_{1}

We focus on the single layer formulation which provides an integral equation of the first kind that is very badly conditioned. The condition number of the unpreconditioned system increases exponentially with the multiscale levels. A remedy utilizing overlapping domain decompositions applied to the Boundary Element Method by means of wavelets is examined. The width of the overlapping of the subdomains plays an important role in the estimation of the eigenvalues as well as the condition number of the additive domain decomposition operator. We examine the convergence analysis of the domain decomposition method which depends on the wavelet levels and on the size of the subdomain overlaps. Our theoretical results related to the additive Schwarz method are corroborated by numerical outputs.

Integral equation simulations have useful applications in synthetic medical design and molecular docking. The challenges to be confronted when treating a BEM (Boundary Element Method) simulation are multiple. First, the resulting BEM-matrix is dense if classical polynomial basis functions are used. Second, the matrix entries are usually integrals admitting 4D integrands which are singular. In addition, the matrix density results in a large memory capacity requirement which leads to the need of a dense linear solver for standard polynomial bases. On the other hand, the advantage of BEM [

We want to highlight here our main contributions in the theoretical and practical significances. We elaborate mathematical proofs which guarantee the convergence of the additive Schwarz method. For a decomposition

such that the single layer bilinear form

The significance of the above upper bound is that the ASM operator with respect to the weakly singular bilinear form

verifies on the maximal level L the eigenvalue lower bound

Our next contribution is the theoretical estimation of the largest eigenvalue of the domain decomposition method. The involvement of the overlap size

That is significant in deducing the upper estimate

The main significance of this study is to provide a rigorous preconditioner which is theoretically demonstrated to reduce the condition number. We have an analytical deduction of the condition number which does not grow exponentially with the multiscale level. Indeed, the condition number admits the upper bound

As for the practical contribution, we present outcomes from computer implementations which originate from molecular patches. We use realistic geometries consisting of molecular surfaces on our domain decomposition. The implementation is complete and not just some part of the theory is illustrated. In particular, the BEM linear system as well as the domain decomposition technique has been implemented completely. We contribute in practically exhibiting that the domain decomposition method admits a significant advantage over the unpreconditioned system. A lot of reduction of the iteration number is achieved. By growing the multiscale levels, the required iteration counts grow only very slowly in contrast to the unpreconditioned system whose iteration counts increase significantly fast. In addition, we contribute in utilizing a graph based approach to practically assemble the domain decomposition for the BEM application.

We will describe now the principal advantages of our approach compared with previous methods. An incomplete Cholesky factorization has been recently used in [

A reverse Schur preconditioning technique for use in hierarchical matrices has been newly described in [

In term of domain decompositions [

This section is occupied by the presentation of the integral equation of first kind which is formulated on a boundary surface

・ We have a covering of the surface by four-sided patches

・ The intersection of two different patches

・ Each patch

・ The patch decomposition has a global continuity: for each pair of patches

・ The manifold

An illustration of the above surface structure is depicted in

where we employ the divided difference

admits an overall smoothness of

We will consider only geometries which are globally smooth and which admit moderate curvature. For each patch

After transformation onto

Upon the whole surface

We will use the next Sobolev space on the manifold

where

We introduce also the dual space

By designating the 3D region enclosed within

We make now the change of unknown by using the density function

Introduce the single layer operator

The continuous problem is to search for

Once the solution u to the integral Equation (11) becomes available, the solution

whose construction will be specified later on. By discretizing (11) in each subspace

which is a boundary integral equation of the first kind where we use the kernel

We are only interested in the solution

The Gram determinant

for

The determination of a matrix entry

That means, the condition number increases exponentially as

whose verification is the purpose of this document.

This section will be occupied by the construction of the nested subspaces (12) on the whole surface

The internal knots on the next level

where

and the inclusion

On each patch

On the whole surface

with the dimensionalities

It is deduced from the above construction that we have the inclusion

Since the single-scale basis functions

with respect to the

For the explicit expression of the wavelet functions

whose relation with the single scale basis is such that

The wavelet functions constitute an orthonormal basis

where the first Dirac

where

so that we have the dimensionalities

A function

The next norm equivalences related to the coefficients are valid [

with constants

The 2D-wavelet spaces on the unit square

We have therefore

With respect to the wavelet basis functions, the integrals in (19) and (20) become

where

Before embarking to the next statement, let us enumerate the 2D-basis

Similarly for level

As a consequence, the basis indices which are exactly on level

where

The following theorem is a collection of properties which enable the subsequent statements.

