Great @Rouben Rostamian  !!! It seems that your code works fine..Thanks!! 

@vv thanks.

As you say, "t it is easy to implement a Lebesgue filling curve in any dimension (just a few lines in Maple". In fact, you are a post with this issue:

https://www.mapleprimes.com/questions/220980-Convert-Peano-Curve-Into-A-Function

@vv thanks for your reply. Your code is for Maple 2020, right? In the paper linked in my post, the code (in C) is for the n-dimensional Hilbert curve, and for its  "pseudo-inverse"

@Carl Love The version is  Maple 17.00

@acer You are right, I am not using the 17v, sorry by the confussion.

@tomleslie Tanks for your reply. In Maple 17 your code returns me the error

Error, `Fractals` does not evaluate to a module
 

However, the attached txt contains procedures to (numerically) compute H(t), H being the (approximation of the ) Hilbert curve and t a point in [0,1], as well as the pseudo inverse of H.

Regards.

@Carl Love Mmm....I am not sure at all, but I would have to review the article in more detail but it is likely that I am wrong.

In the paper entitled "One-Dimensional Global Optimization for Observations with Noise" by Calvin and Zilinskas, says that for the global optimization fo a single vairable function with noise (normal 0,1)  "...a Wiener process is accepted as a statistical model of the objective function."

@Carl Love Great!! Niw the code seems to work fine! Thank you very much.

@Carl Love Thanks!! Your code returns me the following error:

Error, (in Calvin) Insert is not a command in the ArrayTools package

I use Maple 2015, it seems that the "Insert" optiion is not valid for "ArrayTools":

with(ArrayTools);
 [AddAlongDimension, Alias, AllNonZero, AnyNonZeros, Append, BlockCopy, CircularShift, ComplexAsFloat, Compress, Concatenate, Copy, DataTranspose, Diagonal, Dimensions,  ElementDivide, ElementMultiply, ElementPower, Extend, Fill,  FlipDimension, HasNonZero, HasZero, IsEqual, IsZero, LowerTriangle, MultiplyAlongDimension, NumElems, Permute,  PermuteInverse, RandomArray, RegularArray, RemoveSingletonDimensions, Replicate, Reshape, SearchArray,  Size, Uncompress, UpperTriangle]

The option

Is not "compatible" with "Digits:=26"?

Again, thanks,

@Carl Love Thanks, you are right: I have not chosen the appropiate sequence! However, taking theta_n:=log(2+n) or theta_n:=(n)^1/4, the results have not "improved substantially".

Again, thanks.

@Carl Love  I'm very sorry for the distraction, the correct article is this:

  https://www.sciencedirect.com/science/article/pii/S0885064X01905746  

Basically, I have tried to follow the code given in Fig. 1, p 309 of the PDF. In step 4, the maximum of the indicated probability is given explicitly in formula (12) p 312. (the numbers Y_n, rho_n, etc are defined in p 311)

About the "try... catch": really, I think that the algorithm does not return an error.

Again, thank you very much.

@Carl Love Thanks.

The article, in this case, is available for free download.

https://www.sciencedirect.com/science/article/pii/S0898122105002397

Yes, as you say, I see that some variables are not used...Maybe a "debug" that I forgot to delete? Please, find attached the "corrected" file. Even for the simple function (t-0.5)^2, I see that the approximation is not "very fine"....

I reiterate my gratitude for your useful help.

Calvin_Algo_2.mw

@Carl Love @mmcdara @vv If it's not a nuisance, someone might check the above attached file, please? I "suspect" that I am doing something wrong ...

I have tried to implement the algorithm descrbied in https://www.sciencedirect.com/science/article/pii/S0898122105002397

Many thanks.

@mmcdara  Thanks for your comments.

I agree with you. Maybe the best is to implemennt a "more actual" algorithm, such as this by J.M. Calcin

https://www.sciencedirect.com/science/article/pii/S0885064X01905746

I have trieed to encode it into Maple, but I am not sure tat that the code be correct

Alg_noise_v1.mw

@vv Thnaks for your comment.