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88-451: TK Solver (originally TK!Solver ) is a mathematical modeling and problem solving software system based on a declarative , rule-based language , commercialized by Universal Technical Systems, Inc. Invented by Milos Konopasek in the late 1970s and initially developed in 1982 by Software Arts , the company behind VisiCalc , TK Solver was acquired by Universal Technical Systems in 1984 after Software Arts fell into financial difficulty and

176-429: A ) ln(2/0.01)/ ε ≈ 10.6( b – a ) / ε . Despite its conceptual and algorithmic simplicity, the computational cost associated with a Monte Carlo simulation can be staggeringly high. In general the method requires many samples to get a good approximation, which may incur an arbitrarily large total runtime if the processing time of a single sample is high. Although this is a severe limitation in very complex problems,

264-503: A paradigm shift offers radical simplification. For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into

352-400: A prior probability distribution (which can be subjective), and then update this distribution based on empirical data. An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending

440-581: A Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem, and a Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). Here are some examples: Kalos and Whitlock point out that such distinctions are not always easy to maintain. For example, the emission of radiation from atoms is a natural stochastic process. It can be simulated directly, or its average behavior can be described by stochastic equations that can themselves be solved using Monte Carlo methods. "Indeed,

528-425: A certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables , state variables , exogenous variables, and random variables . Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants . The variables are not independent of each other as

616-436: A common approach is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics. Defining a metric to measure distances between observed and predicted data

704-558: A computation on each input (test whether it falls within the quadrant). Aggregating the results yields our final result, the approximation of π . There are two important considerations: Uses of Monte Carlo methods require large amounts of random numbers, and their use benefitted greatly from pseudorandom number generators , which are far quicker to use than the tables of random numbers that had been previously used for statistical sampling. Monte Carlo methods are often used in physical and mathematical problems and are most useful when it

792-540: A computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering , physics models are often made by mathematical methods such as finite element analysis . Different mathematical models use different geometries that are not necessarily accurate descriptions of

880-419: A flow of probability distributions with an increasing level of sampling complexity arise (path spaces models with an increasing time horizon, Boltzmann–Gibbs measures associated with decreasing temperature parameters, and many others). These models can also be seen as the evolution of the law of the random states of a nonlinear Markov chain. A natural way to simulate these sophisticated nonlinear Markov processes

968-401: A human system, we know that usually the amount of medicine in the blood is an exponentially decaying function, but we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use

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1056-468: A mean-field particle interpretation of neutron-chain reactions, but the first heuristic-like and genetic type particle algorithm (a.k.a. Resampled or Reconfiguration Monte Carlo methods) for estimating ground state energies of quantum systems (in reduced matrix models) is due to Jack H. Hetherington in 1984. In molecular chemistry, the use of genetic heuristic-like particle methodologies (a.k.a. pruning and enrichment strategies) can be traced back to 1955 with

1144-412: A particular pattern: For example, consider a quadrant (circular sector) inscribed in a unit square . Given that the ratio of their areas is ⁠ π / 4 ⁠ , the value of π can be approximated using a Monte Carlo method: In this procedure the domain of inputs is the square that circumscribes the quadrant. One can generate random inputs by scattering grains over the square then perform

1232-431: A priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. Usually, it

1320-466: A probabilistic interpretation. By the law of large numbers , integrals described by the expected value of some random variable can be approximated by taking the empirical mean ( a.k.a. the 'sample mean') of independent samples of the variable. When the probability distribution of the variable is parameterized, mathematicians often use a Markov chain Monte Carlo (MCMC) sampler. The central idea

1408-518: A programming language like Fortran or APL was superior for simultaneous solution of linear equations . The magazine concluded that despite limitations, it was a "powerful tool, useful for scientists and engineers. No similar product exists". By version 5.0, TK Solver added Matrix handling functionality. Competitive products appeared by mid-1988: Mathsoft's Mathcad and Borland 's Eureka: The Solver . Dan Bricklin , known for VisiCalc and his Software Arts 's initial development of TK Solver,

1496-451: A question which occurred to me in 1946 as I was convalescing from an illness and playing solitaires. The question was what are the chances that a Canfield solitaire laid out with 52 cards will come out successfully? After spending a lot of time trying to estimate them by pure combinatorial calculations, I wondered whether a more practical method than "abstract thinking" might not be to lay it out say one hundred times and simply observe and count

