on an energy minimizing basis for algebraic …in an algebraic multigrid setting, the construction...

ON AN ENERGY MINIMIZING BASISFOR ALGEBRAIC MULTIGRID METHODS

JINCHAO XU, LUDMIL ZIKATANOV, AND LUDMIL ZIKATANOV

ABSTRACT. This paper is devoted to the study of an energy minimizingbasis first introduced in Wan, Chan and Smith (2000) for algebraic multi-grid methods. The basis will be first obtained in an explicit and compactform in terms of certain local and global operators. The basis functionsare then proved to be locally harmonic functions on each coarse grid “el-ement”. Using these new results, it is illustrated that this basis can benumerically obtained in an optimal fashion. In addition to the intendedapplication for algebraic multigrid method, the energy minimizing basismay also be applied for numerical homogenization.

1. INTRODUCTION

The present paper falls into the category of algebraic multigrid meth-ods for solving large scale algebraic systems arising from the discretiza-tion of partial differential equations. A motivation for this type of researchis a more automatic and less problem dependent construction of coarserspaces in an algebraic multigrid method in comparison with the more tra-ditional geometric multigrid method. Many techniques have been devel-oped in the literature on algebraic multigrid methods, see Brandt [3], Rugeand Stuben [19], Brandt, McCormick and Ruge [4], [5], de Zeew [8],Dandy [9], Chan, Xu and Zikatanov [7], Vanek, Mandel and Brezina [21,18, 20].

The construction of coarse spaces, a key issue in an algebraic multigridalgorithm, has been guided by theoretical as well as heuristic considera-tions. In regard to theoretical consideration, the work of Bramble, Pasciak,Wang and Xu [2] have been often used in the literature (c.f. [21, 18, 23, 7].According to the theory in [2], the coarse space needs to satisfy two majorrequirements in relation to stability and approximation. In [23], an energyminimizing basis was first constructed to facilitate these two requirements.

The authors were supported in part by NSF Grants No. DMS-0074299 and No.DMS-0209497 and Center for Computational Mathematics and Applications, Penn StateUniversity.

1

2 JINCHAO XU, LUDMIL ZIKATANOV, AND LUDMIL ZIKATANOV

The stability is addressed by minimizing the total energy of all (locally sup-ported) basis functions

��:

(1.1) �� where �� is the energy norm (see (2.3) below) associated with the givenelliptic boundary value problem. And the approximation property is main-tained by imposing following constraint to the above optimization problem:

(1.2) � � ��which essentially means that the basis functions will preserve constants lo-cally.

Extensive numerical experiments reported in [23, 22] show that this en-ergy minimizing basis leads to uniformly convergent multigrid methods formany problems of practical interest such as problems with rough coeffi-cients. In addition to the reported efficiency of this energy minimizingbasis, this type of methods are particularly interesting from a theoreticalpoint of view. A recent algorithmic development based on the “smoothed-aggregation” approach in the algebraic multigrid method, as well as theoret-ical framework is reported in works by Vanek, Mandel and Brezina [21, 18,20]. In this approach an explicit relation is drawn between the constructionof a base for the coarse space and the “energy” of the basis functions and aspointed out in [23] and [18], this can be viewed as one (or several) steps to-ward obtaining basis functions minimizing a quadratic (energy) functional.

As mentioned above, the energy minimizing basis is given by a globaloptimization problem (1.1) with a pointwise constraint (1.2). A major con-cern on this approach is the cost for solving this constraint optimizationproblem. This concern served as a major motivation of this paper. One con-clusion we will draw in this paper is that this optimization problem can besolved very easily and efficiently.

By looking at the energy minimizing basis within an appropriate func-tional space framework, in this paper, we are able to shed many new lightson this method. First, we are able to obtain a close-form exact solutionto the constraint optimization problem (1.1)-(1.2) in terms of one globaloperator and some local operators. Using this close-form solution, we areable to illustrate how this optimization can be solved efficiently. Perhapsmore significantly (at least from a theoretical point of view), we discov-ered that these basis functions are locally harmonic within each coarse grid“element”. This newly discovered property of the energy minimizing ba-sis suggests that various “harmonic extension” techniques, used to definecoarse spaces in multigrid method (see [7, 6, 16]) are very closely related

ON AN ENERGY MINIMIZING BASIS FOR ALGEBRAIC MULTIGRID METHODS 3

to the energy minimization algorithms. This property also suggests that theenergy minimizing basis may also be used for numerical homogenizationfor problems having a multiscale nature (see [10, 14]).

