Digital Integration – Loss of Precision in Orthogonalization of Polynomials with Orthogonalize

The context

As a way to understand the growth of the structure in the universe,
I am interested in characterizing the curvature of random fields such as this one:

enter the description of the image here

To do this, I start with a PDF of the eigenvalues ​​of the second derivative of the field (measurement of the local curvature). This PDF file contains a Gaussian random field

Pdf1 = 2 Sqrt[2/[Pi]](x1-x2) Exp[1/2 (-x1 (3 x1-x2)-x2 (3 x2-x1))]

and looks like this:

rg = {{x1, -Infinity, Infinity}, {x2, -Infinity, x1}};
ContourPlot[Pdf1, Sequence @@ rgn // Evaluate,  ImageSize -> Small]

enter the description of the image here

whereas when the field becomes non-Gaussian, the PDF can be very different:

enter the description of the image here

My goal is to use orthogonal polynomials to represent this non-Gaussian PDF file.

Attempt

I have defined a scalar product

Clear[int]; int[a_, b_] : =
To integrate[ a b  Pdf1, Sequence @@ rg // Evaluate]

and a digital integration version of it

    Clear[nint]; nint[a_, b_] : =
NIntegrate[ a b  Pdf1, Sequence @@ rg // Evaluate]

I define my 2D polynomial

p = 2; pol0 =
Table[Table[x1^i x2^(p1 - i), {i, 0, p1}] // Flatten // Union,
{p1, 0, p}]// Flatten // Join

{1, x1, x2, x1 ^ 2, x1 x2, x2}

    pol = Orthogonalize[pol0, int[#1, #2] &]

They look like this:

Map[ContourPlot[# Pdf1  , Sequence @@ rgn // Evaluate, 
   PlotPoints -> 15, PlotRange -> All] &, pol]

enter the description of the image here

If I do the same thing numerically

    pol = Orthogonalize[pol0, intn[#1, #2] &, Method -> "Reorthogonalisation"];

I receive the same answer.

But

If I try to find higher order polynomials numerically

    p = 6; pol0 =
Table[Table[x1^i x2^(p1 - i), {i, 0, p1}] // Flatten // Union,
{p1, 0, p}]// Flatten // Join;
pol = Orthogonalize[pol0, intn[#1, #2] &, Method -> "Reorthogonalisation"];

I understood

(* {1., 1.81473 x1-0.804133, -1.53835 x1 + 2.37903 x2 + 1.73585,2.33148 x1 ^ 2-2.23663 x1 + 0.170409 x2-0.0991482, -2.84065 x1 ^ 2 + 4.36636 x1 x2 + 4.97902 x4.66.66., 1.84722 x1 ^ 2-5.47403 x1 x2-4.89841 x1 + 4.11307 x2 ^ 2 + 6.90647 x2 + 2.25076,0,3.15107 x1 ^ 2 x2 + 1.83935 x1 ^ 2-3.44839 x1 x2-3.23326 x1 + 0.212753 x2 0.219.13 x2 + 0.676981,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0} *)

In other words, the loss of precision in the orthogonalization leads to
zero polynomials of higher order.


I've also tried Gramm Schmitt by hand (somewhere on SE)

gs[vecs_, ip___] : = Module[{ovecs = vecs},
Do[ovecs[[i]]- = projection[ovecs[[i]], ovecs[[j]], ip]{i, 2,
Length[vecs]}, {j, 1, i - 1}

; ovecs];
pol1 = gs[pol0, Function[
    NIntegrate[ ##  Pdf1, Sequence @@ rg // Evaluate]]];

but that gives the same loss of precision.


