When Existence is Enough

1. When Existence is Enough Dan Kalman American University

2. A chemist, a physicist, an engineer and a mathematician …

3. Existence Proofs • Some collection of problems to solve Example: ax2 + bx + c = 0 • Not all instances are solvable • Conditions which assure existence of a solution Example: b2 – 4ac≥ 0 • Not a prescription for finding a solution • WHAT GOOD IS THAT?

4. A Real Practical Example • Example is both real and practical • Worked on in the aerospace industry in LA • Subject area: designing a satellite communication system • General Problem: Can a given system design handle a projected data load? • Resource Allocation problem: who gets to talk to which satellite when? • Existence result tells if the load can be handled, but does not tell how to allocate the resources • Result was used in a very practical way

5. Computer Model Overview • Geometric framework • Earth and satellite motion • Instantaneous visibility • Discrete time step model

6. Geometric Framework • x-y-z coordinate system • Earth = sphere centered at (0,0,0) • Equator in xy plane • Earth rotates around z axis • Ground Stations travel in horizontal circles around z axis • Satellites travel in ellipses, one focus at (0,0,0) • Given initial position and velocity of a satellite, we can calculate its position at any time • Given latitude and longitude of ground station, we can calculate its position at any time

7. Instantaneous Visibility • Geometric Models for visibility • At instant, positions of satellites and stations given by motion models • Constraints described in terms of lines, angles, cones • Line of sight from station to satellite is computed as a vector • Vector methods used to compute angles • Communication possible when satellite can see the station

11. Discrete Time Step Model • Compute positions of all satellites and stations at one fixed time • Determine which satellites can see which stations • Advance time by one minute, repeat all calculations • Repeat many many times • For a 24 hour simulation, repeat 1440 times

12. Design Problem • Fixed ground stations • Predefined connection time requirements • LARGE range of choices for satellite orbits • For a given set of orbits, can all connect time requirements be met?

13. Graph Theory Formulation • Bi-Partite Graph: Two sets of vertices • One vertex for each ground station • A separate vertex for each satellite for each time step • Edges indicate visibility • Visibility graph

15. Problem Formulation • Edge Count = degree • At station vertexdegree = amountof connect time • Assignment Subgraph: Degree 1 at each satellite/time vertex • Given: Visibility graph and required degree at each station vertex • To Find: Assignment subgraph that meets all requirements • Existence Question: does a solution exist?

17. History • Gordan's Problem: Is there a finite set of invariants that can be used to generate all the rest? • David Hilbert gives existence proof of a solution, 1888 • Gordan: ``This is not mathematics. This is theology.'' • Felix Klein: ``wholly simple and, therefore, logically compelling.'' • History sided with Hilbert and Klein • Gordan best remembered for being wrong about existence proofs. His statement ``has echoed in mathematics long after his own mathematical work has fallen silent.'' (Constance Reid)

18. Gordon Klein Hilbert

19. Satellites: Necessary Conditions • ConReq: connection requirement for a station • ConReq for a station must be  visibility graph degree for the station • Total of all ConReqs must be  number of satellite-time nodes • These are two extreme cases of a more general constraint • For any subset of stations, the sum of ConReqs must be  the number of satellite-time nodes connected to the subset

21. Necessary and Sufficient • Chris Reed approach • With n stations, 2n - 1 necessary conditions • Check them all • If one fails, no solution • If all conditions are met .... A solution must exist! • ``Bed rest, plenty of fluids, and a good hard proof.'' • The marriage problem in graph theory

22. Existence Result • Finding the assignment subgraph is computationally prohibitive • Checking all the conditions is computationally feasible • We can easily compute whether the assignment problem is solvable • We cannot find the solution • This is exactly what is meant by an existence result

23. VISREV • Legacy Code • Compute visibility graph AND right graph statistics to check all the necessary conditions • I added logic to do all the tests, and report on the solvability of the assignment problem • Computationally intensive: 1440 time steps, 20 satellites, 10 stations, 210 - 1  1000 conditions to check • Several hours on Cyber 7600 number cruncher • Probably a few minutes on a PC today

24. Minimal Transmission Rate • Chris Reed Idea • Connection requirements inversly proportional to transmission rate • Assignment Problem Unsolvable: try over with increased transmission rate • Assignment Problem Solvable: try over with decreased transmission rate • Find smallest transmission rate that permits assignment problem to be solved • This provides a comparison between system designs • NOTE: never need to actually find the solution to assignment problem. Existence is enough.

25. Brute Force Computation • 109 system designs • Computer time charges assessed by the second • Submit 15 jobs in the morning and tie up computers all day • In one week: $80,000 of computer charges • Chris Reed: don’t worry about computer charges – it’s “funny money”

28. Epilog: Funny Money • Near the end of the brute force attack • Crossing El Segundo Blvd on my way to the LAAFS gym • Met up with Chris Reed • STOP the analysis!!! • ``It's not funny''