The rest of the paper is organized as follows. Some preliminaries andnotation are given in � 2. The energy minimizing basis functions will bestudied in � 3. An application of the basis function for numerical homog-enization will be given in � 4. Discussions on the efficient computation ofthe basis function will be given in � 5. A few numerical examples will bereported in � 6 and a couple of concluding remarks will be made in the lastsection.

2. PRELIMINARIES

We consider the following variational problem: Find �� suchthat

(2.1) � �� for all �� where

(2.2) � �� Here

�! #" $&%( � � � �(')�(* ) be a polygonal domain and �+� �� is the stan-

dard Sobolev space which contains functions having square integrable firstderivatives. It is well-known that (2.1) is uniquely (up to a constant) solv-able if

�� is a strictly positive scalar function and

�is square integrable

such that�,�� .-

.Associated with the above problem, we introduce the so-called energy

norm (semi-norm, more precisely) � �� :(2.3) � � �� /�� 0�We note that, for the purpose of studying the energy minimizing basis func-tions in this paper, the above simple model problem (with natural boundarycondition) is sufficient to illustrate the main ideas. Our results should alsobe easily applied to other boundary value problems without much modifi-cation.

We now consider the finite element space of continuous piecewise lin-ear functions 1324165 �7� �� defined on a triangulation 895 of

�. The

corresponding finite element discretization of (2.1) is: Find ��5:�;1<5 suchthat

(2.4) � �65=�� >5 �� ,5 � for all ,5?�71<5 �We are interested in developing multigrid methods for solving the above

finite element equation. A general multigrid methods can be illustrated


clearly by a simple two-grid method as follows based on an appropriatelychosen subspace 1�� 1<5 and an initial guess � �� to �/5 .Algorithm 2.1 (Two-grid). for � � � �(')� � � �

�� .� � � � �� 5�� where

(2.5) ��5 �71<5=� =5 � � 5=�� >5 �� ,5 �� ,5 � � ,5?��1/5and

(2.6) �� 71�� 5=�� 71�� endfor

In the above algorithm �5 � � � � � is some simple approximation to � � � � �on 1<5 , chosen in a way that (2.5) can be easily solved and high frequencyerror components can be reduced. Most popular choices of 5 are relatedto some local relaxation methods such as Jacobi and Gauss-Seidel itera-tions. This step is often called a smoothing step. The second step in theabove algorithm, known as coarse grid correction, amounts to the construc-tion of appropriate coarse space 1�� . In an algebraic multigrid setting, theconstruction of 1�� (or equivalently, using a more algebraic teminology, theprolongtation matrix) is one central issue. The rest of the paper will bedevoted to the study an energy minimizing basis for 1�� proposed in [23].

3. A CONSTRUCTION OF BASIS BASED ON GLOBAL ENERGY

MINIMIZATION AND LOCAL CONSTANT PRESERVATION

The coarse space 1�� 1<5 can be obtained by constructing a set oflinearly independent functions � � ��! � 1<5 which constitutes a basis of1�� . Motivated by the theory of Bramble, Pasciak, Wang and Xu [2] Wan,Chan and Smith [23] proposed a special construction which we shall nowdescribe.

We start our description with a a given set of mesh-subdomains� �

(con-sisting of a union of elements in 8<5 ) that satisfy:

(3.1)� � �"�#

��

and $��% & "')( � � '+*�,.-�0/ �where the superscript 1 is the standard set-complement.

We aim to construct basis functions � � ��! � that are subject to the fol-lowing restrictions:

(1) They are locally supported:243�5�5 � � �� $��


(2) They form a partition of unity:

(3.2)

�� # �� 2 �

for all� � ��

(3) They have a total minimal energy among all the partition of unitywith the given prescribed supports; namely � � �� ! � is the mini-mizer of:

(3.3) �� ' � � �� subject to

� ' �71 ' � �� ' � � ' � �� Here

(3.4) 1 ' � �� 71<5 � 243�5�5 � � $� ' � �Remark 3.1. We note that the functions

� � �� satisfying the properties men-tioned above are linearly independent due to the second assumption in (3.1)and the partition of unity property (3.2).

Due to the required property (3.3), � � �� ! � will be known as an energyminimizing basis subject to the subdomains (supports)

� � ' � . The propertiesof this basis and their computations have been studied by Wan, Chan andSmith [23]. We will now further study this basis from a different perspectiveto gain some new insights and develop new algorithms.

