Multi-Index Monte Carlo: When Sparsity Meets Sampling

We propose and analyze a novel Multi-Index Monte Carlo (MIMC) method for weak approximation of stochastic models that are described in terms of differential equations either driven by random measures or with random coefficients. The MIMC method is both a stochastic version of the combination technique introduced by Zenger, Griebel and collaborators and an extension of the Multilevel Monte Carlo (MLMC) method first described by Heinrich and Giles. Inspired by Giles's seminal work, we use in MIMC high-order mixed differences instead of using first-order differences as in MLMC to reduce the variance of the hierarchical differences dramatically. This in turn yields new and improved complexity results, which are natural generalizations of Giles's MLMC analysis and which increase the domain of the problem parameters for which we achieve the optimal convergence, $\mathcal{O}(\text{TOL}^{-2}).$ Moreover, in MIMC, the rate of increase of required memory with respect to $\text{TOL}$ is independent of the number of directions up to a logarithmic term which allows far more accurate solutions to be calculated for higher dimensions than what is possible when using MLMC. We motivate the setting of MIMC by first focusing on a simple full tensor index set. We then propose a systematic construction of optimal sets of indices for MIMC based on properly defined profits that in turn depend on the average cost per sample and the corresponding weak error and variance. Under standard assumptions on the convergence rates of the weak error, variance and work per sample, the optimal index set turns out to be the total degree (TD) type. In some cases, using optimal index sets, MIMC achieves a better rate for the computational complexity than the corresponding rate when using full tensor index sets..

Three dimensional visualization of water pipelines

The water pipeline industry is one of the key industries that serve the basic public needs of the society. Just like other essential utilities such as gas and electricity, underground cables are required to transport this vital human need from one part of the city to another. From Dubai to Paris, virtually all cities around the world have underground pipes serving the same purpose and the city of Kuantan isn’t any different. Due to the importance of these water pipes, it is necessary to device reliable means of protecting them from various forms of damages.For a system that affects the human existence so much, a pipe failure would be disastrous, as would an extended repair outage. Experience has shown that many pipelines get damaged when construction workers accidentally strike pipes buried underground, during excavation or other tasks that require digging the earth’s surface. This occurs due to the fact that most existing pipes are represented in two dimensional (2D) format and the information contained in 2D maps isn’t entirely accurate. This limitation in 2D visualization makes it difficult to clearly understand or conceptualize the pipelines below the ground, hence, the need for a more effective way of visualizing the underground water pipes. Three dimensional (3D) visualization is increasingly being used to overcome these limitations, but not many individuals or organisations can afford the huge cost of most of the available packages and there is a growing demand for more affordable 3D visualization tools. This research has devised a means of visualizing underground water pipes using a series of software packages, including the ArcGIS 3D Analyst extension. The integration of datasets with these packages is vital to the actualization of the research objectives. This research will illustrate how such integration is possible using the ESRI geodatabase for data storage, and exporting the dataset into other environments for visualization purposes

In vitro antimicrobial activity of total sesquiterpene lactones and phenols isolated from some Iraqi plants

The antimicrobial potency of the crude ethanolic extracts from different Iraqi plants were evaluated . Further more, total sesquiterpene lactones and phenolic compounds were isolated and their antimicrobial activity attempted. The results indicated that crude extracts have no activity except that of Callistemon lanceolatus. Also, the sesquiterpene lactones and phenolic compounds isolated from Callistemon lanceolatus were the most significant antimicrobial active constituents of the studied plants

Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity

We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDEs) with random data, where the random coefficient is parametrized by means of a countable sequence of terms in a suitable expansion. MISC is a combination technique based on mixed differences of spatial approximations and quadratures over the space of random data and, naturally, the error analysis uses the joint regularity of the solution with respect to both the variables in the physical domain and parametric variables. In MISC, the number of problem solutions performed at each discretization level is not determined by balancing the spatial and stochastic components of the error, but rather by suitably extending the knapsack-problem approach employed in the construction of the quasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy optimization procedure to select the most effective mixed differences to include in the MISC estimator. We apply our theoretical estimates to a linear elliptic PDEs in which the log-diffusion coefficient is modeled as a random field, with a covariance similar to a Mat\'ern model, whose realizations have spatial regularity determined by a scalar parameter. We conduct a complexity analysis based on a summability argument showing algebraic rates of convergence with respect to the overall computational work. The rate of convergence depends on the smoothness parameter, the physical dimensionality and the efficiency of the linear solver. Numerical experiments show the effectiveness of MISC in this infinite-dimensional setting compared with the Multi-index Monte Carlo method and compare the convergence rate against the rates predicted in our theoretical analysis

Study of the Effect of Laser on the Structural and Electrical Properties of the Compound Bi2-(x+y)CdyAgx Sr2Ca2Cu3O10+δ Superconductor at High Temperatures

