SLIDE 1
Big Picture
What is it all about?
Dense Matrix
1 4 3 7 8 6 2 5 9
Sparse Matrix
4 9 Importance?
Efficiency Faster Solutions
Applications?
Sciences Systems of equations Image Processing Data Analysis
Multiresolution Algorithms for Sparse Matrix Representation By: - - PowerPoint PPT Presentation
Multiresolution Algorithms for Sparse Matrix Representation By: - - PowerPoint PPT Presentation
Multiresolution Algorithms for Sparse Matrix Representation By: Mario Barela Mentor: Professor Garcia-Cervera Big Picture What is it all about ? 1 4 3 7 8 6 Dense Matrix 2 5 9 0 4 0 0 0 0 Sparse Matrix 0 0 9 Importance?
SLIDE 2
SLIDE 3
Research Project Goals
Questions:
What algorithms can effectively transform a given dense matrix into
a sparse one?
Will the new representation accurately model the original to a
certain degree of error?
SLIDE 4
Research Project Goals
Project Goals: Learn Algorithms : Modify Algorithms : Run Large Scale Simulations A B
SLIDE 5
Experimental Methods for Multiresolution Algorithm
1 2 3 4 5 6 7 8
}
K+2 K+3
1
- Remove odd
data points.
- Interpolate
between even data points. 2
- Evaluate at odd
data points.
- Subtract odd
data points and store. 3
- Repeat process.
- Obtain coarser
resolution then truncate to achieve compression.
Differences
K K+1
SLIDE 6
1-D Data Compression
200 400 600 800 1000 1200 A(x) B(x) C(x) Original tol=.001 tol=.01 tol=.1
Nonzeros
SLIDE 7
Experimental Data
SLIDE 8
Multiresolution and Standard Form
𝐵
𝐷𝑝𝑛𝑞𝑠𝑓𝑡𝑡𝑗𝑝𝑜 𝐵𝑁𝑆 𝐷𝑖𝑏𝑜𝑓 𝑝𝑔 𝐶𝑏𝑡𝑓 𝐵𝑇
Given an 𝑂𝑦𝑂 matrix 𝐵, and an 𝑂𝑦1 vector 𝑔, we compute 𝐵𝑔 as follows: (𝐵𝑔)𝑁𝑆= 𝐵𝑇𝑔𝑁𝑆
SLIDE 9
Standard Form Data
10000 20000 30000 40000 50000 60000 70000 2-point Linear 3-point Parabolic 4-point Cubic 6-point tol=10⁻⁴ tol=10⁻⁷
Matrix A: 256x256 NONZEROS
SLIDE 10
Matrix-Vector Multiplication
𝐵 𝑗, 𝑘 = log
((𝑗 − 𝑘)2); 𝑗𝑔 𝑗 ≠ 𝑘, 0 𝑓𝑚𝑡𝑓𝑥ℎ𝑓𝑠𝑓 𝑔
𝑗 = sin 2𝜌 𝑗
𝑂
Product (𝐵𝑔) Error ∙ ∞-error= 4.35 ∗ 10−2
SLIDE 11
Results and Conclusions
Results:
𝐵 𝐵𝑁𝑆 𝐵𝑇 The order of interpolation used effected the sparsity of transformed
matrices and accuracy of multiplication.
Achieved similar results to that of Shalom who used six point-
interpolation. Future Plans:
Test the efficiency of our Algorithms in respect to
matrix-vector multiplication.
Use our multiresolution algorithms to efficiently solve problems that
arise in Mathematics and the Sciences.
SLIDE 12
Results and Conclusions
Acknowledgements:
Professor Carlos Garcia-Cervera: Department of Mathematics,
University of California Santa Barbara.
University of California Leadership Excellence through Advanced
Degrees (UC LEADS). References:
Ami Harten and Itai Yad-Shalom “Fast Multiresolution Algorithms for
Matrix-Vector Multiplication”.
[1] Francesc Arandiga and Vicente F. Candela “Multiresolution