I have also tried the eigenvectors of the matrix of scalar products

            mat = ParallelTable[int[i, j], {i, pol0}, {j, pol0}

;
eigs = Eigensystem[mat // N]; 

But the orthogonal polynomials are not of increasing order:

                Map[ContourPlot[#  Pdf1, Sequence @@ rgn // Evaluate, 
   PlotPoints -> 15, PlotRange -> All] And
pol = eigs[[2]].pol0 / Sqrt[eigs[[1]]]// Chop]

enter the description of the image here

Note that although orthogonal, they are not orthogonal "in the same direction".

Question

How can I calculate orthogonal polynomials of higher order precisely?

Annex question

One option would be to stick to the symbolic evaluation of the scalar product,
but it seems like it takes forever for higher order polynomials.

Is it possible to say orthogonalizing do not standardize the polynomials which, symbolically, seem to take the most time?

unit – How to make a user interface on a mesh without distortion or loss of quality?

I create a text adventure game in Unity 2018.2. My plan is that the interface looks like an open book with the user interface displayed on the page, curved with the mesh. After some research, I decided to use a rendering texture for that.

My problem, however, is that the rendering texture is of very poor quality. See the image below – the top line represents the canvas with text, set on Screen Space Camera, as it appears in the world. This camera defines the rendering texture, which is applied to an empty plane mesh (the bottom line). In other words, the top line corresponds to the original and the last to the rendering texture. The bottom line is crushed and distorted, which (if I understand correctly) also leads to the white outline when it samples from the bottom 0 alpha camera.

What I have already tried:

  • change the size of the rendering texture in 1920 * 1080 or 2048 * 2048
  • change the looping mode of the rendering texture, which has no effect
  • changing various settings in the canvas, camera, and texture
  • resizing the plane (enlarges the text but does not make it grow and remains deformed)

Some additional details: the text is made with Text Mesh Pro. The aircraft shader is set to UI / Default, otherwise the background of the camera was also rendered and covered everything behind it.

So, what is the solution to this? Can I apply the render texture to a mesh without it being stretched or distorted? Or is there another way to achieve what I'm looking for?

enter the description of the image here

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Discussion in & BHT Bots & Automation Tools & # 39; started by Rethi Stone, December 9, 2018 at 7:53.


  1. Rethi Stone,
    December 9, 2018 at 7:53

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networking – Random loss of write permissions on Windows 10 when attempting to write to an Ubuntu network share 18.04, configuration via GUI

I've therefore shared a few files via the Ubuntu 18.04 graphical interface. I have a hard drive in the Ubuntu machine, ext4, and I access it and write to it from Windows 10. Everything was working fine, I transferred a lot of data, but suddenly I was I have lost the ability to write in any of the shared folders. Windows. After a while, it started working again, but now I have lost the write permissions in another folder! I do not know what to do or think. I receive a message on Windows: Access to the destination folder denied. You must have permissions to perform this action.

I've tried looking in smb.conf, but it does not contain any entries related to the folders I've shared. Is this normal? How do Nautilus actions work?

My hard drive is mounted in / mnt / hdd

The folders are / mnt / hdd / folder1,2 etc.

Stabilization of the image – Does the OIS reduce in some cases the quality of the image (softness, vignetting, loss of contrast, etc.)?

I always take photos with the optical stabilization enabled (unless I take long exposure photos with a tripod), but I was wondering if the SIO could (in some cases) reduce the quality of the image.

OIS works by moving a glass element inside the lens. Therefore, if I shake the camera while taking a picture, the lens has an off-center glass element, which should soften the final photo, but I do not have it. have never met.

Can optical stabilization reduce image quality? Is this something only visible on certain types of lenses or not at all?

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privacy – What are the security implications of the loss of a paper calendar?

Assuming I have a paper calendar (organizer) containing the following information:

  • dates / places I went and where do I go
  • phone numbers and email addresses of some people
  • my own coordinates so that the calendar can be returned in case of loss

If I lose this schedule and a stranger detects it, is there anything that should worry me?

As an average person, I think the worst that can happen is that a) the searcher does not return the calendar and b) can make annoying phone calls / sending annoying emails, well as really dangerous.

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