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The final step in implementing any solution is to check that the method behaves as desired, despite any simplifying assumptionsthat may have been made in its derivation. This may involve a detailed analysis ofthe resulting methodology, or it may involve simulations. Thorough testing would involve both analysis and simulationin a variety of settings that mimic, as closely as possible, the situations in which the method will be used.

Imagine being lost on a mountainside on a foggy night. Your goal is to get to the village which liesat the bottom of a valley below. Though you cannot see far, you can reach out and feelthe nearby ground. If you repeatedly step in the direction that heads downhill most steeply, you eventually reach a depression inwhich all directions lead up. If the contour of the land is smooth, and without any local depressions that can trap you,then you will eventually arrive at the village. The optimization procedure called “steepest descent”implements this scenario mathematically where the mountainside is defined by the “performance” function andthe optimal answer lies in the valley at the minimum value. Many standard communications algorithms (adaptive elements) can beviewed in this way.

An example of optimization: polynomial minimization

This first example is too simple to be of practical use, but it does show many of the ideas starkly. Suppose that the goal is tofind the value at which the polynomial

J ( x ) = x 2 - 4 x + 4

achieves its minimum value. Thus step (1) is set. As any calculus book will suggest, the direct way to find the minimum is totake the derivative, set it equal to zero, and solve for x . Thus, d J ( x ) d x = 2 x - 4 = 0 is solved when x = 2 , which is indeed the value of x where the parabola J ( x ) reaches bottom. Sometimes (one might truthfully say “often”), however, such directapproaches are impossible. Maybe the derivative is just too complicated to solve (which can happen when the functions involved in J ( x ) are extremely nonlinear). Or maybe the derivative of J ( x ) cannot be calculated precisely from the available data, and instead must beestimated from a noisy data stream.

One alternative to the direct solution technique is an adaptive method called “steepest descent”(when the goal is to minimize), and called “hill climbing” (when the goal is to maximize).Steepest descent begins with an initial guess of the location of the minimum, evaluates which direction from this estimate is most steeply “downhill,”and then makes a new estimate along the downhill direction. Similarly, hill climbing begins with an initialguess of the location of the maximum, evaluates which direction climbs the most rapidly, and then makes a new estimate along theuphill direction. With luck, the new estimates are better than the old. The process repeats, hopefully getting closer to theoptimal location at each step. The key ingredient in this procedure is to recognize that the uphilldirection is defined by the gradient evaluated at the current location, while the downhill direction is the negative of this gradient.

Questions & Answers

what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Software receiver design. OpenStax CNX. Aug 13, 2013 Download for free at http://cnx.org/content/col11510/1.3
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