Propagation of Error bounds to estimate errors for loosely coupleed multi-physics problems.

1. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Probabilistic Error Bounds for Order Reduction
of Smooth Nonlinear Models
Mohammad G. Abdo
and
Hany S. Abdel-Khalik
and
Presented by: Congjian Wang
North Carolina State University
Nuclear Department
mgabdo@ncsu.edu and abdelkhalik@ncsu.edu
June 16, 2014
1 / 27

2. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Motivation
ROM plays a vital role in many desiplines, specially for
computationally intensive applications.
2 / 27

3. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Motivation
ROM plays a vital role in many desiplines, specially for
computationally intensive applications.
It i s mandatory to equip reduced order models with error metrics
to credibly defend the predictions of the reduced model.
2 / 27

4. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Motivation
ROM plays a vital role in many desiplines, specially for
computationally intensive applications.
It i s mandatory to equip reduced order models with error metrics
to credibly defend the predictions of the reduced model.
Probabilistic error bounds are mostly used in the linear moulding.
2 / 27

5. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Motivation
ROM plays a vital role in many desiplines, specially for
computationally intensive applications.
It i s mandatory to equip reduced order models with error metrics
to credibly defend the predictions of the reduced model.
Probabilistic error bounds are mostly used in the linear moulding.
Reduction errors need to be propagated across various
interfaces such as parameter interface(i.e. cross sections), state
function(i.e. flux) and response of interest(i.e. reaction rates,
detector response etc..).
2 / 27

6. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
We will adopt one formal mathematical definition that has been
developed back in the 1960s in the signal processing community.
3 / 27

7. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
We will adopt one formal mathematical definition that has been
developed back in the 1960s in the signal processing community.
Definition
A nonlinear function f with n inputs is said to be reducable and of
intrinsic dimension r (0 ≤ r ≤ n) if there exists a non linear function g
with r inputs and an n × r matrix Q such that r is the smallest integer
satisfying:
f (x) = g ˜x ;
where x ∈ Rn and ˜x = QT x ∈ Rr
3 / 27

8. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Reduction Algorithms
In our context, reduction algorithms refer to two different
algorithms, each is used at a different interface:
4 / 27

9. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Reduction Algorithms
In our context, reduction algorithms refer to two different
algorithms, each is used at a different interface:
Snapshot reduction algorithm (Gradient-free)(Reduces response
interface).
4 / 27

10. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Reduction Algorithms
In our context, reduction algorithms refer to two different
algorithms, each is used at a different interface:
Snapshot reduction algorithm (Gradient-free)(Reduces response
interface).
Gradient-based reduction algorithm(Reduces parameter interface).
4 / 27

11. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Snapshot Reduction
Consider the reducible model under inspection to be described by:
y = f (x) , (1)
The algorithm proceeds as follows:
5 / 27

12. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Snapshot Reduction
Consider the reducible model under inspection to be described by:
y = f (x) , (1)
The algorithm proceeds as follows:
1 Generate k random parameters realizations: {xi }k
i=1.
2 Execute the forward model in Eq.[1] k times and record the
corresponding k variations of the responses: yi = f (xi )
k
i=1
,
referred to as snapshots, and aggregate them in a matrix as
follows:
Y = y1 y2 · · · yk ∈ Rm×k
.
3 Calculate the singular value decomposition (SVD):
Y = U VT
; where U ∈ Rm×k
.
5 / 27

13. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Snapshot Reduction (cont.)
4 Select the dimensionality of the reduced space for the responses
to be ry , such that ry ≤ min (m, k). Identify the active subspace
as the range of the first ry columns of the matrix U, denoted by
Ury . Note that in practice ry is increased until the error
upper-bound in step 5 meets a user-defined error tolerance.
5 For a general response y, calculate the error resulting from the
reduction as: ey = I − Ury Ury
T
y .
6 / 27

14. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Gradient-baised Reduction
This algorithm may be described by the following steps:
7 / 27

15. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Gradient-baised Reduction
This algorithm may be described by the following steps:
1 Execute the adjoint model k times, each time with a random
realization of the input parameters, and aggregate the pseudo
response derivatives in a matrix:
G =
dR
pseudo
1
dx
x1
· · ·
dR
pseudo
k
dx
xk
.
7 / 27

16. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Gradient-baised Reduction
This algorithm may be described by the following steps:
1 Execute the adjoint model k times, each time with a random
realization of the input parameters, and aggregate the pseudo
response derivatives in a matrix:
G =
dR
pseudo
1
dx
x1
· · ·
dR
pseudo
k
dx
xk
.
2 Calculate the SVD: G = WSPT
, and select the first rx columns
of W (denoted by Wrx ) to span the active subspace for the
parameters such that:
ex = I − Wrx WT
rx
x .
7 / 27

17. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Notice that discarding components in the parameter space will
give rise to errors in the response space even if no reduction in
the response space is rendered.
8 / 27

18. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Notice that discarding components in the parameter space will
give rise to errors in the response space even if no reduction in
the response space is rendered.
To distinguish between different errors at different levels we
introduce:
8 / 27

19. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Different Errors
1
f (x) − Qy QT
y f (x)
f (x)
≤
y
y ,
where Qy is a matrix whose orthonormal columns span the
response subspace Sy and
y
y is the user-defined tolerance for
the relative error in response due to reduction in response space
only .
9 / 27

20. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Different Errors
1
f (x) − Qy QT
y f (x)
f (x)
≤
y
y ,
where Qy is a matrix whose orthonormal columns span the
response subspace Sy and
y
y is the user-defined tolerance for
the relative error in response due to reduction in response space
only .
2
f (x) − f Qx QT
x x
f (x)
, ≤ x
y
Similarly, Qx is a matrix whose orthonormal columns span an
active subspace Sx in the parameter space and x
y is the
user-defined tolerance for the relative error in response due to
reduction in parameter space only.
9 / 27

21. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Different Errors (cont.)
3
f (x) − Qy QT
y f Qx QT
x x
f (x)
≤
xy
y ,
where
xy
y is the user-defined tolerance for the relative error in
response due to simultaneous reductions in both spaces.
10 / 27

22. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The previous relative errors can be estimated using Dixon’s
Theory[3].
11 / 27

23. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The previous relative errors can be estimated using Dixon’s
Theory[3].
Dixon’s theory
It all started by Dixon(1983) when he estimated the largest
and/or smallest eigen value and hence the condition number of a
real positive definite matrix A.
11 / 27

24. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The previous relative errors can be estimated using Dixon’s
Theory[3].
Dixon’s theory
It all started by Dixon(1983) when he estimated the largest
and/or smallest eigen value and hence the condition number of a
real positive definite matrix A.
His work relies on a basic set of theorems and lemmas[3, 7] that
we will introduce in the following few slides.
11 / 27

25. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Theorem
If A ∈ Rnxn is a real positive definite matrix whose eigen values
are λ1 ≥ λ2 ≥ · · · ≥ λn > 0.
12 / 27

26. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Theorem
If A ∈ Rnxn is a real positive definite matrix whose eigen values
are λ1 ≥ λ2 ≥ · · · ≥ λn > 0.
Let S := x ∈ Rn; xT x = 1 be a unit hyper sphere.
12 / 27

27. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Theorem
If A ∈ Rnxn is a real positive definite matrix whose eigen values
are λ1 ≥ λ2 ≥ · · · ≥ λn > 0.
Let S := x ∈ Rn; xT x = 1 be a unit hyper sphere.
x = x1 · · · xn
T
; n ≥ 2 and xi ∼ U(−1, 1) over S.
12 / 27

28. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Theorem
If A ∈ Rnxn is a real positive definite matrix whose eigen values
are λ1 ≥ λ2 ≥ · · · ≥ λn > 0.
Let S := x ∈ Rn; xT x = 1 be a unit hyper sphere.
x = x1 · · · xn
T
; n ≥ 2 and xi ∼ U(−1, 1) over S.
Let θ ∈ R > 1.
12 / 27

29. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Theorem
If A ∈ Rnxn is a real positive definite matrix whose eigen values
are λ1 ≥ λ2 ≥ · · · ≥ λn > 0.
Let S := x ∈ Rn; xT x = 1 be a unit hyper sphere.
x = x1 · · · xn
T
; n ≥ 2 and xi ∼ U(−1, 1) over S.
Let θ ∈ R > 1.
⇒
P xT
Ax ≤ λ1 ≤ θxT
Ax ≥ 1 −
2
π
n
θ
. (2)
12 / 27

30. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The next corollary has been explored by many authors[8, 5, 6, 4]
and has been employed in different applications, it gave the
modern texture to Dixon’s bound.
Corollary
if B ∈ Rmxn such that A = LLT = BT B; where L = BT is the cholesky
factor of A.
13 / 27

31. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The next corollary has been explored by many authors[8, 5, 6, 4]
and has been employed in different applications, it gave the
modern texture to Dixon’s bound.
Corollary
if B ∈ Rmxn such that A = LLT = BT B; where L = BT is the cholesky
factor of A.
And if σ1 ≥ · · · ≥ σn > 0 are the singular values of B (i.e. λi = σ2
i ).
13 / 27

32. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The next corollary has been explored by many authors[8, 5, 6, 4]
and has been employed in different applications, it gave the
modern texture to Dixon’s bound.
Corollary
if B ∈ Rmxn such that A = LLT = BT B; where L = BT is the cholesky
factor of A.
And if σ1 ≥ · · · ≥ σn > 0 are the singular values of B (i.e. λi = σ2
i ).
Then the previous theorem can be written as:
P Bx ≤ (σ1 = B ) ≤
√
θ Bx ≥ 1 −
2
π
n
θ
. (3)
13 / 27

33. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
The next corollary has been explored by many authors[8, 5, 6, 4]
and has been employed in different applications, it gave the
modern texture to Dixon’s bound.
Corollary
if B ∈ Rmxn such that A = LLT = BT B; where L = BT is the cholesky
factor of A.
And if σ1 ≥ · · · ≥ σn > 0 are the singular values of B (i.e. λi = σ2
i ).
Then the previous theorem can be written as:
P Bx ≤ (σ1 = B ) ≤
√
θ Bx ≥ 1 −
2
π
n
θ
. (3)
Selecting θ = α2 2
π n ; where α > 1 yields:
P B ≤ α
2
π
√
n max
i=1,2,··· ,k
Bx(i) ≥ 1 − α−k . (4)
13 / 27

34. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Propagating error bounds
Consider a physical model:
y = f (x) where f : Rn
→ Rm
.
14 / 27

35. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Propagating error bounds
Consider a physical model:
y = f (x) where f : Rn
→ Rm
.
The model is subjected to both types of reduction at both
parameter and response interfaces. Thes responses are
aggregated in Yx and Yy respectively.
14 / 27

36. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Propagating error bounds
Consider a physical model:
y = f (x) where f : Rn
→ Rm
.
The model is subjected to both types of reduction at both
parameter and response interfaces. Thes responses are
aggregated in Yx and Yy respectively.
The bound for each case is:
x
y = α1
2
π
√
N max
i=1,2,··· ,k1
Y − Yx
wi ,
y
y = α2
2
π
√
N max
i=1,2,··· ,k2
Y − Yy
wi ,
14 / 27

37. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
ROM
Dixon 1983
Propagating the error bound accross different interfaces
Propagating error bounds(cont.)
Then the response error due to both reductions can be
calculated from:
P Y − Yxy
≤ x
y +
y
y ≥ 1 − α
−k1
1 1 − α
−k2
2 (5)
15 / 27

38. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Case Study 1
The first numerical test is an algebraic prototype nonlinear model
where:
y = f (x) ; f : Rn → Rm; n = 15; m = 10 such that:
















y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
















= B ×


















aT
1 x
(aT
2 x)2
(1.4 ∗ aT
2 x + 1.5 ∗ aT
3 x)2
1
1+exp(−aT
2 x)
cos(0.8aT
4 x + 1.6aT
5 x)
(aT
6 x + aT
7 ) ∗ [(aT
7 )2 + sin(aT
8 x)]
(1 + 0.1exp(−aT
8 x))[(aT
9 x)2 + (aT
10x)2]
aT
9 x + 0.2aT
10x
aT
10x
aT
9 x + 8aT
10x


















where ai ∈ Rn; i = 1, 2, · · · , m and B is a random m × m matrix.
16 / 27

39. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Case Study 2
The second case study involves a realistic neutron transport of a
PWR pin cell model.
17 / 27

40. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Case Study 2
The second case study involves a realistic neutron transport of a
PWR pin cell model.
The objective is to test the proposed probabilistic error bound
due to reductions at both the parameter and response spaces.
17 / 27

41. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Case Study 2
The second case study involves a realistic neutron transport of a
PWR pin cell model.
The objective is to test the proposed probabilistic error bound
due to reductions at both the parameter and response spaces.
The computer code employed is TSUNAMI-2D, a control module
in SCALE 6.1 [1], wherein the derivatives are provided by SAMS,
the sensitivity analysis module for SCALE 6.1.
17 / 27

42. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Case Study 1
The dimension of the parameter space is n = 15, and the response space is
m = 10, and a user defined tolerance of 10−5 is selected. The parameter active
subspace is found to have a size of rx = 9 whereas the response active
subspace is ry = 9.number of tests is 10000.
Fig. 1 shows the function behavior plotted along a randomly selected direction in
the parameter space.
Figure : Function behavior along a random input direction.
18 / 27

43. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Table I. shows the minimum theoretical probabilities
Pact = number of successes
total number of tests predicted by the theorem and the actual
probability resulting from the numerical test.
Table : Algebraic Model Results
Error Bound Pact Ptheo
Y−Yx
Y ≤ x
y 1.0 0.9
Y−Yy
Y ≤
y
y 0.998 0.9
Y−Yxy
Y ≤ x
y +
y
y 1.0 0.81
19 / 27

44. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Relative Errors
Next we show the relative error Y−Yxy
Y due to both reductions
vs. the theoretical upper bound predicted by the theory x
y +
y
y .
Figure : Theoretical and actual error for case study 1.
20 / 27

45. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Case Study 2
For the pin cell model the full input subspace (cross sections) had a size
of n = 1936, whereas the output (material flux) was of size m = 176.
The cross-sections of the fuel, clad, moderator and gap were perturbed
by 30%(relative perturbations). Based on a user defined tolerance of
10−5, the sizes of the input and output active subspaces are rx = 900
and ry = 165, respectively.
Table II shows the minimum theoretical probabilities predicted by the
theorem and the probability resulted from the numerical test.
Table : Algebraic Model Results
Error Bound Pact Ptheo
Y−Yx
Y ≤ x
y 1.0 0.9
Y−Yy
Y ≤
y
y 1.0 0.9
Y−Yxy
Y ≤ x
y +
y
y 1.0 0.81
21 / 27

46. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Case Study 1
Case Study 2
Relative Errors
Next we show the relative error Y−Yxy
Y due to both reductions
vs. theoretical upper bound predicted by the theory x
y +
y
y .
Figure : Theoretical and actual error for case study 2.
22 / 27

47. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Conclusions
This manuscript has equipped our previously developed ROM
techniques with probabilistic error metrics that bound the
maximum errors resulting from the reduction.
Given that reduction algorithms can be applied at any of the
various model interfaces, e.g., parameters, state, and responses,
the developed metric effectively aggregates the associated errors
to estimate an error bound on the response of interest.
The results show that we can start to break the linear moulding
and start to explore nonlinear smooth functions.
These functionality will prove essential in our ongoing work
focusing on extension of ROM techniques to multi-physics
models.
23 / 27

48. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Bibliography I
SCALE:A Comperhensive Modeling and Simulation Suite for
Nuclear Safety Analysis and Design,ORNL/TM-2005/39, Version
6.1, Oak Ridge National Laboratory, Oak Ridge, Tennessee,June
2011. Available from Radiation Safety Information Computational
Center at Oak Rodge National Laboratory as CCC-785.
Y. BANG, J. HITE, AND H. S. ABDEL-KHALIK, Hybrid reduced
order modeling applied to nonlinear models, IJNME, 91 (2012),
pp. 929–949.
J. D. DIXON, Estimating extremal eigenvalues and condition
numbers of matrices, SIAM, 20 (1983), pp. 812–814.
24 / 27

49. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Bibliography II
N. HALKO, P. G. MARTINSSON, AND J. A. TROPP, Finding
structure with randomness:probabilistic algorithms for
constructing approximate matrix decompositions, SIAM, 53
(2011), pp. 217–288.
P. G. MARTINSSON, V. ROKHLIN, AND M. TYGERT, A
randomized algorithm for the approximation of matrices, tech.
report, Yale University.
J. A. TROPP, User-friendly tools for random matrices.
S. S. WILKS, Mathematical statistics, John Wiley, New York,
1st ed., 1962.
25 / 27

50. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Bibliography III
F. WOOLFE, E. LIBERTY, V. ROKHLIN, AND M. TYGERT, A fast
randomized algorithm for the approximation of matrices,
preliminary report, Yale University.
26 / 27

51. Motivation
Background of Supporting Algorithms and Theory
Numerical tests and results
Conclusions
Bibliography
Thanks
Questions/Suggestions?
27 / 27