1584-469: A system algebraically by the principle of consecutive substitution. When multiple rules contain multiple unknowns, the program can trigger an iterative solver which uses the Newton–Raphson algorithm to successively approximate based on initial guesses for one or more of the output variables. Procedure functions can also be used to solve systems of equations. Libraries of such procedures are included with

1672-433: A system and to study the effects of different components, and to make predictions about behavior. Mathematical models can take many forms, including dynamical systems , statistical models , differential equations , or game theoretic models . These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models . In many cases,

1760-538: Is a useful tool for assessing model fit. In statistics, decision theory, and some economic models , a loss function plays a similar role. While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations . Tools from nonparametric statistics can sometimes be used to evaluate how well

1848-403: Is already known from direct investigation of the phenomenon being studied. An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology. It should also be noted that while mathematical modeling uses mathematical concepts and language, it

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1936-527: Is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation . These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on

2024-634: Is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization , numerical integration , and generating draws from a probability distribution . In physics-related problems, Monte Carlo methods are useful for simulating systems with many coupled degrees of freedom , such as fluids, disordered materials, strongly coupled solids, and cellular structures (see cellular Potts model , interacting particle systems , McKean–Vlasov processes , kinetic models of gases ). Other examples include modeling phenomena with significant uncertainty in inputs such as

2112-494: Is for the pseudo-random sequence to appear "random enough" in a certain sense. What this means depends on the application, but typically they should pass a series of statistical tests. Testing that the numbers are uniformly distributed or follow another desired distribution when a large enough number of elements of the sequence are considered is one of the simplest and most common ones. Weak correlations between successive samples are also often desirable/necessary. Sawilowsky lists

2200-413: Is generating draws from a sequence of probability distributions satisfying a nonlinear evolution equation. These flows of probability distributions can always be interpreted as the distributions of the random states of a Markov process whose transition probabilities depend on the distributions of the current random states (see McKean–Vlasov processes , nonlinear filtering equation ). In other instances,

2288-559: Is included, with built-in functions for accessing it. TK Solver is also the platform for engineering applications marketed by UTS, including Advanced Spring Design, Integrated Gear Software, Interactive Roark’s Formulas, Heat Transfer on TK, and Dynamics and Vibration Analysis. Tables, plots, comments, and the MathLook notation display tool can be used to enrich TK Solver models. Models can be linked to other components with Microsoft Visual Basic and .NET tools, or they can be web-enabled using

2376-551: Is listed and stored on its own worksheet—the Rule Sheet, Variable Sheet, Unit Sheet, etc. Within each worksheet, each object has properties summarized on subsheets or viewed in a property window. The interface uses toolbars and a hierarchal navigation bar that resembles the directory tree seen on the left side of the Windows Explorer . The declarative programming structure is embodied in the rules, functions and variables that form

2464-401: Is no consensus on how Monte Carlo should be defined. For example, Ripley defines most probabilistic modeling as stochastic simulation , with Monte Carlo being reserved for Monte Carlo integration and Monte Carlo statistical tests. Sawilowsky distinguishes between a simulation , a Monte Carlo method, and a Monte Carlo simulation: a simulation is a fictitious representation of reality,

2552-640: Is not itself a branch of mathematics and does not necessarily conform to any mathematical logic , but is typically a branch of some science or other technical subject, with corresponding concepts and standards of argumentation. Mathematical models are of great importance in the natural sciences, particularly in physics . Physical theories are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used. It

2640-415: Is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in

2728-452: Is to design a judicious Markov chain model with a prescribed stationary probability distribution . That is, in the limit, the samples being generated by the MCMC method will be samples from the desired (target) distribution. By the ergodic theorem , the stationary distribution is approximated by the empirical measures of the random states of the MCMC sampler. In other problems, the objective

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2816-476: Is to sample multiple copies of the process, replacing in the evolution equation the unknown distributions of the random states by the sampled empirical measures . In contrast with traditional Monte Carlo and MCMC methodologies, these mean-field particle techniques rely on sequential interacting samples. The terminology mean field reflects the fact that each of the samples ( a.k.a. particles, individuals, walkers, agents, creatures, or phenotypes) interacts with

