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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 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 to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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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