This document discusses the minimum fill-in problem for sparse matrices. It begins with an introduction to fill-in that occurs during Gaussian elimination due to the introduction of new non-zero elements. It describes how the minimum fill-in problem is NP-hard and discusses various heuristics to minimize fill-in, including minimum degree ordering and nested dissection. The minimum degree algorithm works by repeatedly eliminating the vertex with minimum degree but does not always produce optimal orderings. The document provides examples to illustrate minimum degree and discusses enhancements like mass elimination to improve its performance.
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
New data structures and algorithms for \\post-processing large data sets and ...Alexander Litvinenko
In this work, we describe advanced numerical tools for working with multivariate functions and for
the analysis of large data sets. These tools will drastically reduce the required computing time and the
storage cost, and, therefore, will allow us to consider much larger data sets or ner meshes. Covariance
matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and
store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a
low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of
Matern- and Slater-type functions with varying parameters and demonstrate numerically that their
approximations exhibit exponentially fast convergence. We prove the exponential convergence of the
Tucker and canonical approximations in tensor rank parameters. Several statistical operations are
performed in this low-rank tensor format, including evaluating the conditional covariance matrix,
spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood,
inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations
reduce the computing and storage costs essentially. For example, the storage cost is reduced from an
exponential O(nd) to a linear scaling O(drn), where d is the spatial dimension, n is the number of
mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed
techniques are the assumptions that the data, locations, and measurements lie on a tensor (axesparallel)
grid and that the covariance function depends on a distance,...
Before you operate a heavy haulage truck, it is essential that you take all of the necessary precautions. While safety precautions when driving are important regardless of what you drive, statistics reveal that heavy haulage truck drivers are at a heightened risk of collision and accidents. If you are a heavy haulage driver, consider these precautionary tips before you get behind the wheel.
Read more: http://www.nationalheavyhaulage.com.au/news/208-preparing-to-drive-a-heavy-haulage-truck
On maximal and variational Fourier restrictionVjekoslavKovac1
Workshop talk slides, Follow-up workshop to trimester program "Harmonic Analysis and Partial Differential Equations", Hausdorff Institute, Bonn, May 2019.
New data structures and algorithms for \\post-processing large data sets and ...Alexander Litvinenko
In this work, we describe advanced numerical tools for working with multivariate functions and for
the analysis of large data sets. These tools will drastically reduce the required computing time and the
storage cost, and, therefore, will allow us to consider much larger data sets or ner meshes. Covariance
matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and
store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a
low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of
Matern- and Slater-type functions with varying parameters and demonstrate numerically that their
approximations exhibit exponentially fast convergence. We prove the exponential convergence of the
Tucker and canonical approximations in tensor rank parameters. Several statistical operations are
performed in this low-rank tensor format, including evaluating the conditional covariance matrix,
spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood,
inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations
reduce the computing and storage costs essentially. For example, the storage cost is reduced from an
exponential O(nd) to a linear scaling O(drn), where d is the spatial dimension, n is the number of
mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed
techniques are the assumptions that the data, locations, and measurements lie on a tensor (axesparallel)
grid and that the covariance function depends on a distance,...
Before you operate a heavy haulage truck, it is essential that you take all of the necessary precautions. While safety precautions when driving are important regardless of what you drive, statistics reveal that heavy haulage truck drivers are at a heightened risk of collision and accidents. If you are a heavy haulage driver, consider these precautionary tips before you get behind the wheel.
Read more: http://www.nationalheavyhaulage.com.au/news/208-preparing-to-drive-a-heavy-haulage-truck
For those who are planning to travel China, here are 6 essential money travel tips that lets you enjoy your upcoming China tour without spending a fortune.
Ambientes interativos e os impactos nos usuários (Aula 1)Andre de Abreu
Conteúdo da disciplina "Ambientes interativos e os impactos nos usuários", ministrada na pós-graduação em comunicação digital da USP (www.digicorpecausp.net). Na primeira aula, o conceito de interatividade e como as interações humanas são potencializadas pelas internet e pelas rede sociais digitais.
Apresentação sobre o conceito de Indisciplinaridade (ou ausência de fronteiras entre campos do conhecimento) feita para o "Encontro de Comunicação Digital da ABRP-SP (http://migre.me/6VfA).
