Computational methods mathematical modeling

Alternative methods include (1) computer-based methods (mathematical models and expert systems) (2) physicochemical methods, in which physical or chemical effects are assessed in systems lacking cells and, most typically, (3) in vitro methods, in which biological effects are observed in cell cultures, tissues, or organs. 

For information on fundamental aspects of mathematical and computational methods in modelling the following titles can be recommended ... 

Abstract. The paper is dedicated to the mathematical model describing dynamics of an artificial heart valve being moved by inhomogeneous incompressible fiuid fiow with variable viscosity, and its computational method. The modeling results of tricuspid valve performance axe presented. 

Figure 4-1 describes some aspects of information flow in the analysis of complicated problems. The boxes in the figure represent specialized methods and tools that contribute to the investigation. These include, for example, experimental techniques, computer codes, numerical methods, mathematical models of well-characterized fundamental scientific systems, and numerical simulations of complex components of technological systems. Several general observations can be made ... 

S. Kocak and H.U. Akay. (2001) Parallel Schur Cointlement Method for Large-Scale Systems on Distributed Memory Computers. AppUed Mathematical Modelling 25. 

Considerable work has been done on mathematic models of the extmsion process, with particular emphasis on screw design. Good results are claimed for extmsion of styrene-based resins using these mathematical methods (229,232). With the advent of low cost computers, closed-loop control of... 

Traditional control systems are in general based on mathematical models that describe the control system using one or more differential equations that define the system response to its inputs. In many cases, the mathematical model of the control process may not exist or may be too expensive in terms of computer processing power and memory. In these cases a system based on empirical rules may be more effective. In many cases, fuzzy control can be used to improve existing controller systems by adding an extra layer of intelligence to the current control method. 

Statistical and algebraic methods, too, can be classed as either rugged or not they are rugged when algorithms are chosen that on repetition of the experiment do not get derailed by the random analytical error inherent in every measurement,i° 433 is, when similar coefficients are found for the mathematical model, and equivalent conclusions are drawn. Obviously, the choice of the fitted model plays a pivotal role. If a model is to be fitted by means of an iterative algorithm, the initial guess for the coefficients should not be too critical. In a simple calculation a combination of numbers and truncation errors might lead to a division by zero and crash the computer. If the data evaluation scheme is such that errors of this type could occur, the validation plan must make provisions to test this aspect. 

The algebraic/iterative and the brute force methods are numerical respectively computational techniques that operate on the chosen mathematical model. Raw residuals r are weighted to reflect the relative reliabilities of the measurements. 

Molecular mechanics is a useful and reliable computational method for structure, energy, and other molecular properties. The mathematical basis for molecular models in MM3 has been described, along with the limitations of the method. One of the major difficulties associated with molecular mechanics, in general, and MM3 in particular is the lack of accurately parameterized diverse functional groups. This lack of diverse functional groups has severely limited the use of MM3 in pharmaceutical applications. 

There also exists an alternative theoretical approach to the problem of interest which goes back to "precomputer epoch" and consists in the elaboration of simple models permitting analytical solutions based on prevailing factors only. Among weaknesses of such approaches is an a priori impossibility of quantitative-precise reproduction for the characteristics measured. Unlike articles on computer simulation in which vast tables of calculated data are provided and computational tools (most often restricted to standard computational methods) are only mentioned, the articles devoted to analytical models abound with mathematical details seemingly of no value for experimentalists and present few, if any, quantitative results that could be correlated to experimentally obtained data. It is apparently for this reason that interest in theoretical approaches of this kind has waned in recent years. 

Le Maguer and Yao (1995) presented a physical model of a plant storage tissue based on its cellular structure. The mathematical equivalent of this model was solved using a finite element-based computer method and incorporated shrinkage and different boundary conditions. The concept of volume average was used to express the concentration and absolute pressure in the intracellular volume, which is discontinuous in the tissue, as a... 

Figure 3. The modeling relation, as adapted from J. L. Casti [90] The encoding operation provides the link between a natural system (real world) and its formal representation (mathematical world). A set of rules and computational methods allows to infer properties (theorems) of the formal system. Using a decoding relation, we can interpret those theorems in terms of the behavior of the natural system. In this sense, the inferred properties of the formal system become predictions about the natural system, allowing us to verify the consistency of the encoding. The modeling process needs to provide the appropriate encoding/decoding relations that translate back and forth between thereal world and the mathematical world.

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