في هذه الدراسة تم تحضير عينات المركب (Bi2-(x+y)CdyAgx Sr2Ca2Cu3O10+δ) وعند تراكيز مختلفة ل x حيث (x=0,0.1,0.2,0.3,0.4) وباستخدام طريقة تفاعل الحالة الصلبة وتحت ضغط هيدروستاتيكي (8ton/cm2) وعند درجة حرارة تلدين (850 C) , ومعرفة تأثير الليزر على الخواص التركيبية والكهربائية للمركب الفائق التوصيل الكهربائي. لوحظ عند فحص تحت حيود الاشعة السينية (XRD) تبين ان افضل نسبة تعويض لx هي 0.3 حيث ان قيمة a=b=5.3799(A), c=36.22(A) حيث تبين ان التركيب من النوع الرباعي القائم.This study included the preparation of composite samples (Bi2-(x+y)CdyAgx Sr2Ca2Cu3O10+δ  )by solid state reaction method and under hydro static pressure (8ton/cm2) interaction and annealing temperature   (850 C) also determine the effect of the laser on the structural and electrical properties in the compound in various concentrations of x where x=(0,0.1,0.2,0.3,0.4) observed by examining the XRD , the best ration of cooperation for (x) is 0.3 as the value of a=b=5.3799(A) , c=36.22(A) showed that the installation of tetragonal structure

Optimization of mesh hierarchies in Multilevel Monte Carlo samplers

We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm. The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional It\^o stochastic differential equation discretized by a Milstein scheme

Nested Multilevel Monte Carlo with Biased and Antithetic Sampling

We consider the problem of estimating a nested structure of two expectations taking the form $U_0 = E[\max\{U_1(Y), \pi(Y)\}]$, where $U_1(Y) = E[X\ |\ Y]$. Terms of this form arise in financial risk estimation and option pricing. When $U_1(Y)$ requires approximation, but exact samples of $X$ and $Y$ are available, an antithetic multilevel Monte Carlo (MLMC) approach has been well-studied in the literature. Under general conditions, the antithetic MLMC estimator obtains a root mean squared error $\varepsilon$ with order $\varepsilon^{-2}$ cost. If, additionally, $X$ and $Y$ require approximate sampling, careful balancing of the various aspects of approximation is required to avoid a significant computational burden. Under strong convergence criteria on approximations to $X$ and $Y$, randomised multilevel Monte Carlo techniques can be used to construct unbiased Monte Carlo estimates of $U_1$, which can be paired with an antithetic MLMC estimate of $U_0$ to recover order $\varepsilon^{-2}$ computational cost. In this work, we instead consider biased multilevel approximations of $U_1(Y)$, which require less strict assumptions on the approximate samples of $X$. Extensions to the method consider an approximate and antithetic sampling of $Y$. Analysis shows the resulting estimator has order $\varepsilon^{-2}$ asymptotic cost under the conditions required by randomised MLMC and order $\varepsilon^{-2}|\log\varepsilon|^3$ cost under more general assumptions.Comment: 28 pages, 2 figure

Power Load Optimization for a Remote Area BTS Site Using MPPT Algorithms

Solar energy is one of the most significant renewable energy sources. It has advantages of no pollution and does not make any noise and for this reasons the using of solar energy is increased rapidly in the last years. Solar energy depends on the environmental factors, such as temperature (T) and irradiation (G), which are dynamic, therefore it requires to obtaining the maximum power at any ambient conditions to support the load with the required power. Maximum power point tracking (MPPT) is one of the methods which widely used to optimize system performance and many MPPT techniques have been published such as hill climbing (HC) and incremental conductance (IncCond) methods. This paper deals with tracking the maximum power point (MPP) of standalone photovoltaic (PV) source used to support the rural area base transceiver station (BTS) that uses the MOTOROLA (Horizon II macro) equipment at changing the ambient conditions. The comparison of the two MPPT algorithm methods (HC and IncCond) to maximize the obtained solar power is achieved. Also an equation for calculating input capacitor value used in buck boost converter (BBC) design was developed in this paper. The KC65T solar module is used as the PV panel in this work. Simulation results are obtained using MATLAB program (Ver.14). The main result of the comparisons shows that the IncCond method is better than the HC method in fast environmental conditions changing

An Efficient Activity Detection System based on Skeleton Joints Identification

The increasing criminal activities in the current world has drawn lot of interest activity recognition techniques which helps to perform the sophistical analytical operations on human activity and also helps to interface the human and computer interactions. From the existing review analysis it is found that most of the existing systems are not emphasize on computational performance but are more application specific by identifying specific problems. Hence, it is found that all the features are not required for accurate and cost effective human activity detection. Thus, the human skelton action can be considered and presented a simple and accurate process to identify the significant joints only. From the outcomes it is found that the proposed system is cost effective and computational efficient activity recognition technique for human actions