Theorem 1. (see for e.g. [

and hence the equivalence

We will focus in this section on the framework of the ASM domain decomposition. In term of geometric structure, the overlapping domain decomposition will be as follows

where

In term of linear spaces, this leads to the decomposition

where

On account of the overlapping condition (57), the space decomposition (58) is not necessarily a direct sum. Denote the orthogonal projection onto

The ASM operator is defined by

The initial problem (13) is identical [

The expression of each term

where

(i) For any function

for a constant

(ii) For an arbitrary representation

If those two criteria (i) and (ii) are satisfied, then we have the following spectral properties of the additive domain decomposition in term of the smallest and largest eigenvalues [

The objective of the next description is to verify those two properties for the BEM bilinear form

Each subdomain

In the construction, we assume additionally that

Theorem 2. Consider an overlapping domain decomposition

such that the single layer bilinear form

Proof. Let us consider any function

By using the above construction of

By using the orthogonal projections

Since

On the other hand, we have

By using

for piecewise constant functions, we obtain

Eventually, we conclude from the equivalence (55)

W

We find in [

The operator

One has the next piecewise constant approximation for

By applying that to

Lemma 1. Consider two different subdomains

The next estimate is valid

where the constant c is independent of the maximum level L.

Proof. We have

where

By using the primitive

By using the boundedness of the functions

We use the expression

By combining (88) and (89), one deduces from (86)

W

Theorem 3. Consider an overlapping domain decomposition

where the constant c is independent of the maximal level L and the overlap widths.

Proof. Let a patch pair

We intend first to estimate

For

In addition, one has [

where

where

On the other hand, one has the estimate

On account of the result in (83), one deduces

Therefore, by using the enumerations of

Consequently, it yields the next estimate

On account of the fact that

we deduce

where the last relation was due to the

As a consequence, we obtain

Since

W

Theorem 4. Consider an overlapping domain decomposition

(56) and (57). Consider also a function

By using the bilinear form

where the constant c is independent on the maximal level L.

Proof. We are showing first that

because

where we used in the last equality

By combining that with (100), we obtain

In the same fashion as in the deduction of (78), we have

Similarly, we have

As a consequence,

By using the

W

Corollary 1. Consider an overlapping domain decomposition

verifies on the maximal level L the eigenvalue range

and the condition number upper bound

where the constants

Proof. Consider an arbitrary representation

because

where the constant c is independent on the level L and the functions u,

W

The spectral range might not be optimal yet but our current objective in this document is mainly to eliminate the exponential dependence

which becomes very small as the maximal level L increases. Therefore, the proposed method reduces the upper bound of the condition number from

In this section, we present some practical results related to the previous theory where we use several molecular models. For the quantum models, we employ Water Clusters and other molecules which are acquired from PDB files. When the molecular dynamic steps attain its equilibrium state where the total energy becomes stable, a water cluster is obtained by extracting the water molecules which are contained in some given large sphere whose radius controls the final size of the Water Cluster. The Hydrogen and Oxygen atoms contained in that large sphere constitute the components of the Water Clusters. The creation of the patch decomposition of the molecular surfaces is performed as described in [

For the practical construction of the domain decomposition on molecular surfaces, we apply a graph partitioning technique. We assemble a graph

We compare in ^{−9} are respectively 83, 145, 339, 804 for levels 1 till 4 by using the direct method. In order to perceive the plots of the domain decomposition results more clearly, we depict in

Although it is not the purpose of this document, we summarize in

Water Cluster) and several right hand-sides. We consider two exact solutions which are respectively

In ^{−9} whereas 1007 iterations are required for the direct method to obtain an error of order

Iteration | Direct method | Domain decomposition | ||
---|---|---|---|---|

Error | Reduction | Error | Reduction | |

0 | 1.288400e+03 | --- | 8.399300e+03 | --- |

3 | 1.081400e+02 | 0.689492 | 2.362800e+03 | 0.485394 |

13 | 8.052800e+00 | 0.917719 | 2.568000e+02 | 0.640958 |

24 | 4.379000e+00 | 0.955863 | 8.671400e−01 | 0.472788 |

35 | 3.490100e+00 | 0.964836 | 3.553900e−03 | 0.430791 |

46 | 2.343700e+00 | 0.967192 | 3.893700e−07 | 0.489318 |

54 | 1.830400e+00 | 0.973203 | 7.446200e−10 | 0.692799 |

200 | 2.023200e−01 | 0.981374 | ||

300 | 5.048100e−02 | 0.981586 | ||

430 | 4.493300e−03 | 0.983647 | ||

560 | 5.759100e−04 | 0.981659 | ||

697 | 4.573200e−05 | 0.978120 | ||

885 | 8.928400e−07 | 0.976550 | ||

1007 | 5.863900e−08 | 0.979455 |

We considered the single layer formulation using multiscale wavelet basis where the resulting system is badly conditioned. The additive version of the domain decomposition was used to circumvent the problem of bad conditioning. We concentrated on the non-overlapping domain decomposition where every subdomain is constituted of several patches. The convergence of the corresponding additive Schwarz method was examined. The smallest and the largest eigenvalues as well as the condition number have been estimated. Practical implementations exhibit satisfactory numerical results corresponding to the proposed theory.

Randrianarivony, M. (2016) Domain Decomposition for Wavelet Single Layer on Geometries with Patches. Applied Mathematics, 7, 1798-1823. http://dx.doi.org/10.4236/am.2016.715151