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

Fri May 5 16:58:11 EDT 2023

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:
> 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.
> 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.
> 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 <mailto:stephen.guerin at simtable.com>
> CEO, https://www.simtable.com <http://www.simtable.com/>
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> 
> On Fri, Apr 28, 2023 at 7:50 PM Angel Edward <edward.angel at gmail.com <mailto: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 <mailto: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 <mailto: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.
>>> 
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