agreement with values reported in the benchmark (within the statistical Monte Carlo uncertainty of 15 pcm). Fission rate results obtained by DS showed full compatibility with results obtained by conventional stochastic sampling methods (such as Random Sampling or Latin Hypercube Sampling) and with those reported in the benchmark. The DS-Simplex showed the highest uncertainty in predicting the uncertainty on fission rate when compared to Random Sampling (with a maximum relative difference of 10% in the uncertainty estimate), whereas other DS methods showed a maximum deviation of 5%. The DS methods offer a remarkable reduction in the number of samples (from thousand to tens or hundreds according to the selected approach) required for an accurate uncertainty quantification for both global and local quantities.

Урбанизация национальных республик Саяно-Алтая сквозь призму теории модернизации

Introduction. In the national republics of the Sayan-Altai Region — Altai, Tuva and Khakassia — urbanization has been essentially delayed and is, hence, accelerated. The first urban settlements appeared here only in the 20th century, and many of them were referred to as urban de jure only. Despite the growth of publications studying various aspects of the urbanization in Siberia, there are almost no research works dealing with the development of this historical, cultural, complicated social process and compiled from materials of this region proper. The paper provides a first attempt to assess the features and patterns of urbanization in the Sayan-Altai Region through the prism of modernization theory. Goals. The article aims to analyze urbanization as the most important aspect of modernization through materials collected across Altai, Tuva and Khakassia in the 20th – early 21st centuries. Materials and Methods. The article primarily investigates official statistical data that make it possible to trace the actual urbanization dynamics (for both populations and territories), as well as the process of transformation experienced by material components of city life. Empirically, the study rests on materials of surveys conducted in the Republic of Khakassia (2018) and in the Republics of Tuva and Altai (2019). The total number of interviewees is 2000. The study employs deterministic sampling methods. Results. The work concludes the urbanization development pattern examined in the region is non-linear, which was determined by the difficult and indirect modernization processes nationwide and those in its certain parts in particular. In the history of the ethnic federal subjects of the Sayan-Altai Region, local features of urbanization are more evident than those in other regions of the country, which is largely due to the influence of the ethnic factor, specifics of indigenous cultures and worldviews. Urbanization as a process is studied on the regional materials in two main perspectives: on the one hand, the paper provides analysis of quantitative indicators of urbanization, such as the dynamics of urban population and growth of urban settlements; on the other hand, an important role in the study of urbanization is assigned to its qualitative indicators, the latter including the transformed structure of urban population and evolution of its living standards.

Research Progresses of Uncertainty Quantification Methods for High Fidelity Numerical Nuclear Reactor

Numerical reactor based on high fidelity model and method is with the characteristics of high precision and high resolution, but the inherent uncertainty of nuclear data and other parameters will seriously affect the uncertainty of numerical reactor analysis results. Based on the review of the research progresses of numerical reactors and its uncertainty quantification in the world, this paper focuses on the research progresses in NECP Laboratory of Xi’an Jiaotong University in recent years, including the development of one-step high fidelity numerical reactor program NECP-X, the generation of nuclear data covariance database, the uncertainty propagation method based on deterministic method and sampling method, and the uncertainty quantification in transient calculation. An advanced sampling method COST is proposed. Based on the high fidelity numerical reactor program, the uncertainty propagation of covariance of various nuclear parameters in the steady-state and transient analysis of the reactor core is quantified for the first time, which is of great significance for engineering application of numerical reactors.

Novel Deterministic Angular Sampling Methods for 3D Channel Models

For the purpose of emulating realistic ray-based channels more exactly, two novel sum-of-cisoids (SOC) channel simulator parameters computation strategies, i.e., the bidirectional allocation (BA) and the simplified forward allocation (FA) methods, are proposed in this letter. Compared with the classical equal power method, the asymmetric computing property of these two methods further improves the spatial correlation emulation accuracy and significantly mitigates the intra-cluster correlation of sub-paths. As shown in the simulation results, both BA and FA methods enable the statistical spatial-temporal characteristics of emulated channel match extremely well with that of the target channel. More important, the proposed processes are deterministic, and will not introduce randomness in different realizations. Hence the proposed methods are always ergodic with arbitrary angle spreads.

Application of deterministic sampling methods for uncertainty quantification in manufacturing tolerances in neutron physics

In this work, an application of the deterministic sampling (DS) methodology for uncertainty quantification of manufacturing tolerances is assessed. The DS methodology is applied to eight benchmark cases, from two criticality safety benchmark series, extracted from the International Handbook of Evaluated Criticality Safety Benchmark Experiments (i.e., LCT-007 & LCT-039). The DS results are compared to two other sampling techniques: Random Sampling and Latin Hypercube Sampling. Ten input parameters such as isotopic composition of the fuel, geometry properties of the assembly, pitch array, etc. were considered. Only the multiplicative factor (keff) was considered as output quantity. Evaluation of the keff was obtained using the Monte Carlo code MCNP for the LCT-007 case, and the Serpent code for LCT-039.Results show very good agreement among all methodologies for the calculated keff values and their corresponding uncertainties. The largest uncertainty difference with DS, compared to the reference values, is 46 pcm for LCT-007 and 9 pcm for LCT-039. The use of DS remarkably reduces the number of samples needed to perform an accurate uncertainty quantification (from thousands of samples down to a dozen).This study provides insights in the application of DS methodology in the neutron physics domain and showcases its benefits and limitations.

