[FRIAM] links for this morning's FRIAM: Special Unitary Groups and Quaternions

Fri May 5 17:22:49 EDT 2023

I think that's the same as when I said "I knew how to solve n-body systems
with particle N^2/2 forces (corrected) with some quadtree or octree
optimizations to get from n^2 to nlog(n)." . Or are you saying something
different?

On Fri, May 5, 2023 at 2:58 PM Angel Edward <edward.angel at gmail.com> wrote:

> Here’s another connection I had forgotten. Consider particles on a 2D
> rectangle  with 1/r^2 repulsion. If you break up the rectangle into smaller
> rectangles in which particles can only stay in their own rectangles or move
> to neighbor rectangles, the N^2 force calculation comes down to N log N,
> same as the limit on good sorting algorithms. This technique came up when
> we were using particles to form an isosurface in 3D.
>
> Ed
> __________
>
> Ed Angel
>
> Founding Director, Art, Research, Technology and Science Laboratory (ARTS
> Lab)
> Professor Emeritus of Computer Science, University of New Mexico
>
> 1017 Sierra Pinon
> Santa Fe, NM 87501
> 505-984-0136 (home)   edward.angel at gmail.com
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>
> On May 5, 2023, at 2:31 PM, Stephen Guerin <stephen.guerin at simtable.com>
> wrote:
>
> Thanks Roger and Ed!
>
> I've spent some time with Ed and Frank discussing this and I've really
> filled in some gaps in my knowledge of parallel algorithms. eg, I knew how
> to solve n-body system with particle N^2/2 focus with some quadtree or
> octree optimizations to get from n^2 to nlog(n). But the FFT transform on
> laplacians solving Poisson equation was new to me and I can now see the
> beauty. Today, Ed quickly threw out the Kronecker Operator/Product which
> Frank knew but I didn't. Frank flashed me a wikipedia article
> <https://en.wikipedia.org/wiki/Kronecker_product> on his phone with
> symbolics that I couldn't immediately grok. But asking chatGPT to explain
> the operator to a 3D graphics person I immediately got it and had the
> benefit that I would usually implement this function with two inner loops
> over rows and columnts instead of using Kronecker available in optimized
> linear algebra/graphics libraries. Often this was happening under the hood
> of my tools but didn't realize it.
>
> As a 3D graphics developer, understanding the Kronecker matrix can be very
> useful. The Kronecker product is often used in computer graphics and
> computer vision applications, such as texture mapping, geometric
> transformations, and image processing. Here are a few specific ways in
> which Kronecker matrix can be useful to a 3D graphics developer:
>
>    1. Texture mapping: The Kronecker product can be used to create
>    repetitive patterns in textures, such as brick walls, tiles, or grass. By
>    creating a base texture and applying a Kronecker product with a smaller
>    texture, a developer can create a seamless and repeating texture that
>    covers a larger surface.
>    2. Geometric transformations: The Kronecker product can be used to
>    perform geometric transformations, such as scaling, rotation, and
>    translation, on 3D objects. By creating a Kronecker matrix with a
>    transformation matrix, a developer can apply the transformation to every
>    vertex of an object, resulting in a transformed object.
>    3. Image processing: The Kronecker product can be used to perform
>    image processing operations, such as blurring, sharpening, or edge
>    detection, on 3D images. By creating a Kronecker matrix with a filter
>    matrix, a developer can apply the filter to every pixel of an image,
>    resulting in a processed image.
>
> In summary, the Kronecker matrix is a powerful tool that can be used in
> various ways by 3D graphics developers. Whether it's creating textures,
> transforming objects, or processing images, understanding the Kronecker
> matrix can help a developer achieve their desired results more efficiently
> and effectively.
>
>
>
> _______________________________________________________________________
> Stephen.Guerin at Simtable.com <stephen.guerin at simtable.com>
> CEO, https://www.simtable.com <http://www.simtable.com/>
> 1600 Lena St #D1, Santa Fe, NM 87505
> office: (505)995-0206 mobile: (505)577-5828
>
>
> On Fri, Apr 28, 2023 at 7:50 PM Angel Edward <edward.angel at gmail.com>
> wrote:
>
>> Most of my dissertation (1968) was on numerical solution of potential
>> problems. One of the parts was a proof that some of the known iterative
>> methods converged. The argument loosely went something like this. Consider
>> the 2D Poisson equation on a square. If you use an N x N approximation with
>> the usual discretization of the Laplacian
>>
>> u_ij = (u_i(j-1) + u_i(j+1) + u_(i_1)j + i_(j+1))/4
>>
>> i.e, the average of the surrounding points, the problem reduces to the
>> solution of a set of N^2 linear equations
>>
>> Ax = b
>>
>> where x in a vector of the unknown {u_ij} arranged by rows or columns, b
>> is determined by the boundary conditions and the right side of the Poisson
>> equation. The interesting part is A which is block tridiagonal. With only a
>> small error A can be made block cyclic. You can then diagonalize A with a
>> sine transform and I was able to use that for proofs.
>>
>> A few years later when the FFT came about, we realized that we could use
>> the FFT to do the sine transform and the resulting numerical method was as
>> least as efficient as any other method people had come up with.
>>
>> Ed
>>
>> Here’s a reference from 1986 that I think was based on paper at a Bellman
>> Continuum
>>
>> ``From Dynamic Programming to Fast Transforms,'' E. Angel, J. Math. Anal.
>> Appl.,119,1986.
>>
>> Ed
>> __________
>>
>> Ed Angel
>>
>> Founding Director, Art, Research, Technology and Science Laboratory (ARTS
>> Lab)
>> Professor Emeritus of Computer Science, University of New Mexico
>>
>> 1017 Sierra Pinon
>> Santa Fe, NM 87501
>> 505-984-0136 (home)   edward.angel at gmail.com
>> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>>
>> On Apr 28, 2023, at 8:18 AM, Stephen Guerin <stephen.guerin at simtable.com>
>> wrote:
>>
>> Special Unitary Groups and Quaternions
>>
>> Mostly for Ed from the context of last week's Physical Friam if you're
>> coming today.
>>
>> Discussion was around potential ways of visualizing the dynamics of
>> SU(3), SU(2), (SU1) that highlights Special Unitary Groups. (wiki link
>> from Frank <https://en.wikipedia.org/wiki/Special_unitary_group>), and
>> can we foreground how quaternions are used in this process.
>>
>> and a related bit on forces, I'm searching for ways to
>> visualize/understand how FFTs with Poisson equation
>> <https://www.codeproject.com/Articles/5308623/Solving-Poisson-Equation>
>> are used to compute the forces from scalar fields (eg gravitational force
>> from mass density, electric force from charge, etc) and if there's any
>> relation to Special Unitary Groups.
>>
>> -S
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