Let us first introduce some notation. We first define a discrete operator� � 1�� 1 as

(3.5) � �� 1 �and its restriction

� �of�

on each subspace

(3.6)� � � � � �� 71 � � � ��

Let � �� 1 � be the standard � � projection defined as� � �� 71 �9�We note that the domain of the � � projection can be extended beyond� � �� , say, 1� � , the dual of 1 � .

Define

(3.7) � � �� ! �� 9�

Thanks to (3.1), it is easy to see that the operator � � 1�� 1 is an isomor-phism.

We are now in a position to state and prove the first major result of thispaper.


Theorem 3.2. There exist a unique energy minimizing basis� � �� subject

to the subdomain��

and, furthermore these basis functions are given by

(3.8)� ��

Proof. Our main task is to solve the constrained minimization problem(3.3). Naturally, we can use the Lagrangian multiplier approach by seek-ing for the critical point of the following quadratic functional:

� � � ' & �' �� ' �� ' �0� *

Here�

is the Lagrangian multiplier. Differentiating the above functionalgives that

�� 71 � �

Hence the the � -th component of the critical point� � �� is given by

(3.9) � �� 1 � � � � � ��

From the above equations we obtain that

� �� Summing all these up leads to:

� � � � � �This gives a derivation of (3.8).

It is obvious that this unique critical point� � �� is indeed the unique

global minimizer of (3.3) that has a convex objective functional and a con-vex constraint.

There is actually a more direct way of verifying that�� given by (3.8)

is indeed the global minimizer of� * � * � . In fact, setting � � � � �� , we


have

� �� ! ��

�� ! ��

� �� ! ��

� �� ! �� ' � � ��

� �� ! ��

� �# ��

� �� ! ��

� � � � � � � �0�

Remark 3.3. we note that the last line of arguments in the above proof canalso be used to derive the following more general identity:

(3.10)� � � /�� 71 �

This type of identity has been used often in the multigrid literature, see [24,25].

We note that in the original paper [23], a Lagrangian multiplier methodwas also attempted to solve the optimization problem (3.3) within an alge-braic framework. Thanks to the use of functional space framework here,we are able to obtain more precise and explicit results. More significantly,these explicit expressions of

�� lead us to reveal some very importantproperties of these basis functions. One such properties is that each suchbasis function is locally discrete a-harmonic. We say that a function ��1is discrete a-harmonic on a subdomain � if

� ?�� - � for all �71/5�� 2 �� 71<5 � 2 3�5�5 � � $� � �3.1. Coarse grid elements. To define an analogue of coarse grid elements(an analogue to a finite element coarse grid), we first consider the followingopen set

� � � & �"�# �� * , % ��


Given� � � � , define the following function with values in the subsets of� � � � � � � � � :

(3.11) � �� To rule out any ambiguity we shall assume that for any

� � � � the set � � ��is ordered in ascending order. We then define

(3.12) �� The following simple proposition will lead us to an appropriate definitionof coarse grid elements.

Proposition 3.4. For the sets �� defined in (3.12) we have

(a) �� ,� � � � �� .(b) Either �� /

or �� , � � � � , � � � � .(c) There are finite number � of different sets �� ,

� � � � .Proof. The

�� direction in (a) follows from the fact that

� �� ,and hence � � �� . The other direction follows from the definition of�� .

To prove (b) let us assume that there exists � � � � , such that � �� and� �� . The definition of �� and �� then gives that � �� .By (a), �� . This proves (b).

The conclusion (c) follow directly from (b).

Let �� denote the finite collection of � sets in (c) from the aboveProposition 3.4. We have

� � � "��

�� "� ��

� �

As it is obvious that $� � � $� ,

(3.13) $� � $� � � "� ��!�

� � "� ��!�

$� �This means that the collection of �� forms a non-overlapping partition of�

. Each element in � � will be called a coarse grid element.

Remark 3.5. It is tempting to show how these macroelements will look onan unstructured grid, and in Fig 1, we have depicted three such supportstogether with their intersection. But let us point out that an essential featureof the technique we present here is that the coarse elements need not bedefined it explicitly and they might have quite complicated shape.

Lemma 3.6. Let� � � � � . Then, for each coarse element � �"�� - � for all

� �71/5�� 0�


FIGURE 1. A piece of triangular grid and supports of threebasis functions. On the right bottom picture, the intersec-tions are plotted. The darker colored domain correspond toa coarse element and is intersection of all three supports. Thelighter shaded domains are intersections of two supports andthe white area corresponds to no intersection.