Robust CUR Decomposition: Theory and Imaging Applications

This paper considers the use of Robust PCA in a CUR decomposition framework and applications thereof. Our main algorithms produce a robust version of column-row factorizations of matrices $\mathbf{D}=\mathbf{L}+\mathbf{S}$ where $\mathbf{L}$ is low-rank and $\mathbf{S}$ contains sparse outliers. These methods yield interpretable factorizations at low computational cost, and provide new CUR decompositions that are robust to sparse outliers, in contrast to previous methods. We consider two key imaging applications of Robust PCA: video foreground-background separation and face modeling. This paper examines the qualitative behavior of our Robust CUR decompositions on the benchmark videos and face datasets, and find that our method works as well as standard Robust PCA while being significantly faster. Additionally, we consider hybrid randomized and deterministic sampling methods which produce a compact CUR decomposition of a given matrix, and apply this to video sequences to produce canonical frames thereof.

Uncertainty Analysis on k-ε Turbulence Model in the Prediction of Subcooled Boiling in Vertical Pipes

Computational fluid dynamics (CFD) has become an effective method for researching two-phase flow in reactor systems. However, the uncertainty analysis of Computational fluid dynamics simulation is still immature. The effects of uncertainties from two-phase models and boundary conditions have been analyzed in our previous work. In this work, the uncertainties from a turbulence model on the prediction of subcooled boiling flow were analyzed with the DEBORA benchmark experiments by a deterministic sampling method. Seven parameters in the standard k-ε model, which interrelated momentum, energy, turbulent kinetic energy, and dissipation rate, were studied as uncertainty sources, including Cμ, Cμ,g, C1ε, C2ε, σk, σε, and Prt. Radial parameters were calculated to study the effects of uncertainties from the turbulence model. The contributions of each uncertainty source on void fraction and liquid temperature were also analyzed. It was found that the models can simulate subcooled boiling flow accurately and uncertainty analysis by deterministic sampling can give a reference interval to increase the reliability of results. The C2ε and C1ε, parameters in the production term and dissipation term of transport equations, dominate the radial distributions of void fraction and liquid temperature.

Deterministic Completion of Rectangular Matrices Using Asymmetric Ramanujan Graphs: Exact and Stable Recovery

In this paper we study the matrix completion problem: Suppose $X \in {\mathbb {R}}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$ , is it possible to recover $X$ exactly with noise-free measurements, or approximately with noisy measurements? Existing solutions to these problems involve sampling the elements uniformly and at random. In this paper, we present a deterministic sampling method for matrix completion, based on choosing the sampling set as the edge set of an asymmetric Ramanujan bigraph, and derive sufficient conditions under which the unknown matrix is completed exactly with noise-free measurements, and is approximately completed with noisy measurements. To study how close they are to being necessary, we conducted numerical simulations on randomly generated low rank matrices, using the LPS families of Ramanujan graphs. These simulations demonstrate two facts: (i) In order to achieve exact completion, it appears sufficient to choose the degree $d$ of the Ramanujan graph to be $\geq 3r$ . (ii) There is a “phase transition,” whereby the likelihood of success suddenly drops from 100% to 0% if the rank is increased by just one or two beyond a critical value. The phase transition phenomenon is well-known and well-studied in vector recovery using $\ell _1$ -norm minimization. However, it is less studied in matrix completion and nuclear norm minimization, and not much understood.

Trajectory prediction of cylinders freely dropped into water in two dimensions based on state space model

Experiments show that the behavior of dropped cylindrical objects may be stochastic if small disturbances exist at the initial moment. An envelope of various trajectories can be obtained from experimental tests. In addition, numerical results also indicate that small changes of initial states, e.g., drop angle, rolling frequency, and the selection of drag coefficient, affect the trajectory greatly. If the initial state follows the Gaussian distribution,it can be considered as a nonlinear prediction problem. Then the Unscented Transformation (UT) method, as a deterministic sampling method, can be used to predict the trajectory. In this paper, we firstly present a systematic formulation of the cylinder's three-degree-of-freedom (3DOF) of motions as a state space model. Secondly, the UT method and the interval estimation method are employed to solve this prediction problem. Finally, the predicted trajectories will be compared with the experimental data. Meanwhile, two performance parameters are first proposed to evaluate the accuracy of predicted trajectory. Both parameters can be directly applied to the salvage and recovery of dropped objects in the offshore operations. In addition, the numerical results also show that the UT method is an efficient way to predict the trajectory of cylinders freely dropped into water. • The state space model of a dropped cylinder has been successfully constructed in this paper. • The trajectory of a dropped cylinder is considered as the stochastic process due to the uncertainty at the initial moment. • The Unscented Transformation (UT) method is employed to solve the nonlinear trajectory prediction. • Two performance measures R A and R L have been first proposed to evaluate the accuracy of numerical trajectory prediction, in order to meet the requirement from the offshore engineering industry.