2904-634: Is to use randomness to solve problems that might be deterministic in principle. The name comes from the Monte Carlo Casino in Monaco, where the primary developer of the method, mathematician Stanislaw Ulam , was inspired by his uncle's gambling habits. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from a probability distribution. They can also be used to model phenomena with significant uncertainty in inputs, such as calculating

2992-471: Is within ε of μ . If n > k , then n simulations can be run “from scratch,” or, since k simulations have already been done, one can just run n – k more simulations and add their results into those from the sample simulations: An alternate formula can be used in the special case where all simulation results are bounded above and below. Choose a value for ε that is twice the maximum allowed difference between μ and m. Let 0 < δ < 100 be

3080-404: The embarrassingly parallel nature of the algorithm allows this large cost to be reduced (perhaps to a feasible level) through parallel computing strategies in local processors, clusters, cloud computing, GPU, FPGA, etc. Before the Monte Carlo method was developed, simulations tested a previously understood deterministic problem, and statistical sampling was used to estimate uncertainties in

3168-530: The simulations required for further postwar development of nuclear weapons, including the design of the H-bomb, though severely limited by the computational tools at the time. Von Neumann, Nicholas Metropolis and others programmed the ENIAC computer to perform the first fully automated Monte Carlo calculations, of a fission weapon core, in the spring of 1948. In the 1950s Monte Carlo methods were used at Los Alamos for

3256-403: The speed of light , and we study macro-particles only. Note that better accuracy does not necessarily mean a better model. Statistical models are prone to overfitting which means that a model is fitted to data too much and it has lost its ability to generalize to new events that were not observed before. Any model which is not pure white-box contains some parameters that can be used to fit

3344-670: The LAAS-CNRS in a series of restricted and classified research reports with STCAN (Service Technique des Constructions et Armes Navales), the IT company DIGILOG, and the LAAS-CNRS (the Laboratory for Analysis and Architecture of Systems) on radar/sonar and GPS signal processing problems. These Sequential Monte Carlo methodologies can be interpreted as an acceptance-rejection sampler equipped with an interacting recycling mechanism. From 1950 to 1996, all

3432-569: The Monte Carlo method while studying neutron diffusion, but he did not publish this work. In the late 1940s, Stanislaw Ulam invented the modern version of the Markov Chain Monte Carlo method while he was working on nuclear weapons projects at the Los Alamos National Laboratory . In 1946, nuclear weapons physicists at Los Alamos were investigating neutron diffusion in the core of a nuclear weapon. Despite having most of

3520-494: The NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to

3608-630: The RuleMaster product or linked with Excel spreadsheets using the Excel Toolkit product. There is also a DesignLink option linking TK Solver models with CAD drawings and solid models. In the premium version, standalone models can be shared with others who do not have a TK license, opening them in Excel or the free TK Player. BYTE in 1984 stated that "TK!Solver is superb for solving almost any kind of equation", but that it did not handle matrices , and that

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3696-432: The algorithm to obtain m is Suppose we want to know how many times we should expect to throw three eight-sided dice for the total of the dice throws to be at least T. We know the expected value exists. The dice throws are randomly distributed and independent of each other. So simple Monte Carlo is applicable: If n is large enough, m will be within ε of μ for any ε > 0. Let ε = | μ – m | > 0. Choose

3784-478: The calculation of risk in business and, in mathematics, evaluation of multidimensional definite integrals with complicated boundary conditions . In application to systems engineering problems (space, oil exploration , aircraft design, etc.), Monte Carlo–based predictions of failure, cost overruns and schedule overruns are routinely better than human intuition or alternative "soft" methods. In principle, Monte Carlo methods can be used to solve any problem having

3872-570: The case that, the results of these experiments are not well known. Monte Carlo simulations are typically characterized by many unknown parameters, many of which are difficult to obtain experimentally. Monte Carlo simulation methods do not always require truly random numbers to be useful (although, for some applications such as primality testing , unpredictability is vital). Many of the most useful techniques use deterministic, pseudorandom sequences, making it easy to test and re-run simulations. The only quality usually necessary to make good simulations

3960-408: The coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability. In general, model complexity involves a trade-off between simplicity and accuracy of

