Can you help me solve this problem? The problem lies in the following:
Problem: nonlinear diffusion for image processing:
img1
img2
img3
img4
img5
img6
img7
k is a user-defined parameter.
a) write method of lines for the problem above.
b) RK method according to the following Butcher's table.
Butcher's table
c) A-stability condition of the method.
Can you help me solve this?
I tried and solved solution b) but I couldn't understand how to solve other parts.
Related
I want to compare two smoothing methods for a bigram model:
Add-one smoothing
Interpolated Absolute Discounting
For the first method, I found some codes.
def calculate_bigram_probabilty(self, previous_word, word):
bigram_word_probability_numerator = self.bigram_frequencies.get((previous_word, word), 0)
bigram_word_probability_denominator = self.unigram_frequencies.get(previous_word, 0)
if self.smoothing:
bigram_word_probability_numerator += 1
bigram_word_probability_denominator += self.unique__bigram_words
return 0.0 if bigram_word_probability_numerator == 0 or bigram_word_probability_denominator == 0 else float(
bigram_word_probability_numerator) / float(bigram_word_probability_denominator)
However, I found nothing for the second method except for some references for 'KneserNeyProbDist'. However, this is for trigrams!
How can I change my code above to calculate it? The parameters of this method must be estimated from a development-set.
In this answer I just clear up a few things that I just found about your problem, but I can't provide a coded solution.
with KneserNeyProbDist you seem to refer to a python implementation of that problem: https://kite.com/python/docs/nltk.probability.KneserNeyProbDist
There exists an article about Kneser–Ney smoothing on wikipedia: https://en.wikipedia.org/wiki/Kneser%E2%80%93Ney_smoothing
The article above links this tutorial: https://nlp.stanford.edu/~wcmac/papers/20050421-smoothing-tutorial.pdf but this has a small fault on the most important page 29, the clear text is this:
Modified Kneser-Ney
Chen and Goodman introduced modified Kneser-Ney:
Interpolation is used instead of backoff. Uses a separate discount for one- and two-counts instead of a single discount for all counts. Estimates discounts on held-out data instead of using a formula
based on training counts.
Experiments show all three modifications improve performance.
Modified Kneser-Ney consistently had best performance.
Regrettable the modified Version is not explained in that document.
The original documentation by Chen & Goodman luckily is available, the Modified Kneser–Ney smoothing is explained on page 370 of this document: http://u.cs.biu.ac.il/~yogo/courses/mt2013/papers/chen-goodman-99.pdf.
I copy the most important text and formula here as screenshot:
So the Modified Kneser–Ney smoothing now is known and seems being the best solution, just translating the description beside formula in running code is still one step to do.
It might be helpful that below the shown text (above in screenshot) in the original linked document is still some explanation that might help to understand the raw description.
I am currently trying to simulate ballistics on an object, that is otherwise not affected by physics. To be precise, I have a rocket-like projectile, that is following an parabolic arc from origin to target with a Lerp. To make it more realistic, I want it not to move at constant speed, but to slow down towards the climax and speed up on its way back down.
I have used the Mathf.Smoothstep function to do the exact opposite of what i need on other objects, i.e. easing in and out of the motion.
So my question is: How do I get an inverted Smoothstep?
I found out that what i would need is actually the inverted formula to smoothstep [ x * x*(3 - 2*x) ], but being not exactly a math genius, I have no idea how to do that. All I got from online calculators was some pretty massive new function, which I'm afraid would not be very efficient.
So maybe there is a function that comes close to an inverted smoothstep, but isn't as complex to compute.
Any help on this would be much appreciated
Thanks in advance,
Tux
Correct formula is available here:
https://www.shadertoy.com/view/MsSBRh
Solution by Inigo Quilez and TinyTexel
Flt SmoothCubeInv(Flt y)
{
if(y<=0)return 0;
if(y>=1)return 1;
return 0.5f-Sin(asinf(1-2*y)/3);
}
I had a similar problem. For me, mirroring the curve in y = x worked:
So an implementation example would be:
float Smooth(float x) {
return x + (x - (x * x * (3.0f - 2.0f * x)));
}
This function has no clamping, so that may have to be added if x can go outside the 0 to 1 interval.
Wolfram Alpha example
If you're moving transforms, it is often a good idea to user iTween or similar animation libraries instead of controlling animation yourself. They have a an easy API and you can set up easing mode too.
But if you need this as a math function, you can use something like this:
y = 0.5 + (x > 0.5 ? 1 : -1) * Mathf.Pow(Mathf.Abs(2x - 1),p)/2
Where p is the measure of steepness that you want. Here's how it looks:
You seem to want a regular parabola. See the graph of this function:
http://www.wolframalpha.com/input/?i=-%28x%2A2-1%29%5E2%2B1
Which is the graph that seems to do what you want: -(x*2-1)^2+1
It goes from y=0 to y=1 and then back again between x=0 and x=1, staying a bit at the top around x=0.5 . It's what you want, if I understood it correctly.
