Introduction Outline Sensor Placement: . . . Towards Analytical Data and Knowledge . . . Knowledge . . . Techniques for Optimizing Resulting Geometric . . . How to Use the . . . Knowledge Acquisition, Future Work Processing, Propagation, and Home Page Title Page Use in Cyberinfrastructure ◭◭ ◮◮ L. Octavio Lerma ◭ ◮ Computational Science Program Page 1 of 60 University of Texas at El Paso 500 W. University Go Back El Paso, TX 79968, USA lolerma@episd.org Full Screen Close Quit
Introduction 1. Introduction Outline Sensor Placement: . . . • Knowledge-related processes are important: we rely on Data and Knowledge . . . them when we drive, communicate, etc. Knowledge . . . • Surprisingly, the very process of acquiring and propa- Resulting Geometric . . . gating information is the least automated. How to Use the . . . Future Work • At present, to decide on the best way to place sensors Home Page or propagate data, we mostly use numerical models. Title Page • These models are very resource-consuming, rely on su- percomputers, not ready for everyday applications. ◭◭ ◮◮ ◭ ◮ • We therefore need analytical models – which would al- low easier optimization and application. Page 2 of 60 • Developing such models is our main objective. Go Back Full Screen Close Quit
Introduction 2. Outline Outline Sensor Placement: . . . • We describe analytical models for all stages of knowl- Data and Knowledge . . . edge processing. Knowledge . . . • We start with knowledge acquisition: optimal sensor Resulting Geometric . . . placement for stationary and mobile sensors. How to Use the . . . Future Work • We then deal with data and knowledge processing: how Home Page to best organize computing power and research teams. Title Page • We deal with knowledge propagation and resulting knowl- edge enhancement; we analyze: ◭◭ ◮◮ ◭ ◮ – how early stages of idea propagation occur; – how to assess the initial knowledge level; Page 3 of 60 – how to present the material and how to provide Go Back feedback. Full Screen • Finally, we analyze how knowledge is used. Close Quit
Introduction 3. Sensor Placement: Case Study Outline Sensor Placement: . . . • Biological weapons are difficult and expensive to de- Data and Knowledge . . . tect. Knowledge . . . • Within a limited budget, we can afford a limited num- Resulting Geometric . . . ber of bio-weapon detector stations. How to Use the . . . Future Work • It is therefore important to find the optimal locations Home Page for such stations. Title Page • A natural idea is to place more detectors in the areas with more population. ◭◭ ◮◮ ◭ ◮ • However, such a commonsense analysis does not tell us how many detectors to place where. Page 4 of 60 • To decide on the exact detector placement, we must Go Back formulate the problem in precise terms. Full Screen Close Quit
Introduction 4. Towards Precise Formulation of the Problem Outline Sensor Placement: . . . • The adversary’s objective is to kill as many people as Data and Knowledge . . . possible. Knowledge . . . • Let ρ ( x ) be a population density in the vicinity of the Resulting Geometric . . . location x . How to Use the . . . Future Work • Let N be the number of detectors that we can afford Home Page to place in the given territory. Title Page • Let d 0 be the distance at which a station can detect an outbreak of a disease. ◭◭ ◮◮ ◭ ◮ • Often, d 0 = 0 – we can only detect a disease when the sources of this disease reach the detecting station. Page 5 of 60 • We want to find ρ d ( x ) – the density of detector place- Go Back ment. Full Screen � • We know that ρ d ( x ) dx = N. Close Quit
Introduction 5. Optimal Placement of Sensors Outline Sensor Placement: . . . • We want to place the sensors in an area in such a way Data and Knowledge . . . that Knowledge . . . – the largest distance D to a sensor Resulting Geometric . . . – is as small as possible. How to Use the . . . Future Work • It is known that the smallest such number is provided Home Page by an equilateral triangle grid: Title Page h ✛ ✲ ◭◭ ◮◮ r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ◭ ◮ ❆ ❆ ❑ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ h ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆❆ Page 6 of 60 ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❯ ❆ ❆ ✁ ❆ ✁ ❆ ✁ r r r ✁ ❆ ✁ ❆ ✁ ❆ Go Back ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Full Screen ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Close Quit
