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

[Russell Standish]

On Fri, May 05, 2023 at 03:22:49PM -0600, Stephen Guerin wrote:
> 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?
> 

Similar. Quad/Octrees are good for inhomgenous simulations. Particle
in cell methods are for homogenous simulations.

Note a technique I developed in conjuction with Duraid Madina in the
early noughties which we called GraphCode is effectively a
particle-in-cell method on an arbitrary graph. The graph can change
dynamically, and the cells redistributed to rebalance a calculation
using a graph partitioning algorithm - it makes for something more
general than quad/octrees.

I have written this up a couple of times, but have moved on to other
things now, and the work has had little impact. But I did discover
that a Spanish group took my code to develop an ABM system that was as
least twice as performant as the equivalent model in RepastHPC, so we
obviously did somethig right (J. Supercomputing, 2018,
doi:10.1007/s11227-018-2688-8).


> 
> On Fri, May 5, 2023 at 2:58 PM Angel Edward  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.an...@gmail.com
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
> 
> 
> On May 5, 2023, at 2:31 PM, Stephen Guerin 
>  > 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 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 

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

[Stephen Guerin]

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  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.an...@gmail.com
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>
> On May 5, 2023, at 2:31 PM, Stephen Guerin 
> 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
>  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.gue...@simtable.com 
> CEO, https://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 
> 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 

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

[Angel Edward]

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.an...@gmail.com
505-453-4944 (cell) http://www.cs.unm.edu/~angel

> On May 5, 2023, at 2:31 PM, Stephen Guerin  
> 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 
>  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.gue...@simtable.com 
> CEO, https://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  > 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 

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

[Stephen Guerin]

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
 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.gue...@simtable.com 
CEO, https://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  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.an...@gmail.com
> 505-453-4944 (cell)  http://www.cs.unm.edu/~angel
>
> On Apr 28, 2023, at 8:18 AM, Stephen Guerin 
> 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
> ), and can we
> foreground how 

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

[Angel Edward]

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.an...@gmail.com
505-453-4944 (cell) http://www.cs.unm.edu/~angel

> On Apr 28, 2023, at 8:18 AM, Stephen Guerin  
> 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 
> ), 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 
>  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
> -. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. .
> FRIAM Applied Complexity Group listserv
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Re: [FRIAM] links for this morning's FRIAM: Special Unitary Groups and Quaternions

[Roger Frye]

a quaternion version of Euler's formula
https://www.youtube.com/watch?v=dvpAFWBVgy0

A bit late for the discussion, and kind of sketchy when you get into it.

-Roger


> On Apr 28, 2023, at 8:18 AM, Stephen Guerin  
> 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 
> ), 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 
>  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
> -. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. .
> FRIAM Applied Complexity Group listserv
> Fridays 9a-12p Friday St. Johns Cafe   /   Thursdays 9a-12p Zoom 
> https://bit.ly/virtualfriam
> to (un)subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
> FRIAM-COMIC http://friam-comic.blogspot.com/
> archives:  5/2017 thru present 
> https://redfish.com/pipermail/friam_redfish.com/
>  1/2003 thru 6/2021  http://friam.383.s1.nabble.com/

-. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. .
FRIAM Applied Complexity Group listserv
Fridays 9a-12p Friday St. Johns Cafe   /   Thursdays 9a-12p Zoom 
https://bit.ly/virtualfriam
to (un)subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
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[FRIAM] links for this morning's FRIAM: Special Unitary Groups and Quaternions

[Stephen Guerin]

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
), 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
 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
-. --- - / ...- .- .-.. .. -.. / -- --- .-. ... . / -.-. --- -.. .
FRIAM Applied Complexity Group listserv
Fridays 9a-12p Friday St. Johns Cafe   /   Thursdays 9a-12p Zoom 
https://bit.ly/virtualfriam
to (un)subscribe http://redfish.com/mailman/listinfo/friam_redfish.com
FRIAM-COMIC http://friam-comic.blogspot.com/
archives:  5/2017 thru present https://redfish.com/pipermail/friam_redfish.com/
  1/2003 thru 6/2021  http://friam.383.s1.nabble.com/