Other ways to write this function, according to wolfram alpha, would be -(4*(x-1)*x) and (4-4*x)*x
Hope it helps.
I am using Freefem++ to solve the poisson equation
Grad^2 u(x,y,z) = -f(x,y,z)
It works well when I have an analytical expression for f, but now I have an f numerically defined (i.e. a set of data defined on a mesh) and I am wondering if I can still use Freefem++.
I.e. typical code (for a 2D problem in this case), looks like the following
mesh Sh= square(10,10); // mesh generation of a square
fespace Vh(Sh,P1); // space of P1 Finite Elements
Vh u,v; // u and v belongs to Vh
func f=cos(x)*y; // analytical function
problem Poisson(u,v)= // Definition of the problem
int2d(Sh)(dx(u)*dx(v)+dy(u)*dy(v)) // bilinear form
-int2d(Sh)(f*v) // linear form
+on(1,2,3,4,u=0); // Dirichlet Conditions
Poisson; // Solve Poisson Equation
plot(u); // Plot the result
I am wondering if I can define f numerically, rather than analytically.
Mesh & space Definition
We define a square unit with Nx=10 mesh and Ny=10 this provides 11 nodes on x axis and the same for y axis.
int Nx=10,Ny=10;
int Lx=1,Ly=1;
mesh Sh= square(Nx,Ny,[Lx*x,Ly*y]); //this is the same as square(10,10)
fespace Vh(Sh,P1); // a space of P1 Finite Elements to use for u definition
Conditions and problem statement
We are not going to use solve but we ll handle matrix (a more sophisticated way to solve with FreeFem).
First we define CL for our problem (Dirichlet ones).
varf CL(u,psi)=on(1,2,3,4,u=0); //you can eliminate border according to your problem state
Vh u=0;u[]=CL(0,Vh);
matrix GD=CL(Vh,Vh);
Then we define the problem. Instead of writing dx(u)*dx(v)+dy(u)*dy(v) I suggest to use macro, so we define div as following but pay attention macro finishes by // NOT ;.
macro div(u) (dx(u[0])+dy(u[1])) //
So Poisson bilinear form becomes:
varf Poisson(u,v)= int2d(Sh)(div(u)*div(v));
After we extract Stifness Matrix
matrix K=Poisson(Vh,Vh);
matrix KD=K+GD; //we add CL defined above
We proceed for solving, UMFPACK is a solver in FreeFem no much attention to this.
set(KD,solver=UMFPACK);
And here what you need. You want to define a value of function f on some specific nodes. I'm going to give you the secret, the poisson linear form.
real[int] b=Poisson(0,Vh);
You define value of the function f at any node you want to do.
b[100]+=20; //for example at node 100 we want that f equals to 20
b[50]+=50; //and at node 50 , f equals to 50
We solve our system.
u[]=KD^-1*b;
Finally we get the plot.
plot(u,wait=1);
I hope this will help you, thanks to my internship supervisor Olivier, he always gives to me tricks specially on FreeFem. I tested it, it works very well. Good luck.
The method by afaf works in the case when the function f is a free-standing one. For the terms like int2d(Sh)(f*u*v), another solution is required. I propose (actually I have red it somewhere in Hecht's manual) an approach that covers both cases. However, it works only for P1 finite elements, for which the degrees of freedom are coincided with the mesh nodes.
fespace Vh(Th,P1);
Vh f;
real[int] pot(Vh.ndof);
for(int i=0;i<Vh.ndof;i++){
pot[i]=something; //assign values or read them from a file
}
f[]=pot;
if I have an "AND gate" with three inputs "A,B and control line C"
can I control switching AND gate on or off .. just like this if C == 1 then AND gates works with input A,B if C==0 then nothing happens
can this be done by any method ??
This is how you would do it in pseudo-code(if I understood correctly):
if((c == 1) && (A || B))
Forgive me if I've got the wrong end of the stick- logic gates are just a way of making a simple concept confusing to me.
Achieve the same output from simpler (2-input) AND gates:
Yeah, hook a switch to the chip to toggle whether it is powered on or not. You don't need any fancy equipment or doubled gates, just turn the thing on to have it function as normal, turn it off and no matter what inputs you apply you'll get 0v on the output pin.
I'm trying to add functionality to these datapaths so that it can read the following instruction:
ADDNEW X, Y, Z
if (Y > Z) X = Y+ Z
else X = MEM[Y+Z]
Not really sure where to start.. so any hints are appreciated. I think I need to rewrite the result of the slt somewhere to tell it which way to go.. but not sure how? Yes is for homework so not expecting answers.. just hopefully a direction. Thank you..
second http://www.utdallas.edu/~cantrell/ee4304/Pipe-data+control.jpg
The top design is a single cycle and the bottom design is a pipeline design.