quasi-Monte Carlo Approach Research Articles

Quantification of Polymorphic Uncertainties: A Quasi-Monte Carlo Approach

Quantification of Polymorphic Uncertainties: A Quasi-Monte Carlo Approach

Probabilistic evaluation of dynamic positioning operability with a Quasi-Monte Carlo approach

During the design phase of an offshore unit, estimating the station-keeping capabilities of the dynamic positioning (DP) system is mandatory. This means, in conventional offshore applications, to determine the maximum sustainable wind speed as a function of the encounter heading, which the unit may counteract by employing the onboard actuators or mooring lines only. Besides the deterministic estimation of DP capability, it is possible to assess the operability of the DP system following a non-deterministic probabilistic process by employing the site-specific joint wind-wave distributions to model the environment. In such a case, the operability results from a Monte Carlo integration process. Here it is proposed to enhance the applicability of the probabilistic analysis of DP operability, investigating the application of a Quasi-Monte Carlo method. In this sense, the procedure uses quasi-random samplings following a Sobol sequence instead of employing random samples of the joint distributions. In this paper, the Quasi-Monte Carlo process is tested and compared on a reference ship, highlighting the improvements to the established probabilistic DP prediction concerning the number of calculations needed to estimate operability. The significant reduction of computational time makes the newly implemented method suitable for the early design stage applications.

Pricing of Arithmetic Average Asian Option by Combining Variance Reduction and Quasi-Monte Carlo Method

Financial derivatives have developed rapidly over the past few decades due to their risk-averse nature, with options being the preferred financial derivatives due to their flexible contractual mechanisms, particularly Asian options. The Black–Scholes stock option pricing model is often used in conjunction with Monte Carlo simulations for option pricing. However, the Black–Scholes model assumes that the volatility of asset returns is constant, which does not square with practical financial markets. Additionally, Monte Carlo simulation suffers from slow error convergence. To address these issues, we first correct the asset volatility in the Black–Scholes model using a GARCH model. Then, the low error convergence rate of the Monte Carlo method is improved using variance reduction techniques. Meanwhile, the quasi-Monte Carlo approach based on low discrepancy sequences is used to refine the error convergence rate. We also provide a simulation experiment and result analysis to validate the effectiveness of our proposed method.

Quasi Monte Carlo time-frequency analysis

We study signal processing tasks in which the signal is mapped via some generalized time-frequency transform to a higher dimensional time-frequency space, processed there, and synthesized to an output signal. We show how to approximate such methods using a quasi-Monte Carlo (QMC) approach. We consider cases where the time-frequency representation is redundant, having feature axes in addition to the time and frequency axes. The proposed QMC method allows sampling both efficiently and evenly such redundant time-frequency representations. Indeed, 1) the number of samples required for a certain accuracy is log-linear in the resolution of the signal space, and depends only weakly on the dimension of the redundant time-frequency space, and 2) the quasi-random samples have low discrepancy, so they are spread evenly in the redundant time-frequency space. One example of such redundant representation is the localizing time-frequency transform (LTFT), where the time-frequency plane is enhanced by a third axis. This higher dimensional time-frequency space improves the quality of some time-frequency signal processing tasks, like the phase vocoder (an audio signal processing effect). Since the computational complexity of the QMC is log-linear in the resolution of the signal space, this higher dimensional time-frequency space does not degrade the computation complexity of the proposed QMC method. The proposed QMC method is more efficient than standard Monte Carlo methods, since the deterministic QMC sample points are optimally spread in the time-frequency space, while random samples are not.

Reliability analysis of structures with multimodal distributions based on direct probability integral method

For practical structures, some input random variables follow multimodal distributions, and conventional reliability analysis methods may result in large computational errors. In this paper, a novel direct probability integral method (DPIM), which decouples governing equation of structure and the probability density integral equation (PDIE), is proposed to address the static and dynamic reliability assessment of structures involving random variables with multimodal distributions. Firstly, the multimodal probability density functions (PDFs) of input random variables are established by the Gaussian mixture model. Then, using numerical integration and smoothing technique of Dirac delta function, the PDFs of structural responses with multimodal random variables are achieved by solving the PDIE, in which three numerical integration algorithms, namely, the Quasi-Monte Carlo approach, the sparse grid approach, and Generalized F-discrepancy-based point selection approach are employed. Further, the reliability of static structure can be readily obtained by integrating the PDF of response function, while the dynamic reliability can be evaluated by DPIM combining with extreme value distribution of stochastic process. Finally, several examples demonstrate the superiority of DPIM, and the Generalized F-discrepancy-based point selection approach has the highest accuracy and efficiency for solving PDIE. The characteristics of uncertainty propagation in structures involving multimodal distributions of input random variables are revealed.