Life as the King of the Road can provide a rewarding career. But no-one said being in control of a heavy vehicle carrying expensive cargo and meeting critical deadlines was easy. To see if you’re cut out for truck driving, have a look at the Top 10 qualities that all great truck drivers share.
Read more: http://www.nationalheavyhaulage.com.au/news/213-top-10-qualities-of-a-great-truck-drive
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
Review for the Third Midterm of Math 150 B 11242014Probl.docxjoellemurphey
Review for the Third Midterm of Math 150 B 11/24/2014
Problem 1
Recall that 1
1−x =
∑∞
n=0 x
n for |x| < 1.
Find a power series representation for the following functions and state the radius of
convergence for the power series
a) f(x) = x
2
(1+x)2
.
b) f(x) = 2
1+4x2
.
c) f(x) = x
4
2−x.
d) f(x) = x
1+x2
.
e) f(x) = 1
6+x
.
f) f(x) = x
2
27−x3 .
Problem 2
Find a Taylor series with a = 0 for the given function and state the radius of conver-
gence. You may use either the direct method (definition of a Taylor series) or known
series.
a) f(x) = ln(1 + x)
b) f(x) = sin x
x
c) f(x) = x sin(3x).
Problem 3
Find the radius of convergence and interval of convergence for the series
∑∞
n=1
(x+2)n
n4n
.
Ans. Radius r=2,
√
2 − 2 < x <
√
2 + 2. Problem 4
Find the interval of convergence of the following power series. You must justify your
answers.
∑∞
n=0
n2(x+4)n
23n
.
Ans. −12 < x < 4.
Problem 5
For the function f(x) = 1/
√
x, find the fourth order Taylor polynomial with a=1.
Problem 6
A curve has the parametric equations
x = cos t, y = 1 + sin t, 0 ≤ t ≤ 2π
a) Find dy
dx
when t = π
4
.
b) Find the equation of the line tangent to the curve at t = π/4. Write it in y = mx+b
form.
c) Eliminate the parameter t to find a cartesian (x, y) equation of the curve.
d) Using (c), or otherwise identify the curve.
Problem 7
State whether the given series converges or diverges
a)
∑∞
n=0 (−1)
n+1 n22n
n!
.
b)
∑∞
n=0
n(−3)n
4n−1
.
c)
∑∞
n=1
sin n
2n2+n
.
Problem 8
1
Approximate the value of the integral
∫ 1
0
e−x
2
dx with an error no greater than 5×10−4.
Ans.
∫ 1
0
e−x
2
dx = 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ ... +
(−1)n
(2n+1)n!
+ .... n ≥ 5,
for n=5
∫ 1
0
e−x
2
dx ≈ 1 − 1
3
+ 1
5.2!
− 1
7.3!
+ 1
9.4!
− 1
11.5!
≈ 0.747.
Problem 9
Find the radius of convergence for the series
∑∞
n=1
nn(x−2)2n
n!
.
Ans. R = 1√
e
.
Problem 10
Let f(x) =
∑∞
n=0
(x−1)n
n2+1
.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate the domain of f ′.
Problem 11
Let f(x) =
∑∞
n=0
cos n
n!
xn.
a) Calculate the domain of f.
b) Calculate f ′(x).
c) Calculate
∫
f(x)dx.
Problem 12
Using properties of series, known Maclaurin expansions of familiar functions and their
arithmetic, calculate Maclaurin series for the following.
a) ex
2
b) sin 2x
c)
∫
x5 sin xdx
d) cos x−1
x2
e)
d((x+1) tan−1(x))
dx
Problem 13
Calculate the Taylor polynomial T5(x), expanded at a=0, for
f(x) =
∫ x
0
ln |sect + tan t|dt.
Ans. T5(x) =
x2
2
+ x
4
4!
.
Problem 14
Suppose we only consider |x| ≤ 0.8. Find the best upper bound or maximum value
you can for∣∣∣sin x − (x − x33! + x55! )∣∣∣
Same question: If
(
x − x
3
3!
+ x
5
5!