Proof. By definition � � �� for some �� . Thus

1<5 � � � � �� %� �� 1 � and ��

Thus, by (3.9), we have

�� 1/5�� and �� - �The desired result then follows.

The above lemma basically say that� � � � � is a discrete edge

�-

function with respect to the coarse elements (namely�

is supported around� � ). Figure 2 is an example profile of this function.

Combining the above result with the identity (3.9), we immediately ob-tain our second main result in this paper as follows.

Theorem 3.7. Each basis function� �� is discrete a-harmonic on each coarse

element � �"� � , namely

(3.14) � � �� - � �71/5�� 0�


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

0

0.1

0.2

0.3

0.4

epsilon= 1.00000, h= 0.01562

FIGURE 2. The profile of� � � ��

In one space dimension ( � � �), the above result is rather trivial and it

is in fact already contained in [23]. In this case, the basis function�� is

analogous to the generalized finite element basis function in Babuska andKellog [1]

There are many results in multiple dimensions that may be related topiecewise a-harmonic functions. Examples include RFB (residual-free-bubble) method [11, 12, 13, 15] (for convection dominated problems) andthe multiscale finite element methods by Hou et al [10, 14] (for numericalhomogenization). Methods based on a-harmonic extension methods havealso been commonly used in domain decomposition methods (see [17]).

In fact harmonic extension and similar techniques have been used exten-sively in multigrid literature for the construction of coarse spaces for prob-lems with rough coefficients, see, for example, Hackbusch (1985), Brandt(1996), Zeeuw (1990), Chan, Xu and Zikatanov (1998).

The local harmonic properties in all aforementioned literature are ob-tained by construction from local element boundaries. It is interesting tonote that the energy minimizing basis studied here is a result of a moreglobal construction and the local harmonic properties is a by product fromthe construction.

4. HOMOGENIZATION EFFECT

The construction of an effective coarse grid basis functions for problemswith rough coefficients is totally analogous to a homogenization process. It


00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

10

0.2

0.4

0.6

0.8

1

FIGURE 3. The profile of a typical basis function� ��

� � �� /5 � �� Ratio' � * � - ' -�� - � �' � � � ' � * � ��- � ' � *' �� - � ' � '

TABLE 4.1. Convergence history of the numerical approx-imation �� to �/5 by energy minimization basis. The Ratioin the third column, refers to the ratio of two consequtivenumbers in the second column.

is natural to try how the energy minimizing basis functions work for numer-ical homogenization.

We applied the basis for the problem (1.1) with the following data:�� - � �� 2 �� 2 � � �� - � - � � � � ��

We first compute an accurate numerical approximation ��5 using a suf-ficiently fine grid ( � � ' �� ). With this fine grid finite element space, weconstruct the energy minimizing basis on subsequent coarser meshes andcompute the corresponding approximation � � . The convergence history isshown in the table 4.1

It appears that � �� /5 � �� . This kind of convergence rate isactually better than what we would predict theoretically. Since the energy


minimizing basis locally only preserve constant (not linears), convergencerate in � � norm is then not expected.

It is possible to construct energy minimizing basis functions that preservelinears or higher order polynomials to give better approximation properties,but this is entirely a different story. For optimal efficiency for multigridmethods for second order elliptic equations, it is sufficient for the basisfunction to locally preserve constants. This simple example here is only anillustration on the possible potential of applying this line of techniques forother applications such as numerical homogenization.

5. COMPUTATIONAL ASPECTS OF THE ENERGY MINIMIZING BASIS

One major early concern for the energy minimizing basis is the compu-tational cost for its construction. More specifically, based on the explicitexpression (3.8), the question is if it is possible to economically computethe global Lagrangian multiplier

� � � � � . In this section we will give apositive answer to this critical question.

The main point we will make is that the operator � is either well con-ditioned or it can be easily preconditioned by local operations. Let us firstpoint out, with appropriate assumptions, we can show that

The condition number of � , cond� � � , is bounded uniformly

with respect to � when � �� .Let us know sketch a proof of the above statement with getting into thedetails of specific assumptions of

�� . We first use the identity (3.10),

namely � � � /�� By Poincare inequality � � � � � � �� . Hence� � � /�� 1�� On the other hand, using a partition of unity � ��

(with� � �;1 � and� ��

� � � � and a standard argument,� � � /�� 1 � � � � � � � � � � � ��

This completes the justfication of the statement.The above simple theoretical result can be easily confirmed with numer-

ical experiments, as shown in table 5.1 (middle column) with � � '�� .These results for the simple Poisson equation and the numerical exampleconfirms that the bound of cond

� � � does not depend on the mesh size.