4048-584: The core of a mathematical model. All rules are entered in the Rule Sheet or in user-defined functions. Unlike a spreadsheet or imperative programming environment, the rules can be in any order or sequence and are not expressed as assignment statements. "A + B = C / D" is a valid rule in TK Solver and can be solved for any of its four variables. Rules can be added and removed as needed in the Rule Sheet without regard for their order and incorporated into other models. A TK Solver model can include up to 32,000 rules, and

4136-441: The data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data. The question of whether

4224-477: The desired confidence level – the percent chance that, when the Monte Carlo algorithm completes, m is indeed within ε of μ . Let z be the z -score corresponding to that confidence level. Let s be the estimated variance, sometimes called the “sample” variance; it is the variance of the results obtained from a relatively small number k of “sample” simulations. Choose a k ; Driels and Shin observe that “even for sample sizes an order of magnitude lower than

4312-636: The desired confidence level, expressed as a percentage. Let every simulation result r 1 , r 2 , … r i , … r n be such that a ≤ r i ≤ b for finite a and b . To have confidence of at least δ that | μ – m | < ε /2, use a value for n such that n ≥ 2 ( b − a ) 2 ln ⁡ ( 2 / ( 1 − ( δ / 100 ) ) ) / ϵ 2 {\displaystyle n\geq 2(b-a)^{2}\ln(2/(1-(\delta /100)))/\epsilon ^{2}} For example, if δ = 99%, then n ≥ 2( b –

4400-491: The development of the hydrogen bomb , and became popularized in the fields of physics , physical chemistry , and operations research . The Rand Corporation and the U.S. Air Force were two of the major organizations responsible for funding and disseminating information on Monte Carlo methods during this time, and they began to find a wide application in many different fields. The theory of more sophisticated mean-field type particle Monte Carlo methods had certainly started by

4488-526: The empirical measures of the process. When the size of the system tends to infinity, these random empirical measures converge to the deterministic distribution of the random states of the nonlinear Markov chain, so that the statistical interaction between particles vanishes. Suppose one wants to know the expected value μ of a population (and knows that μ exists), but does not have a formula available to compute it. The simple Monte Carlo method gives an estimate for μ by running n simulations and averaging

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4576-445: The equals sign. Software Arts also released a series of " Solverpacks " - "ready-made versions of some of the formulas most commonly used in specific areas of application." The New York Times described TK Solver as doing "for science and engineering what word processing did for corporate communictions [sic] and calc packages did for finance." Lotus, which had acquired Software Arts, including TK Solver, in 1984 sold its ownership of

4664-438: The first rigorous analysis of these particle algorithms were written by Pierre Del Moral in 1996. Branching type particle methodologies with varying population sizes were also developed in the end of the 1990s by Dan Crisan, Jessica Gaines and Terry Lyons, and by Dan Crisan, Pierre Del Moral and Terry Lyons. Further developments in this field were described in 1999 to 2001 by P. Del Moral, A. Guionnet and L. Miclo. There

4752-401: The geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean. Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how

4840-447: The idea to John von Neumann , and we began to plan actual calculations. Being secret, the work of von Neumann and Ulam required a code name. A colleague of von Neumann and Ulam, Nicholas Metropolis , suggested using the name Monte Carlo , which refers to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money from relatives to gamble. Monte Carlo methods were central to

4928-522: The library that ships with the current version includes utilities for higher mathematics, statistics, engineering and science, finances, and programming. Variables in a rule are automatically posted to the Variable Sheet when the rule is entered and the rule is displayed in mathematical format in the MathLook View window at the bottom of the screen. Any variable can operate as an input or an output, and

5016-611: The mid-1960s, with the work of Henry P. McKean Jr. on Markov interpretations of a class of nonlinear parabolic partial differential equations arising in fluid mechanics. An earlier pioneering article by Theodore E. Harris and Herman Kahn, published in 1951, used mean-field genetic -type Monte Carlo methods for estimating particle transmission energies. Mean-field genetic type Monte Carlo methodologies are also used as heuristic natural search algorithms (a.k.a. metaheuristic ) in evolutionary computing. The origins of these mean-field computational techniques can be traced to 1950 and 1954 with

5104-449: The model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning , the optimization of parameters is called training , while the optimization of model hyperparameters is called tuning and often uses cross-validation . In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting . A crucial part of