)
is used to approximate sin x for |x| ≤ 0.8. What is
the maximum error? Explain what method you are using.
Problem 15
The Taylor polynomial T5(x) of degree 5 for (4 + x)
3/2 is
(4 + x)3/2 ≈ 8 + 3x + 3
16
x2 − 1
128
x3 + 3
4096
x4 − 3
32768
x5.
a) Use this polynomial to find Taylor polynomials for (4 + ...
Gauss jordan and Guass elimination methodMeet Nayak
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
Joel Spencer – Finding Needles in Exponential Haystacks Yandex
We discuss two recent methods in which an object with a certain property is sought. In both, using of a straightforward random object would succeed with only exponentially small probability. The new randomized algorithms run efficiently and also give new proofs of the existence of the desired object. In both cases there is a potentially broad use of the methodology.
(i) Consider an instance of k-SAT in which each clause overlaps (has a variable in common, regardless of the negation symbol) with at most d others. Lovasz showed that when ed < 2k (regardless of the number of variables) the conjunction of the clauses was satisfiable. The new approach due to Moser is to start with a random true-false assignment. In a WHILE loop, if any clause is not satisfied we ”fix it” by a random reassignment. The analysis of the algorithm is unusual, connecting the running of the algorithm with certain Tetris patterns, and leading to some algebraic combinatorics. [These results apply in a quite general setting with underlying independent ”coin flips” and bad events (the clause not being satisfied) that depend on only a few of the coin flips.]
(ii) No Outliers. Given n vectors rj in n-space with all coefficients in [−1,+1] one wants a vector x = (x1, ..., xn) with all xi = +1 or −1 so that all dot products x · rj are at most K √ n in absolute value, K an absolute constant. A random x would make x · rj Gaussian but there would be outliers. The existence of such an x was first shown by the speaker. The first algorithm was found by Nikhil Bansal. The approach here, due to Lovett and Meka, is to begin with x = (0, ..., 0) and let it float in a kind of restricted Brownian Motion until all the coordinates hit the boundary.
Solving connectivity problems via basic Linear Algebracseiitgn
Directed reachability and undirected connectivity are well studied problems in Complexity Theory. Reachability/Connectivity between distinct pairs of vertices through disjoint paths are well known but hard variations. We talk about recent algorithms to solve variants and restrictions of these problems in the static and dynamic settings by reductions to the determinant.
2. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
3. Sparse Systems of Linear Equations Ax = b
sparse, symmetric, positive definite matrices arise frequently
in physical applications such as
• finite element method
• electrical network analysis
• analysis of structural systems
famous direct methods to solve a linear system Ax = b are
• Gauss Elimination
• Cholesky Factorization
8. Disadvantages of Fill-In
Fill-In increases the storage requirement
Fill-In increases the arithmetic operations needed to solve the system
Questions that arise:
1. Are there instances where no fill is introduced?
2. Is there a way to minimize the Fill-In introduced during elimination ?
9. Modelling the Elimination Process on a Graph
Let the graph G = (V, E) correspond to A and let the vertex vi correspond to xi.
Theorem: Elimination Graph [Parter 1961] Eliminating xi from the subsequent
equations, the new graph Gvi
of the remaining system is formed by:
1. eliminating the vertex vi that corresponds to xi from Gvi−1
2. pair-wise connecting the vertices of N(vi).
v1
v2 v2
v3 v3 v3
v4v4 v4v4
G Gv1
Gv2
Gv3
→→ →
10. Modelling the Elimination Process on a Graph
Proof:
assume aii = 0, then xi = −1/aii k=i aikxk + yi/aii
if aij = 0 no substitution need be made
if aij = 0 and aik = 0 then
• introduce xj to the kth equation and
• introduce xk to the jth equation
12. Solving Ax = b with Cholesky
We can decompose A into A = LLT
We know aij =< li, lj >
constructing L
• Algorithm at blackboard
A = LLT
= LU, where LT
= U
14. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
15. Perfect Elimination
Definition: [Rose 1972] A is a perfect elimination matrix if there exists a permutation matrix
P such that ˜A = PAPT
= LLT
and aij = 0 ⇒ lij = 0, i < j.
Definition: For a graph G = (V, E) with |V | = n an ordering of V is a bijection
σ : {1, 2, . . . , n} → V .