Introduction For the equilateral triangle placement, points which are Outline closest to a given detector forms a hexagonal area: Sensor Placement: . . . h ✛ ✲ Data and Knowledge . . . r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ Knowledge . . . ❑ ❆ ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ✟ ❍ ✟ ❍ ❆ h ❆ ✁ ✟ ❆ ✁ ❍ ❆ ✁ ❆ ✟ ❍ ❆ Resulting Geometric . . . ❆ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❯ ❆ ❆ ✁ ❆ ✁ ❆ ✁ How to Use the . . . r r r ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Future Work ❍ ✟ ❍ ✟ ✁ ❆ ✁ ❍ ✟ ❆ ✁ ❆ ✁ ✟ ❍ Home Page ✁ ❆ ✁ ❆ ✁ ❆ ✁ ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Title Page This hexagonal area consists of 6 equilateral triangles: ◭◭ ◮◮ h ✛ ✲ ◭ ◮ r r r r ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❆ ❆ ❑ ❆ ✁ ❆ ✁ ❆ ✁ ❆ Page 7 of 60 ✟ ❆ ❍ ✟ ❍ ❆ h ❆ ✁ ✟ ❆ ✁ ❍ ❆ ✁ ❆ ✟ ❍ ❆ ❆❆ ❩ ✚ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ❩ ✚✚ ❯ ❆ Go Back ❆ ✁ ❩ ❆ ✁ ❆ ✁ ✚ ❩❩ r r r ✁ ❆ ✁ ❆ ✁ ❆ ✚ ✚ ❩ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Full Screen ✟ ❍ ❍ ✟ ✁ ❆ ✁ ❍ ✟ ❆ ✁ ❆ ✁ ❍ ✟ ✁ ❆ ✁ ❆ ✁ ❆ ✁ Close ✁ ❆ ✁ ❆ ✁ ❆ ✁ r r r r Quit
Introduction 6. Optimal Placement of Sensors (cont-d) Outline Sensor Placement: . . . • In each △ , the height h/ 2 is related to the side s by √ √ Data and Knowledge . . . the formula h 3 3 2 = s · cos(60 ◦ ) = s · 2 , hence s = h · 3 . Knowledge . . . Resulting Geometric . . . • Thus, the area A t of each triangle is equal to How to Use the . . . √ √ A t = 1 2 · s · h 2 = 1 3 · 1 3 3 2 · h 2 = Future Work 12 · h 2 . 2 · Home Page • So, the area A s of the whole set is equal to 6 times the √ Title Page 3 2 · h 2 . triangle area: A s = 6 · A t = ◭◭ ◮◮ ◭ ◮ • In a region of area A , there are A · ρ d ( x ) sensors, they cover area ( A · ρ d ( x )) · A s . Page 8 of 60 √ 3 Go Back 2 · h 2 • The condition A = ( A · ρ d ( x )) · A s = ( A · ρ d ( x )) · � 2 Full Screen c 0 def √ implies that h = � , with c 0 = 3 . Close ρ d ( x ) Quit
Introduction 7. Estimating the Effect of Sensor Placement Outline Sensor Placement: . . . • The adversary places the bio-weapon at a location which Data and Knowledge . . . is the farthest away from the detectors. Knowledge . . . • This way, it will take the longest time to be detected. Resulting Geometric . . . How to Use the . . . • For the grid placement, this location is at one of the vertices of the hexagonal zone. Future Work Home Page • At these vertices, the distance from each neighboring √ Title Page 3 detector is equal to s = h · 3 . ◭◭ ◮◮ c 0 c 1 • By know that h = � , so s = � , with ◭ ◮ ρ d ( x ) ρ d ( x ) Page 9 of 60 √ √ √ 4 3 3 · 2 c 1 = 3 · c 0 = . Go Back 3 Full Screen • Once the bio-weapon is placed, it starts spreading until Close it reaches the distance d 0 from the detector. Quit
Introduction 8. Effect of Sensor Placement (cont-d) Outline Sensor Placement: . . . c 1 • The bio-weapon is placed at a distance s = � Data and Knowledge . . . ρ d ( x ) Knowledge . . . from the nearest sensor. Resulting Geometric . . . • Once the bio-weapon is placed, it starts spreading until How to Use the . . . it reaches the distance d 0 from the detector. Future Work • In other words, it spreads for the distance s − d 0 . Home Page • During this spread, the disease covers the circle of ra- Title Page dius s − d 0 and area π · ( s − d 0 ) 2 . ◭◭ ◮◮ • The number of affected people n ( x ) is equal to: ◭ ◮ � � 2 Page 10 of 60 c 1 n ( x ) = π · ( s − d 0 ) 2 · ρ ( x ) = π · � − d 0 · ρ ( x ) . Go Back ρ d ( x ) Full Screen Close Quit
Introduction 9. Precise Formulation of the Problem Outline Sensor Placement: . . . • For each location x , the number of affected people n ( x ) Data and Knowledge . . . is equal to: Knowledge . . . � � 2 c 1 Resulting Geometric . . . n ( x ) = π · � − d 0 · ρ ( x ) . ρ d ( x ) How to Use the . . . Future Work • The adversary will select a location x for which this Home Page number n ( x ) is the largest possible: Title Page � � 2 c 1 π · . ◭◭ ◮◮ n = max � − d 0 · ρ ( x ) x ρ d ( x ) ◭ ◮ • Resulting problem: Page 11 of 60 – given population density ρ ( x ), detection distance d 0 , Go Back and number of sensors N , Full Screen – find a function ρ d ( x ) that minimizes the above ex- � Close pression n under the constraint ρ d ( x ) dx = N . Quit
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