5192-467: The model describes well the properties of the system between data points is called interpolation , and the same question for events or data points outside the observed data is called extrapolation . As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics , we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles traveling at speeds close to

5280-603: The model will be solved for the output variables depending on the choice of inputs. A database of unit conversion factors also ships with TK Solver, and users can add, delete, or import unit conversions in a way similar to that for rules. Each variable is associated with a "calculation" unit, but variables can also be assigned "display" units and TK automatically converts the values. For example, rules may be based upon meters and kilograms, but units of inches and pounds can be used for input and output. TK Solver has three ways of solving systems of equations. The "direct solver" solves

5368-700: The model's user. Depending on the context, an objective function is also known as an index of performance , as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, economists often apply linear algebra when using input–output models . Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables. Mathematical modeling problems are often classified into black box or white box models, according to how much

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5456-553: The model. In black-box models, one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively,

5544-493: The model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability . Thomas Kuhn argues that as science progresses, explanations tend to become more complex before

5632-427: The model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below

5720-408: The modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. Usually, the easiest part of model evaluation is checking whether a model predicts experimental measurements or other empirical data not used in the model development. In models with parameters,

5808-433: The most important and influential ideas of the 20th century, and they have enabled many scientific and technological breakthroughs. Monte Carlo methods also have some limitations and challenges, such as the trade-off between accuracy and computational cost, the curse of dimensionality, the reliability of random number generators, and the verification and validation of the results. Monte Carlo methods vary, but tend to follow

5896-611: The necessary data, such as the average distance a neutron would travel in a substance before it collided with an atomic nucleus and how much energy the neutron was likely to give off following a collision, the Los Alamos physicists were unable to solve the problem using conventional, deterministic mathematical methods. Ulam proposed using random experiments. He recounts his inspiration as follows: The first thoughts and attempts I made to practice [the Monte Carlo Method] were suggested by

5984-406: The noise of the system. Another pioneering article in this field was Genshiro Kitagawa's, on a related "Monte Carlo filter", and the ones by Pierre Del Moral and Himilcon Carvalho, Pierre Del Moral, André Monin and Gérard Salut on particle filters published in the mid-1990s. Particle filters were also developed in signal processing in 1989–1992 by P. Del Moral, J. C. Noyer, G. Rigal, and G. Salut in

6072-418: The number of successful plays. This was already possible to envisage with the beginning of the new era of fast computers, and I immediately thought of problems of neutron diffusion and other questions of mathematical physics, and more generally how to change processes described by certain differential equations into an equivalent form interpretable as a succession of random operations. Later [in 1946], I described

6160-591: The number required, the calculation of that number is quite stable." The following algorithm computes s in one pass while minimizing the possibility that accumulated numerical error produces erroneous results: Note that, when the algorithm completes, m k is the mean of the k results. n is sufficiently large when n ≥ s 2 / ( z ϵ ) 2 {\displaystyle n\geq s^{2}/(z\epsilon )^{2}} If n ≤ k , then m k = m ; sufficient sample simulations were done to ensure that m k

6248-1060: The program and can be merged into files as needed. A list solver feature allows variables to be associated with ranges of data or probability distributions, solving for multiple values, which is useful for generating tables and plots and for running Monte Carlo simulations . The premium version now also includes a "Solution Optimizer" for direct setting of bounds and constraints in solving models for minimum, maximum, or specific conditions. TK Solver includes roughly 150 built-in functions : mathematical, trigonometric , Boolean , numerical calculus , matrix operations, database access, and programming functions, including string handling and calls to externally compiled routines. Users may also define three types of functions: declarative rule functions; list functions, for table lookups and other operations involving pairs of lists; and procedure functions, for loops and other procedural operations which may also process or result in arrays (lists of lists). The complete NIST database of thermodynamic and transport properties

6336-432: The publications on Sequential Monte Carlo methodologies, including the pruning and resample Monte Carlo methods introduced in computational physics and molecular chemistry, present natural and heuristic-like algorithms applied to different situations without a single proof of their consistency, nor a discussion on the bias of the estimates and on genealogical and ancestral tree based algorithms. The mathematical foundations and