Improving Approximate Bayesian Computation via Quasi-Monte Carlo

ABSTRACTABC (approximate Bayesian computation) is a general approach for dealing with models with an intractable likelihood. In this work, we derive ABC algorithms based on QMC (quasi-Monte Carlo) sequences. We show that the resulting ABC estimates have a lower variance than their Monte Carlo counter-parts. We also develop QMC variants of sequential ABC algorithms, which progressively adapt the proposal distribution and the acceptance threshold. We illustrate our QMC approach through several examples taken from the ABC literature.

Investigation of Global Sensitivity Analysis on Operating Revenue and CO 2 Emissions from Crude Oil Distillation using Two-Stage Uncertainty Quantification Method

This paper presents a global sensitivity analysis based strategy for uncertainty quantification of the operating revenue and the carbon dioxide emissions in a crude oil distillation column. Aspen HYSYS is used for rigorous simulation of the column. A two-stage approach, which is executed by an interface between MATLAB and HYSYS, is implemented in this study. A multiplicative dimensional reduction method is applied to first identify which factors have the most influential to the model outputs, e.g. the operating revenue and the CO2 emissions. In the second stage, the uncertainty quantification regressed by Gaussian prediction regression method is exploited. As a result, there is a good agreement between the predicted results by the Gaussian prediction regression method with that of the conventional Quasi-Monte Carlo approach, which shows that the computational efforts was reduced significantly compared to the conventional study (about less than 10 times). Furthermore, the Kernel points estimated by the Gaussian prediction regression method are generated to highlight influential factors identified in the first stage.

Reliability Analysis of Nonlinear Vibratory Systems Under Non-Gaussian Loads

A general methodology is presented for time-dependent reliability and random vibrations of nonlinear vibratory systems with random parameters excited by non-Gaussian loads. The approach is based on polynomial chaos expansion (PCE), Karhunen–Loeve (KL) expansion, and quasi Monte Carlo (QMC). The latter is used to estimate multidimensional integrals efficiently. The input random processes are first characterized using their first four moments (mean, standard deviation, skewness, and kurtosis coefficients) and a correlation structure in order to generate sample realizations (trajectories). Characterization means the development of a stochastic metamodel. The input random variables and processes are expressed in terms of independent standard normal variables in N dimensions. The N-dimensional input space is space filled with M points. The system differential equations of motion (EOM) are time integrated for each of the M points, and QMC estimates the four moments and correlation structure of the output efficiently. The proposed PCE–KL–QMC approach is then used to characterize the output process. Finally, classical MC simulation estimates the time-dependent probability of failure using the developed stochastic metamodel of the output process. The proposed methodology is demonstrated with a Duffing oscillator example under non-Gaussian load.

Quasi-Monte Carlo simulation of Brownian sheet with application to option pricing

ABSTRACTMonte Carlo and quasi-Monte Carlo methods are widely used in scientific studies. As quasi-Monte Carlo simulations have advantage over ordinary Monte Carlo methods, this paper proposes a new quasi-Monte Carlo method to simulate Brownian sheet via its Karhunen–Loéve expansion. The proposed new approach allocates quasi-random sequences for the simulation of random components of the Karhunen–Loéve expansion by maximum reducing its variability. We apply the quasi-Monte Carlo approach to an option pricing problem for a class of interest rate models whose instantaneous forward rate driven by a different stochastic shock through Brownian sheet and we demonstrate the application with an empirical problem.

An analog circuit synthesis tool based on efficient and reliable yield estimation

Analog circuit design has become a very challenging and time consuming process for circuit designers due to increased non-idealities and worsening variability phenomena. In order to facilitate the design process, several analog circuit sizing tools have been proposed in the literature. These tools have then led to yield-aware ones, where a certain yield is targeted. However, the type of variability analysis to be employed is still a topic of discussion due to the challenging trade-off between accuracy and efficiency of the yield analysis. Quasi-Monte Carlo (QMC) approach is one of the efficient techniques that provides efficient variability analysis via deterministic, and more importantly homogeneous sampling. The major bottleneck of the conventional QMC is that there is no practical way to calculate the estimation error. Scrambled-QMC has been utilized to obtain the error bounds of the estimation, thanks to multiple runs of randomized sample sets. However, the requirement of multiple runs substantially increases the synthesis time. To overcome this problem, this paper proposes a novel yield-aware analog circuit sizing tool, where an adaptive sample sizing algorithm for scrambled-QMC is employed in the yield estimation part.

Visual Analysis of Multi-Run Spatio-Temporal Simulations Using Isocontour Similarity for Projected Views.

