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Monte Carlo method

Monte Carlo method
Monte Carlo methods (or Monte Carlo experiments) are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results; typically one runs simulations many times over in order to obtain the distribution of an unknown probabilistic entity. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to obtain a closed-form expression, or infeasible to apply a deterministic algorithm. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration and generation of draws from a probability distribution. The modern version of the Monte Carlo method was invented in the late 1940s by Stanislaw Ulam, while he was working on nuclear weapons projects at the Los Alamos National Laboratory. Introduction[edit] Monte Carlo method applied to approximating the value of π. Monte Carlo methods vary, but tend to follow a particular pattern: History[edit] Definitions[edit] Related:  mathematics

Multidisciplinary design optimization - Wikipedia, the free ency MDO allows designers to incorporate all relevant disciplines simultaneously. The optimum of the simultaneous problem is superior to the design found by optimizing each discipline sequentially, since it can exploit the interactions between the disciplines. However, including all disciplines simultaneously significantly increases the complexity of the problem. These techniques have been used in a number of fields, including automobile design, naval architecture, electronics, architecture, computers, and electricity distribution. However, the largest number of applications have been in the field of aerospace engineering, such as aircraft and spacecraft design. History[edit] Since 1990, the techniques have expanded to other industries. Origins in structural optimization[edit] Gradient-based methods[edit] There were two schools of structural optimization practitioners using gradient-based methods during the 1960s and 1970s: optimality criteria and mathematical programming. Constraints[edit] find

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BBC Four - Climate Change by Numbers Latin Hypercube Sampling | Mathieu Fenniak Introduction Latin hypercube sampling (LHS) is a form of stratified sampling that can be applied to multiple variables. The method commonly used to reduce the number or runs necessary for a Monte Carlo simulation to achieve a reasonably accurate random distribution. LHS can be incorporated into an existing Monte Carlo model fairly easily, and work with variables following any analytical probability distribution. Monte-Carlo simulations provide statistical answers to problems by performing many calculations with randomized variables, and analyzing the trends in the output data. The concept behind LHS is not overly complex. The sampling algorithm ensures that the distribution function is sampled evenly, but still with the same probability trend. Figure 1. Figure 2. Process Sampling To perform the stratified sampling, the cumulative probability (100%) is divided into segments, one for each iteration of the Monte Carlo simulation. Grouping Example Sampling: Conclusion

matplotlib: python plotting — Matplotlib 1.2.1 documentation Computational geometry Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry. Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry. Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points. The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature, and may come from mathematical visualization. The main branches of computational geometry are: Combinatorial computational geometry, also called algorithmic geometry, which deals with geometric objects as discrete entities. Combinatorial computational geometry[edit]

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Dr Hannah Fry: the mathematical models that underpin our sexual success | Science What are the odds? Or how mathematician Peter Backus weighed up his chances of finding love… Just as it’s not possible to calculate precisely how many alien life forms there are, it’s also not possible to calculate exactly how many potential partners you may have. It also applies to maths student Peter Backus’s well-publicised quest to see whether there were intelligent, socially advanced women of the same species out there for him to date. 1. 2. 3. 4. 5. 6. 7. Leaving him with just 26 women in the whole world he would be willing to date. Personally, I think that he is being a little picky. I think there’s room to be a bit more generous. 1. 5. 6. 7. Almost a thousand potential partners across a city, then. But there is another issue. Strangely though, opening our minds to all potential partners seems to be the opposite of what we do when we’re single. The statistics of sex In 1996 Swedish scientists asked 2,810 people how many sexual partners they’d had. The happily ever after