Model Order Reduction for Large Scale Dynamical Systems Lecture 2: Tools From Matrix Theory I what y

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Model Order Reduction for Large Scale Dynamical
SystemsLecture 2 Tools From Matrix Theory
I(what you ought to know already)
NXP Semiconductors
Tamara Bechtold
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Outline

• Vector Spaces with Scalar Product
• Linear Mapping
• Projectors
• Order Reduction via Projection
• Summary

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Outline

• Vector Spaces with Scalar Product
• Linear Mapping
• Projectors
• Order Reduction via Projection
• Summary

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Matrices and Vectors in Rn

• DEFINITION An m x n matrix in Rn is a
  rectangular scheme of mn real numbers, which are
  called elements and are ordered in m rows and n
  columns. An m x 1 matrix is called column vector.
  An 1 x n matrix is called row vector.
• But what is it really?
• In the real world you start from the physical
  phenomena, which might take place in each point
  of the space (e. g. thermal flow) or in some
  points of space (e. g. electrical current flow).
  In both cases you might end up in an equation
  system of the form
• Hence, for us the matrix is something that
  multiplies the solution/state vector.

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Remember
control
simulation
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Scalar (Inner) Product

• DEFINITION Scalar product or inner product of
  two vectors x,y from Rn is the number
• THEOREM Scalar product in Rn is
• (S1) Linear
• (S2) Symmetric
• (S3) Positive definite

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Vector Spaces

• DEFINITION Vector space V over R is a non-empty
  assembly upon which an addition
  and a scalar multiplication
  
  are defined with the following rules
• Elements of V are called vectors. is
  called zero vector. The elements of R are called
  scalars.

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Subspaces spanned by

• DEFINITION A non-empty sub-assembly U from the
  vector space V is called subspace, if it is
  closed for addition and scalar multiplication
• THEOREM A subspace is also a vector space.
• DEFINITION All linear combinations of vectors
  a1, ,al from V are called subspace spanned by
  a1, ,al (spanning set). We write
• In connection with the dynamical system
• We will define Krylov or reachability subspaces

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Linear Independancy, Basis, Dimension

• DEFINITION lgt2 vectors a1, ,al are linear
  dependant if one of them can be represented as a
  linear combination of others.
• DEFINITION A linear independent spanning set of
  space V is called basis of V. Hence, n columns
  of a nxn matrix B are
  a basis of Rn, if B is regular.
• DEFINITION Number of basis vectors of the vector
  space V (with finite spanning set) is called the
  dimension of V.
• THEOREM Each vector can be uniquely represented
  as a linear combination of the basis vectors b1,
  ,bn of V
• DEFINITION Coefficients are called the
  coordinates of x with respect to the basis b1,
  ,bn . Vector is a
  coordinate vector of x.

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Orthonormal Basis

• DEFINITION A basis is called orthogonal if each
  two basis vectors are mutually orthogonal
• It is called orthonormal if the basis
  vectors are unity vectors
• Note Coordinates in respect to the orthonormal
  basis are
• In other words only in the orthonormal basis,
  one can set
• It follows that for matrix Bb1, ,bn holds
• COROLLARY In each vector space with scalar
  product and finite dimension, there exist an
  orthonormal basis. Gram-Schmidt algorithm
  constructs it from an arbitrary basis (we will
  come back to it later).

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Change of Basis, Coordinate Transformation

• THEOREM Let be a
  coordinate vector of arbitrary vector with
  respect to the old basis Bb1, ,bn and let
  be a coordinate vector of
  x with respect to some new basis Bb1, ,bn
  
• Than the coordinate transformation can be
  expressed via the regular transformation matrix T
  as
• Change of basis from B to B can be expressed as

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Orthogonal Change of Basis

• How does the transformation matrix T look like,
  if both basis B and B are orthonormal?
• THEOREMTransformation matrix of the change
  between two orthonormal basis is orthogonal
• Note An orthogonal matrix is square and has
  orthonormal columns. if it is not square it is
  called a matrix with orthogonormal columns if
  the columns are not normalized it is called a
  matrix with orthogonal columns.

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Range, Nullspace, Rank

• DEFINITION The range of a matrix A (map F),
  written range(A) or R(A), is the set of vectors
  that can be expressed as Ax for some x.
• THEOREM range(A) is the space spanned by the
  columns of A.
• DEFINITION The null-space (kernel) of A, written
  null(A) or N(A), is the set of vectors x that
  satisfy Ax 0, where 0 is the 0 - vector in Rm.
• DEFINITION The column rank of a matrix is the
  dimension of its column space. Similarly, the row
  rank of a matrix is the dimension of the space
  spanned by its rows. They are always equal!
• An m x n matrix of full rank is one that has the
  maximal possible rank, that is min(m,n). Hence, a
  matrix of full rank with m n must have n
  linearly independent columns.
• THEOREM A matrix with m n
  has full rank if and only if it maps no two
  distinct vectors to the same vector.