� �� , Laplace oper-ator

�� , Jump coeffi-cient

' � � �� - �' � � � � � � � ��- �' �� - �

TABLE 5.1. Condition number of � for various values of �and � � ' � . On the left is the Laplace operator case and onthe right is jump coefficient case as in (5.1)

� �� , �� ,' � � �� - � � ' �' � � �� - � � ' �' �� - � � ' �

TABLE 5.2. Comparision between the condition number of� and the preconditioned operator by Jacobi method � � .The coefficient

� � ��is as in equation (5.1).

Of course, more interesting applications of the energy minimizing basisare for problems with rough coefficients. So it is important to understandhow cond

� � � depends on the properties of the diffusion coefficients. Ournext numerical example shows that cond

� � � depends very sensitively onthe discontinuity jumps within the diffusion coefficients. In this example,we consider the following discontinuous diffusion coefficient:

(5.1)�� -��

in � - � - � ' �� - � -)� ' �� and�

everywhere else�

The results are shown in the Table 5.1 (right column). Although cond� � �

does not depend on mesh size � , it does, however, depend on the discontinu-ous jump. The good news is that the operator � is essentially locally definedand, as a result, it can be efficiently preconditioned by local operations. Forexample, we can simply apply a single level domain decomposition precon-ditioner (without coarse space) and the corresponding results are shown inthe table 5.2

6. NUMERICAL EXAMPLES

For completeness, we will include a couple of numerical examples in thissection to illustrate the efficiency of energy minimizing multigrid methods.For more numerical examples, we refer to the original papers [23, 22].


The numerical examples we shall consider are variants of the followingboundary value problem� � � �� in

�� - � on � � �

We have performed 4 numerical tests to show independent convergence ofthe conjugate gradient method with respect to the jumps, for computing theaction of � � . In all of these

� ��is piecewise constant and has jumps

acros lines depicted in Fig 4.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 4. Interface lines (left)

The setting for the first 3 examples is varying size of the coefficient. Thejumps across these lines is chosen to be

��-�� ' � , � - � � * � and� - � ��

. Forthe last example we have considered a random coefficient in each elementof the mesh varying in the range � ��- �� - � � . As it can be seen in Fig 5,the convergence of the conjugate gradient method is independent of thejumps. It also is independent of the mesh size, although we do not presentsuch an example here, because the focus in on handling jumps in the co-efficients and highly oscillatory coefficients. A vast amount of numericalexamples related to the convergence of multigrid method based on the en-ergy minimization can be found in papers by Chan, Wan and Smith [23] andWan [22]. The point we are trying to make here is that once the action of� � is computed in optimal way, an uniformly convergent multigrid method


is obtained and many numerical evidences showing this can be found in theliterature.

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 610

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Jumps in the coefficients: N=263169

Jump=10−2

Jump=10−3

Jump=10−4

Random

FIGURE 5. Convergence history of PCG for computing theaction of � � . The examples include jump in the coefficientsand random coefficient in every thiangle

Moreover the computation of� � � �� is not expensive, especially

when � is preconditioned with Jacobi method. It never took more than 5preconditioned conjugate gradient iterations to compute

�.

7. CONCLUDING REMARKS

With the new results from this paper, several new conclusions can bedrawn on the energy minimizing basis functions. Let us now give a coupleof examples. First, the cost of numerical obtaining this basis should not bea matter of major concern any longer. We have theoretically justified andexperimently verified that the constrainted optimization problem that de-fines the basis can be solved in parallel by solving various local problems inconjunction with preconditioned conjugate gradient method. Secondly thenewly discovered local harmonic property of the basis functions in multipledimensions are of particular theoretical interests. This property links themethod with various other methods (in various applications) that make useof local harmonic extensions. This property also opens the door of applyingthe ideas from the energy minimizing basis to numerical homogenization.


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DEPARTMENT OF MATHEMATICS, THE PENNSYLVANIA STATE UNIVERSITY, UNI-VERSITY PARK, PA 16802

E-mail address: [email protected]: http://www.math.psu.edu/xu/

DEPARTMENT OF MATHEMATICS, THE PENNSYLVANIA STATE UNIVERSITY, UNI-VERSITY PARK, PA 16802

E-mail address: [email protected]: http://www.math.psu.edu/ltz/

on an energy minimizing basis for algebraic …in an algebraic multigrid setting, the construction...

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