Oblique Projection Methods

In an oblique projection method we are given two subspaces_Land_Kand seek an approximationu˜ ∈ Kand an element˜λofC_{that satisfy the Petrov-Galerkin} condition,

((A−λI˜ )˜u, v) = 0 ∀v∈ L. (4.41) The subspace_Kwill be referred to as the right subspace and_Las the left subspace. A procedure similar to the Rayleigh-Ritz procedure can be devised by again trans- lating in matrix form the approximate eigenvectoru˜in some basis and expressing the Petrov-Galerkin condition (4.41). This time we will need two bases, one which we denote byV for the subspace_Kand the other, denoted byW,for the subspace

L. We assume that these two bases are biorthogonal, i.e., that(vi, wj) =δij,or

WH_{V = I}

whereIis the identity matrix. Then, writingu˜=V yas before, the above Petrov- Galerkin condition yields the same approximate problem as (4.20) except that the matrixBmis now defined by

Bm = WHAV.

We should however emphasize that in order for a biorthogonal pairV, W to exist the following additional assumption for_Land_Kmust hold.

For any two basesV andW of_Kand_Lrespectively,

det(WHV)6= 0 . (4.42)

In order to interpret the above condition in terms of operators we will define the oblique projector_QL_K onto_Kand orthogonal to_L. For any given vectorxin Cn_,the vector_QL_Kxis defined by

( QL

Kx∈ K x− QL

Kx⊥ L.

Note that the vector_QL_Kxis uniquely defined under the assumption that no vector of the subspace_Lis orthogonal to_K. This fundamental assumption can be seen to be equivalent to assumption (4.42). When it holds the Petrov-Galerin condition (4.18) can be rewritten as

QLK(Au˜−˜λu˜) = 0 (4.43) or

QLKAu˜ = ˜λ˜u .

Thus, the eigenvalues of the matrixAare approximated by those ofA′_=QL

KA|K. We can define an extensionAmofA′

manalogous to the one defined in the pre-

occurrences ofu˜in the above equation would lead toAm=QL

KAQ

L

K. In order to be able to utilize the distance_k(I− PK)uk2in a-priori error bounds a more useful extension is

Am = QLKAPK.

With this notation, it is trivial to extend the proof of Proposition 4.3 to the oblique projection case. In other words, when_Kis invariant, then no matter which left subspace_Lwe choose, the oblique projection method will always extract exact eigenpairs.

We can establish the following theorem which generalizes Theorem 4.3 seen for the orthogonal projection case.

Theorem 4.7 Letγ=kQL

K(A−λI)(I−PK)k2. Then the following two inequal-

ities hold: k(Am−λI)PKuk2≤γk(I− PK)uk2 (4.44) k(Am−λI)uk2≤ p |λ|2_+γ2_{k(I− P} K)uk2. (4.45)

Proof. For the first inequality, since the vector_PKybelongs toKwe haveQ

L KPK = PK and therefore (Am−λI)PKu = Q L K(A−λI)PKu = QL K(A−λI)(PKu−u) = −QL K(A−λI)(I− PK)u . Since(I− PK)is a projector we now have

(Am−λI)PKu=−Q

L

K(A−λI)(I− PK)(I− PK)u.

Taking Euclidean norms of both sides and using the Cauchy-Schwarz inequality we immediately obtain the first result.

For the second inequality, we write

(Am−λI)u = (Am−λI) [PKu+ (I− PK)u]

= (Am−λI)PKu+ (Am−λI)(I− PK)u . Noticing thatAm(I− PK) = 0this becomes

(Am−λI)u= (Am−λI)PKu−λ(I− PK)u .

Using the orthogonality of the two terms in the right hand side, and taking the Euclidean norms we get the second result.

In the particular case of orthogonal projection methods,_QL_Kis identical with

PK,and we havekQ

L

Kk2 = 1. Moreover, the termγcan then be bounded from above by_kAk2. It may seem that since we obtain very similar error bounds for both the orthogonal and the oblique projection methods, we are likely to obtain similar errors when we use the same subspace. This is not the case in general.

One reason is that the scalarγcan no longer be bounded by_kAk2since we have kQL

Kk2≥1andkQ

L

Kk2is unknown in general. In fact the constantγcan be quite large. Another reason which was pointed out earlier is that residual norm does not provide enough information. The approximate problem can have a much worse condition number if non-orthogonal transformations are used, which may lead to poorer results. This however is only based on intuition as there are no rigorous results in this direction.

The question arises as to whether there is any need for oblique projection methods since dealing with oblique projectors may be numerically unsafe. Meth- ods based on oblique projectors can offer some advantages. In particular they may allow to compute approximations to left as well as right eigenvectors simul- taneously. There are methods based on oblique projection techniques that require also far less storage than similar orthogonal projections methods. This will be illustrated in Chapter 4.