1 1
22
3 3
4 4
v1v1 v1v1
v2v2 v2v2
v3v3 v3v3
v4 v4v4 v4
×
×
×
× ×
×
×
××
×
×
×
×
×
×
×
×
×
×
×
×
×
××
×
×
×
××
×
×
×××
•
••
→
LT
= LT
=
16. Chordal Graph (Hanjal und Suranyi [1958])
Definition: A Graph G = (V, E) is chordal if every cycle of length ≥ 4 has a chord.
A chord is an edge connecting two nonconsecutive vertices of the cycle.
G1
chordal
G2
not chordal
G3
not chordal
The class of chordal graphs is the class of perfect elimination graphs.
17. Definition: The ordered graph Gσ is monotone transitive when
∀v ∈ V (v, u1) ∈ E and (v, u2) ∈ E =⇒ (u1, u2) ∈ E,
with σ−1
(v) < σ−1
(u1) and σ−1
(v) < σ−1
(u2)
Theorem: (Rose [1972]) For G = (V, E) the following statements are equivalent:
A is a perfect elimination matrix.
∃ an ordering σ of V such that Gσ is monotone transitive
σ is a perfect elimination order
G is chordal
36. The filled graph G+
v1 v2 v3
v4 v5 v6
G+
σ1
The implicit ordering σ1 of the vertices causes a fill of 4.
G+
σ1
= (V, E ∪ F) is the filled graph, where |F| is the size of the fill.
38. Chordal Completion
G+
σ is monotone transitive
Eliminate vertices from G+
σ according to the same order σ used to obtain G+
σ from G
G+
= (V, E ∪ F) is chordal.
G+
is the chordal completion of G
v1 v1v2 v2v3 v3
v4 v4v5 v5v6 v6
G+
σ2
G+
σ1
39. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
40. Problem Formulation
Minimum Fill-In Problem
Instance: Graph G = (V, E)
Task: Find a set of edges F of smallest size such that
G = (V, E ∪ F) is chordal.
also known as Minimum chordal graph completion Problem
41. Minimum Fill-In is NP-hard
SAT
↓
3 - SAT
↓
Simple Max Cut
↓
Simple Optimal Linear Arrangement ⇒ Optimal Linear Arrangement
↓
Minimum Fill-In
42. Minimum Fill-In vs. Minimal Fill-In
G minimal
chordal completion
minimum
chordal completion
• minimal chordal completion (inclusion-minimal set of edges) can be computed in poly-
nomial time
• can be far from minimum
43. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
44. Minimum Degree Algorithm
repeatedly finds a vertex of minimum degree and eliminates it
Enhancements
• Mass Elimination using Supernodes [George, Liu 1980]
• Multiple Elimination [Liu 1985]
• External Degree [Liu 1985]
• Approximate Degree [Amestoy, Davis, Duff 1996]
fast algorithm, runs in O(nm+
), where m+
= |E ∪ F|
52. Advantages - Disadvantages Minimum Degree Algorithm
Advantages Disadvantages
fast algorithm does not recognize chordal graphs
does not in general produce a minimal ordering
triangulation produced is arbitrarily
greater than a minimum triangulation
74. Minimum Degree Algorithm
repeatedly finds a vertex of minimum degree and eliminates it
Enhancements
• Mass Elimination using Supernodes [George, Liu 1980]
• Multiple Elimination [Liu 1985]
• External Degree [Liu 1985]
• Approximate Degree [Amestoy, Davis, Duff 1996]
fast algorithm
75. Quotientgraph and Mass Elimination
1. Quotientgraph - Supernodes and Enodes, Reachable Set
2. Mass Elimination - u, v ∈ V are indistinguishable in G if NG(u)∪{u} = NG(v)∪{v}
76. Minimum Degree Algorithm
repeatedly finds a vertex of minimum degree and eliminates it
Enhancements
• Mass Elimination using Supernodes [George, Liu 1980]
• Multiple Elimination [Liu 1985]
• External Degree [Liu 1985]
• Approximate Degree [Amestoy, Davis, Duff 1996]
fast algorithm
77. Multiple Elimination
Determine the minimum degree
Choose the vertices that have the minimum degree
Mark their neighbors as unknown degree
98. Minimum Degree Algorithm
repeatedly finds a vertex of minimum degree and eliminates it
Enhancements
• Mass Elimination using Supernodes [George, Liu 1980]
• Multiple Elimination [Liu 1985]
• External Degree [Liu 1985]
• Approximate Degree [Amestoy, Davis, Duff 1996]
fast algorithm
99. External and Approximate Degree
For a set of vertices X ⊂ V , N(X) = v∈X N(v) − X
external degree of X is |N(X)|, that is the number of adjacent vertices that belong to
other supernodes. NOT the number of adjacent supernodes