Novel training algorithms for long short‐term memory neural network

More recently, due to the enormous potential of long short-term memory (LSTM) neural network in various fields, some efficient training algorithms have been developed, including the extended Kalman filter (EKF)-based training algorithm and particle filter (PF)-based training algorithm. However, it should be noted that if the system is highly non-linear, the linearisation employed in the EKF may cause instability. Moreover, the PF usually suffers from the particle degeneracy. Therefore, the PF-based training algorithm may only find a poor local optimum. To solve these problems, an unscented Kalman filter (UKF)-based training algorithm is proposed. The UKF employs a deterministic sampling method; hence, there is no linearisation in it and it does not have the degeneracy problem. Moreover, the computational complexity of the UKF is the same order as that of the EKF. To further reduce the computational complexity, the authors propose a minimum norm UKF (MN-UKF) to obtain a good trade-off between performance and complexity. To the best of the authors’ knowledge, this is the first reported solution to this problem. Simulations using both benchmark synthetic signal and real-world signal illustrate the potential of the algorithms developed.

Deterministic Sampling of Expensive Posteriors Using Minimum Energy Designs

Markov chain Monte Carlo (MCMC) methods require a large number of samples to approximate a posterior distribution, which can be costly when the likelihood or prior is expensive to evaluate. The number of samples can be reduced if we can avoid repeated samples and those that are close to each other. This is the idea behind deterministic sampling methods such as quasi-Monte Carlo (QMC). However, the existing QMC methods aim at sampling from a uniform hypercube, which can miss the high probability regions of the posterior distribution and thus the approximation can be poor. Minimum energy design (MinED) is a recently proposed deterministic sampling method, which makes use of the posterior evaluations to obtain a weighted space-filling design in the region of interest. However, the existing implementation of MinED is inefficient because it requires several global optimizations and thus numerous evaluations of the posterior. In this article, we develop an efficient algorithm that can generate MinED samples with few posterior evaluations. We also make several improvements to the MinED criterion to make it perform better in high dimensions. The advantages of MinED over MCMC and QMC are illustrated using an example of calibrating a friction drilling process.

Uncertainty analysis of condensation heat transfer benchmark using CFD code GASFLOW-MPI

Abstract With the development of the computational capability, Computational Fluid Dynamics (CFD) is more and more widely used as the best estimate approach for the design and safety issues related to reactor thermal hydraulics. The conjugate condensation heat transfer is a widespread phenomenon in nuclear safety analysis, such as a key heat transfer process for Passive Containment Cooling Systems (PCCS) in advanced passive reactors. However, the uncertainty analysis of CFD predictions applied to reactor thermal hydraulics is still immature. In this work, the uncertainty quantification of the condensation heat transfer in COPAIN experiment was performed using a well-validated parallel CFD code GASFLOW-MPI. Six sources of uncertainty were considered here, including the inlet velocity/temperature/turbulent intensity, outlet pressure and condensation heat transfer coefficient, as well as the computational mesh size. For the first five uncertainty sources, the deterministic sampling method was employed in this work to reduce the required number of sampling points. Unlike the ensemble in the random sampling Monte Carlo method, the deterministic sampling method represents the probability density function with an ensemble that has the same statistical moments but contains much fewer samples. Two ensembles which were 2nd and 4th order statistical moments accuracy, respectively, were employed in this work and their performances were compared. The Richardson extrapolation method was used to quantify the uncertainty propagated from the computational mesh size. The contributions of each uncertainty source to the condensation heat flux and temperature distribution were also performed. The results of the uncertainty quantification are consistent well with the experimental data in COPAIN facility, suggesting that the deterministic sampling method and Richardson extrapolation method are powerful tools for uncertainty quantification of CFD calculations.

Hybrid Trajectory Planning for Autonomous Driving in Highly Constrained Environments

In this paper, we introduce a novel and efficient hybrid trajectory planning method for autonomous driving in highly constrained environments. The contributions of this paper are fourfold. First, we present a trajectory planning framework that is able to handle geometry constraints, nonholonomic constraints, and dynamics constraints of cars in a humanlike and layered fashion and generate curvature-continuous, kinodynamically feasible, smooth, and collision-free trajectories in real time. Second, we present a derivative-free global path modification algorithm to extract high-order state information in free space for state sampling. Third, we extend the regular state-space sampling method widely used in on-road autonomous driving systems to a multi-phase deterministic state-space sampling method that is able to approximate complex maneuvers. Fourth, we improve collision checking accuracy and efficiency by using a different car footprint approximation strategy and a two-phase collision checking routine. A range of challenging simulation experiments show that the proposed method returns high-quality trajectories in real time and outperforms existing planners, such as hybrid A* and conjugate-gradient descent path smoother in terms of path quality, efficiency, and computation resources used.