Multi-run simulations are widely used to investigate how simulated processes evolve depending on varying initial conditions. Frequently, such simulations model the change of spatial phenomena over time. Isocontours have proven to be effective for the visual representation and analysis of 2D and 3D spatial scalar fields. We propose a novel visualization approach for multi-run simulation data based on isocontours. By introducing a distance function for isocontours, we generate a distance matrix used for a multidimensional scaling projection. Multiple simulation runs are represented by polylines in the projected view displaying change over time. We propose a fast calculation of isocontour differences based on a quasi-Monte Carlo approach. For interactive visual analysis, we support filtering and selection mechanisms on the multi-run plot and on linked views to physical space visualizations. Our approach can be effectively used for the visual representation of ensembles, for pattern and outlier detection, for the investigation of the influence of simulation parameters, and for a detailed analysis of the features detected. The proposed method is applicable to data of any spatial dimensionality and any spatial representation (gridded or unstructured). We validate our approach by performing a user study on synthetic data and applying it to different types of multi-run spatio-temporal simulation data.

Quasi-Monte Carlo Approach to Asian Options Pricing

Asian options is a contract which gives right to the holder to buy/sell the underlying asset for its average price over prescribed period and it has a lower unpredictability, hence exposing cheaper relative to their European counterparts. Asian options are commonly traded on currencies and commodity products which have low trade volumes. Therefore, the pricing of such options become one of the most interesting fields. Much research work has been done on the European and American options using various techniques like Black-Scholes, binomial tree, finite difference, Monte Carlo and Quasi-Monte Carlo model. But none of the studies compares and identified the best model for options pricing, particularly for Asian options pricing. This article aims to evaluate the effectiveness of existing models on Asian options pricing and to suggest a suitable model for forecasting Asian options prices. Findings of the study indicate that the Quasi-Monte Carlo technique is more superior to any other techniques due to its results with high precision and low standard deviation.

A Modified Least-Squares Simulation Approach to Value American Barrier Options

This paper presents the total least squares quasi-Monte Carlo approach (TLSQM) for valuing American barrier options, which modifies the least-squares Monte Carlo method (LSM). The total least squares are applied to estimate the conditional expected payoff to the option holder from continuation, and the quasi-Monte Carlo method is used to simulate the paths of stock prices. We show that the approximations of a set of conditional expectation functions in the TLSQM method converge to the true expectation function under general assumptions. It provides the mathematical foundation for the use of the TLSQM method in the derivatives research. In order to illustrate the high efficiency of the proposed approach, several examples are given to price American barrier options using TLSQM, LSM, and binomial trees, respectively. The numerical analysis demonstrates that the TLSQM method is accurate, and it makes improvements in computational speed and efficiency, compared with the LSM method.

The accuracy of prostate volume measurement from ultrasound images: A quasi-Monte Carlo simulation study using magnetic resonance imaging

Prostate volume is an important parameter to guide management of patients with benign prostatic hyperplasia (BPH) and to deliver clinical trial endpoints. Generally, simple 2D ultrasound (US) approaches are favoured despite the potential for greater accuracy afforded by magnetic resonance imaging (MRI) or complex US procedures. In this study, different approaches to estimate prostate size are evaluated with a simulation to select multiple organ cross-sections and diameters from 22 MRI-defined prostate shapes. A quasi-Monte Carlo (qMC) approach is used to simulate multiple probe positions and angles within prescribed limits resulting in a range of dimensions. The basic ellipsoid calculation which uses two scanning planes compares well to the MRI volume across the range of prostate shapes and sizes (R=0.992). However, using an appropriate linear regression model, accurate volume estimates can be made using prostate diameters calculated from a single scanning plane.

A New Algorithm for Stochastic Variational Inequality with an Application

Recently stochastic variational inequality has been extensively studied. However there are few methods can be effectively realized. This study considers to solve stochastic variational inequality by combining quasi-Monte Carlo approach and interior point method. The global convergence is established for the new algorithm. An application for the synergies analysis of the supply chain after M&A from the literature is discussed.

Bootstrap variance estimation for complex survey data: a Quasi Monte Carlo approach

This paper presents a quasi Monte Carlo approach in the generation of a replication design for the bootstrap variance estimation of stratified sampling designs. The proposed method makes use of the shuffled Halton sequence, has a faster rate of convergence among the existing bootstrap variance estimators, and performs comparably and well with already established methods such as the rescaled Rao-Wu bootstrap method and the Taylor Linearization method. The method can easily be implemented and extended to other sampling designs. The performance of the proposed method is evaluated through limited simulation studies with application on artificial populations. A stratified simple random element sampling design was used in the simulation studies.