approximate degree computes an upper bound on the degrees inexpensively instead of
computing the exact degrees
101. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
102. Minimum Deficiency
The deficiency of v is the number of edges needed to turn the neighborhood
of a vertex v into a clique
The deficiency corresponds to the number of fill edges introduced
by eliminating v
Try to minimize the overall amount of fill by, at each step,
choosing to eliminate a vertex with minimum deficiency
110. Advantages - Disadvantages Minimum Deficiency Algorithm
Advantages Disadvantages
good quality ordering implementation too expensive
recognizes chordal graphs not always a minimal triangulation
1
1
1
1
1
11
33
111. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
112. Nested Dissection [George 1973]
Nested Dissection examines the graph as a whole before ordering
• orders the vertices of the graph backwards
• begins by deciding which vertices should be eliminated last
works by selecting a balanced separator
• a set of vertices, that when removed from the graph, partitions it into connected
components
vertices in the separator are placed last in the elimination order
recursively orders each of the connected components
118. Performance Bound
At most O(n log n) fill for graphs with small separators (O(
√
n))[Lipton, Rose, Tarja 1979]
• planar graphs
• graphs with bounded genus
• graphs with bounded degree
Approximation algorithm [Agraval, Klein, Ravi 1990] with performance bound O(
√
d log4
n)
• for bounded degree graphs
Theoretical result [Gilbert 1989]: for any graph there exisits a ND algorithm whose fill is
within O(d log n) of minimum, where d is the maximum degree.
119. Advantages - Disadvantages Nested Dissection
Advantages Disadvantages
good ordering for limited practical use
solving Ax = b in parallel
finding a minimum balanced separator
is NP-hard
no fast algorithms for
finding approximate separators
120. Questions left
What properties should the vertex separator have?
What algorithms should be used to find the vertex separator?
When should the nested recursion be halted?
How should the separator vertices be ordered?
In Practice
When connected component small enough use Minimum Degree Algorithm
121. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
122. Hybrid Algorithms
take advantage of of the best characteristics of
Minimum Degree and Nested Dissection
using a few levels of separators to control the fill
introduced by minimum degree orders
Hendrickson and Rothberg[1996] is the current champion
of ordering algorithms - BEND (Bruce and Ed’s Nested Dissection)
algorithm has neither known worst case fill nor work analysis
123. BEND - Combining Nested Dissection and Minimum Degree
compress the graph
Nested Dissection finds a few levels of separators until
connected components have size at most n/32 vertices
• using a multilevel algorithm to find vertex separators,
allowing some imbalance
invoke Minimum Degree on the subgraphs
finally apply Minimum Degree to reorder all separator vertices
124. Difference between Nested Dissection and BEND
Nested Dissection BEND
select a balanced separator use multilevel to find separator
allowing some imbalance
recursively order each subgraph recursively order each subgraph
until small enough until n/32
vertices in separator are reorder separator vertices
placed last in any order using Minimum Degree
125. Overview
Introduction
• What is Fill-In ?
• Perfect Elimination and Chordal Graphs
• The Minimum Fill-In Problem
Heuristics
• Minimum Degree
• Minimum Deficiency
• Nested Dissection
• Hybrid Algorithm
Approximation Algorithm
126. Approximation Algorithm
Approximation Ratio is O(8 · OPT)
G = (V, E) has minimum fill size k
identify in G a kernel set of vertices A ⊆ V , |A| ≤ 4k
• using ideas of the partition algorithm
•
triangulate G by adding only edges between vertices of A
no prior knowledge of k