Quantitative Analysis
Parallel Processing
Numerical Analysis
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Printable PDF file
I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
VII. Implementation tools II.
1. Calculational Linear Algebra.
2. Wavelet Analysis.
3. Finite element method.
4. Construction of approximation spaces.
A. Finite element.
B. Averaged Taylor polynomial.
C. Stable space splittings.
D. Frames.
E. Tensor product splitting.
F. Sparse tensor product. Cure for curse of dimensionality.
5. Time discretization.
6. Variational inequalities.
VIII. Bibliography
Notation. Index. Contents.

Stable space splittings.


ondition

(Space splitting setup) Let $H$ be a separable Hilbert space. Let $A$ be a symmetric linear operator $A:H\rightarrow H$ with the properties MATH for some constants $C_{1},C_{2}>0$ and any $x,y\in H$ . We use the notation MATH and treat the form MATH as the principal norm on $H$ used for definition of closeness, duality and adjointness.

Let MATH be an at most countable collection of Hilbert spaces. Let $\tilde{H}$ be a "Hilbert sum" of the collection MATH : MATH

Let MATH be a collection of bounded linear operators MATH . We define the operator $R$ : MATH

We assume that $R$ is surjective: MATH

Definition

(Stable splitting) In context of the condition ( Space splitting setup ) we say that the collection MATH is a "stable splitting" of $H$ if there exist two constants MATH MATH such that MATH where MATH The quantities MATH are called "lower and upper stability constants" and MATH is "condition number" of the splitting.

Proposition

(Boundedness of surjection) In context of the condition ( Space splitting setup ) and for a stable splitting, the operator $R$ is bounded and MATH

Proof

For any MATH we estimate MATH

Proposition

(Adjoint of surjection) In context of the condition ( Space splitting setup ), the adjoint operator $R^{\ast}$ acts MATH where the $R_{j}^{\ast}$ the adjoint operator of $R_{j}$ defined by MATH

Proof

By definition of adjointness MATH thus $y\in\tilde{H}$ , MATH , $x\in H$ , and, by definitions of $R$ and $\tilde{H}$ , MATH

Proposition

(Boundedness of adjoint of surjection) In context of the condition ( Space splitting setup ) and for a stable splitting, the operator $R^{\ast}$ is bounded and MATH

Proof

The statement is a consequence of boundedness of $R$ , see the proposition ( Boundedness of surjection ).

Definition

(Schwarz operator) For a stable splitting MATH we define the bounded operators MATH MATH The operator $\QTR{cal}{P}$ is called "Schwarz operator" and the operator MATH is called "extended Schwarz operator".

Proposition

(Form of Schwarz operator) In context of the condition ( Space splitting setup ), MATH MATH

Proof

According to the definition of $R$ within the condition ( Space splitting setup ) and according to the proposition ( Adjoint of surjection ), MATH MATH We put these together: MATH MATH

Proposition

(Properties of Schwarz operator) If $\Sigma$ is a stable splitting (see the definition ( Stable splitting )) then the associated Schwarz operator is a symmetric positive definite operator: MATH Thus, there exists $\QTR{cal}{P}^{-1}$ and MATH

Proof

The symmetry follows from the proposition ( Form of Schwarz operator ) and generic properties of taking adjoint: MATH

For any $x\in H$ we introduce a class MATH . The class MATH is never empty by surjectivity of $R$ , see the condition ( Space splitting setup ). Then, for any $x\in H$ and MATH we calculate MATH where the equality takes place for MATH and MATH . We obtained MATH or MATH with equality reached at MATH . Thus MATH Next, we establish the upper bound for $\QTR{cal}{P}$ . We calculate, using the stability of splitting and MATH , MATH thus MATH or MATH

Next, we establish the lower bound for $\QTR{cal}{P}$ . We use again the surjectivity of $R$ , see the condition ( Space splitting setup ), to introduce a non empty class MATH . Then, for any $x\in H$ and any MATH MATH At this point we take infimum with respect to $\tilde{y}$ and use the definition of MATH , MATH We use stability of splitting, MATH Thus MATH or MATH Therefore, $\QTR{cal}{P}$ is invertible. The formula $\left( \#\right) $ now follows from MATH .





Notation. Index. Contents.


















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