6424-451: The purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what

6512-540: The quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the physical sciences , a traditional mathematical model contains most of the following elements: Mathematical models are of different types: In business and engineering , mathematical models may be used to maximize

6600-592: The risk of a nuclear power plant failure. Monte Carlo methods are often implemented using computer simulations, and they can provide approximate solutions to problems that are otherwise intractable or too complex to analyze mathematically. Monte Carlo methods are widely used in various fields of science, engineering, and mathematics, such as physics, chemistry, biology, statistics, artificial intelligence, finance, and cryptography. They have also been applied to social sciences, such as sociology, psychology, and political science. Monte Carlo methods have been recognized as one of

6688-477: The same computer code can be viewed simultaneously as a 'natural simulation' or as a solution of the equations by natural sampling." Convergence of the Monte Carlo simulation can be checked with the Gelman-Rubin statistic . The main idea behind this method is that the results are computed based on repeated random sampling and statistical analysis. The Monte Carlo simulation is, in fact, random experimentations, in

6776-603: The seminal work of Marshall N. Rosenbluth and Arianna W. Rosenbluth . The use of Sequential Monte Carlo in advanced signal processing and Bayesian inference is more recent. It was in 1993, that Gordon et al., published in their seminal work the first application of a Monte Carlo resampling algorithm in Bayesian statistical inference. The authors named their algorithm 'the bootstrap filter', and demonstrated that compared to other filtering methods, their bootstrap algorithm does not require any assumption about that state-space or

6864-457: The simulations. Monte Carlo simulations invert this approach, solving deterministic problems using probabilistic metaheuristics (see simulated annealing ). An early variant of the Monte Carlo method was devised to solve the Buffon's needle problem , in which π can be estimated by dropping needles on a floor made of parallel equidistant strips. In the 1930s, Enrico Fermi first experimented with

6952-405: The simulations’ results. It has no restrictions on the probability distribution of the inputs to the simulations, requiring only that the inputs are randomly generated and are independent of each other and that μ exists. A sufficiently large n will produce a value for m that is arbitrarily close to μ ; more formally, it will be the case that, for any ε > 0, | μ – m | ≤ ε . Typically,

7040-520: The software to Universal Technical Systems less than two years later. Release 5 was still considered "one of the longest–standing mathematical equation solvers on the market today" in 2012. TK Solver's core technologies are a declarative programming language, algebraic equation solver, an iterative equation solver, and a structured, object-based interface, using a command structure. The interface comprises nine classes of objects that can be shared between and merged into other TK files: Each class of object

7128-442: The speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics. Many types of modeling implicitly involve claims about causality . This is usually (but not always) true of models involving differential equations. As

7216-404: The state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of

7304-494: The system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations . A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, Boolean values or strings , for example. The variables represent some properties of

7392-508: The system, for example, the measured system outputs often in the form of signals , timing data , counters, and event occurrence. The actual model is the set of functions that describe the relations between the different variables. General reference Philosophical Monte Carlo simulations Monte Carlo methods , or Monte Carlo experiments , are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept

7480-427: The underlying process, whereas neural networks produce an approximation that is opaque. Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition , experience , or expert opinion , or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify

7568-613: The work of Alan Turing on genetic type mutation-selection learning machines and the articles by Nils Aall Barricelli at the Institute for Advanced Study in Princeton, New Jersey . Quantum Monte Carlo , and more specifically diffusion Monte Carlo methods can also be interpreted as a mean-field particle Monte Carlo approximation of Feynman – Kac path integrals. The origins of Quantum Monte Carlo methods are often attributed to Enrico Fermi and Robert Richtmyer who developed in 1948

7656-458: Was quoted as saying that the market "wasn't as big as we thought it would be because not that many people think in equations." Mathematical model The use of mathematical models to solve problems in business or military operations is a large part of the field of operations research . Mathematical models are also used in music , linguistics , and philosophy (for example, intensively in analytic philosophy ). A model may help to explain

7744-473: Was sold to Lotus Software . Konopasek's goal in inventing the TK Solver concept was to create a problem solving environment in which a given mathematical model built to solve a specific problem could be used to solve related problems (with a redistribution of input and output variables) with minimal or no additional programming required: once a user enters an equation, TK Solver can evaluate that equation as is—without isolating unknown variables on one side of

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