Evaluating the impact of caprock and reservoir properties on potential risk of CO2 leakage after injection

Numerical models are essential tools in fully understanding the fate of injected CO2 for commercial-scale sequestration projects and should be included in the life cycle of a project. Common practice involves modeling the behavior of CO2 during and after injection using site-specific reservoir and caprock properties. Little has been done to systematically evaluate and compare the effects of a broad but realistic range of reservoir and caprock properties on potential CO2 leakage through caprocks. This effort requires sampling the physically measurable range of caprock and reservoir properties, and performing numerical simulations of CO2 migration and leakage. In this study, factors affecting CO2 leakage through intact caprocks are identified. Their physical ranges are determined from the literature from various field sites. A quasi-Monte Carlo sampling approach is used such that the full range of caprock and reservoir properties can be evaluated without bias and redundant simulations. For each set of sampled properties, the migration of injected CO2 is simulated for up to 200 years using the water–salt–CO2 operational mode of the STOMP simulator. Preliminary results show that critical factors determining CO2 leakage rate through caprocks are, in decreasing order of significance, the caprock thickness, caprock permeability, reservoir permeability, caprock porosity, and reservoir porosity. This study provides a function for prediction of potential CO2 leakage risk due to permeation of intact caprock and identifies a range of acceptable seal thicknesses and permeability for sequestration projects. The study includes an evaluation of the dependence of CO2 injectivity on reservoir properties.

Global sensitivity analysis of transmission line fault-locating algorithms using sparse grid regression

Computation of distance to fault on an electrical transmission line is affected by many sources of uncertainty, including parameter setting errors, measurement errors, as well as absence of information and incomplete modelling of a system under fault condition. In this paper we propose an application of the variance-based global sensitivity measures for evaluation of fault location algorithms. The main goal of the evaluation is to identify factors and their interactions that contribute to the fault locator output variability. This analysis is based on the results of Sparse Grid Regression. The method compiles the Functional ANOVA model to represent fault locator output as a function of uncertain factors. The ANOVA model provides a tool for interpretation and sensitivity analysis. In practice, such analysis can help in functional performance tests, especially in: selection of the optimal fault location algorithm (device) for a specific application, calibration process and building confidence in a fault location function result. The paper concludes with an application example which demonstrates use of the proposed methodology in testing and comparing some commonly used fault location algorithms. This example is also used to demonstrate numerical efficiency for this type of application of the proposed Sparse Grid Regression method in comparison to the Quasi-Monte Carlo approach.

Robustness analysis and behavior discrimination in enzymatic reaction networks.

Characterizing the behavior and robustness of enzymatic networks with numerous variables and unknown parameter values is a major challenge in biology, especially when some enzymes have counter-intuitive properties or switch-like behavior between activation and inhibition. In this paper, we propose new methodological and tool-supported contributions, based on the intuitive formalism of temporal logic, to express in a rigorous manner arbitrarily complex dynamical properties. Our multi-step analysis allows efficient sampling of the parameter space in order to define feasible regions in which the model exhibits imposed or experimentally observed behaviors. In a first step, an algorithmic methodology involving sensitivity analysis is conducted to determine bifurcation thresholds for a limited number of model parameters or initial conditions. In a second step, this boundary detection is supplemented by a global robustness analysis, based on quasi-Monte Carlo approach that takes into account all model parameters. We apply this method to a well-documented enzymatic reaction network describing collagen proteolysis by matrix metalloproteinase MMP2 and membrane type 1 metalloproteinase (MT1-MMP) in the presence of tissue inhibitor of metalloproteinase TIMP2. For this model, our method provides an extended analysis and quantification of network robustness toward paradoxical TIMP2 switching activity between activation or inhibition of MMP2 production. Further implication of our approach is illustrated by demonstrating and analyzing the possible existence of oscillatory behaviors when considering an extended open configuration of the enzymatic network. Notably, we construct bifurcation diagrams that specify key parameters values controlling the co-existence of stable steady and non-steady oscillatory proteolytic dynamics.

A Markov Chain Quasi-Monte Carlo Method for Bayesian Estimation of Stochastic Volatility Model

In this paper, we propose a Markov Chain Quasi-Monte Carlo (MCQMC) approach for Bayesian estimation of a discrete-time version of the stochastic volatility (SV) model. The Bayesian approach represents a feasible way to estimate SV models. Under the conventional Bayesian estimation method for SV models, pseudo-random numbers are usually used. Here we develop an algorithm to construct the Markov chain using a quasi-Monte Carlo sequence, or a low discrepancy sequence. We conjecture that the quasi-Monte Carlo sequence gives more precise estimates of the parameters in the SV model. We demonstrate the proposed method and justify our conjecture using both simulated and real nancial returns data.