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Inverse

• DEFINITION A nonsingular or invertable matrix A
  is a square matrix of full rank. Its inverse is
  marked with A-1 and it fulfills
• THEOREM For , the following
  conditions are equivalent
• (a) A has an inverse A-1
• (b) rank(A)n
• (c) range(A) Rn
• (d) null(A) 0
• (e) 0 is not an eigenvalue of A
• (f) 0 is not a singular value of A
• (g) det(A) ? 0
• Moore-Penrose pseudoinverse given
  find X that solves
• is called
  generalized (Moore-Penrose) inverse that
  satisfies

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Outline

• Vector Spaces with Scalar Product
• Linear Mapping
• Projectors
• Order Reduction via Projection
• Summary

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Matrices as Linear Mapping

• DEFINITION A mapping
  between two vector spaces X and Y (over
  R) is called linear, if for all
  and all
• X is called domain and Y is called
  image(range) of F.
• Let be a m x n matrix. The
  assignment is linear mapping
  from Rn to Rm, which we denote with A
• EXAMPLE Coordinate mapping

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Mapping with Coordinate Transformation

• Let X and Y be a vector spaces of dimensions n
  and m and let

(coordinate mapping with respect to the old
bases)
(coordinates with respect to the old bases)
(coordinate transformations)
(coordinates with respect to the new bases)
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Main Question

• How far can we simplify the mapping matrix by
  choosing the proper bases?

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Outline

• Vector Spaces with Scalar Product
• Linear Mapping
• Projectors
• Order Reduction via Projection
• Summary

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Projectors

• DEFINITION A linear mapping
  (square matrix) is called projection if it
  fulfills

Oblique projection
Orthogonal projection
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Complementary Projectors

• DEFINITION If P is a projector, I P is also a
  projector, as
• The matrix I P is called the complementary
  projector to P.
• Onto what space does I P project?
• We can also see that
  . This shows that P separates
  Rn into two spaces, S1 range(P) and S2
  null(P), such that
• S1 and S2 are said to be the complementary
  subspaces.
• We say that P is the projector onto S1 along S2.

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Orthogonal Projectors

• DEFINITION An orthogonal projector is one that
  projects onto a subspace S1 along S2, where, S1
  and S2 are orthogonal. (Warning orthogonal
  projectors are not orthogonal matrices!)
• THEOREM A projector P is orthogonal if and only
  if P PT.
• THOREM An orthogonal projection
  onto the column
  space R(A), of a matrix with
  rank n(m) is given by
• THOREM An orthogonal projection
  onto the column
  space R(Q), of a m x n matrix Q (q1qn) with
  orthonormal columns is given by
• THEOREM For an orthogonal projection is vallied

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Rank-One Orthogonal Projection

• THEOREM An m x n matrix A has rank 1 if and only
  if it can be expressed as an outer product of
  and
• THEOREM The orthogonal projection Pq of
  onto direction defined by a unit vector q is
  given by
• For arbitrary nonzero direction vector a, the
  analogous formula is
• Note that any higher rank projection
  can be made as a sum of rank-one
  projections

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DEMO Orthogonale Projections

• The following demo (with German comments) can be
  found at
• http//ismi.math.uni-frankfurt.de/analysisfuerinfo
  rmatiker/Vorlesung8a.pdf

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Gram-Schmidt Orthogonalization

• ALGORITHM Let a1,a2, be a finite, linear
  independent assembly of vectors. We compute the
  same number of vectors b1, b2, as
• THEOREM After k steps of Gram-Schmidt algorithm
  we have
• If a1, ,ak is a basis of V, so is b1, ,bk
  an othonormal basis of V.

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Gram-Schmidt in R2
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Gram-Schmidt in R3
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Outline

• Vector Spaces with Scalar Product
• Linear Mapping
• Projectors
• Order Reduction via Projection
• Summary

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Recapitulation of Lecture 1

• Projection constitutes a unifying feature of MOR
  methods
• Projection can be seen as a truncation in an
  appropriate basis

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Transformation

• Remember, we need to project
• The change of basis
  leads to
• In general we can always partition x, T and T-1
  as
• This turns the original system into
• Attention still no approximation!

Remember T is regular
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Truncation

• The approximation occurs if it turns out that x2
  can be neglected (if e. g. due to the change of
  base ) and so the T1 and T2
  entries can be truncated as well
• Note the solution vector has been
  approximated by
• In case when T is orthonormal

(Petrov-Galerkin approximation)
(Galerkin approximation)
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Where is the Projection?

• If we replace into the
  above equation, we get

/V (from the left)
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In General Case We Have Oblique Projection

• If we replace into the
  above equation, we get
• Remember the complementary subspaces S1
  range(P) and S2 null(P). In this case P VWT,
  S1 range(VWT) range(V), S2 null(VWT)
  null(WT), because V has a full rank r.

/V (from the left)
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Orthogonal Vs Oblique

• Main question why the oblique projection at all?
• Remember WTAV and VTAV are the matrices that
  multiply the state/solution vector! Hence, we
  want to transform(decompose) them to the simplest
  possible form. In some cases Petrov-Galerkin
  approximation might be easier to compute than the
  Galerkin one.

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MOR as Mapping?
(coordinate transformations)
Homework how can MOR be interpreted via mapping?
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Outline

• Vector Spaces with Scalar Product
• Linear Mapping
• Projectors
• Order Reduction via Projection
• Summary

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Summary

• Linear algebra tools are crucial for MOR
• Unifying feature of MOR is a projection (Galerkin
  or Petrov-Galerkin approximation)
• Matrix x state vector should be simple to compute
• Next lecture Matrix approximations

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Thank you