Matrix maximum principles and their application ´ n Farago ´ Istva Department of Applied Analysis and Computational Mathematics E¨otv¨ os Lor´and University 1117 Budapest, P´azm´ any P. s. 1/c., Hungary e-mail address:
[email protected] Abstract: In the literature, for partitioned matrices with a certain structure some qualitative properties have been introduced (e.g., [2, 7, 9]). In this paper we define some of the basic properties, we give their motivation and analyze their relation. Keywords: system of linear algebraic equations, qualitative properties, maximum principle, non-negativity preservation
1
Introduction
Most of the problems in applied analysis lead to the solution of system of linear algebraic equations of the form Ax = f , with given non-singular quadratic matrix A ∈ IRk×k and vector f ∈ IRk . Hence, there exists the unique solution vector x ∈ IRk , and the classical numerical analysis deals with the construction of efficient algorithm to its calculation. On the other hand, usually from the original (typically continuous) problem we know some information about the qualitative properties of the solution, which mirror some basic physical low and its preservation should be taken into our consideration. E.g. we know that a harmonic function takes its maximum and minimum on the boundary. This well-known result can be generalized to the boundary-value problems as follows. Example 1 Consider the following boundary-value problem of elliptic type: Find a function u ∈ C 2 (Ω) such that −∆u = f in Ω and u = g on ∂Ω, (1) where Ω ⊂ Rd is a bounded domain with Lipschitz continuous boundary ∂Ω for problem (1). If the function f ∈ C(Ω), and f ≤ 0 then for the classical solution of problem (1) the following relation is valid for any x ∈ Ω: u(x) ≤ max g(s). (2) s∈∂Ω
Analogically, for the case f ≥ 0 we have the relation u(x) ≥ min g(s). s∈∂Ω
∗
(3)
Research is supported by OTKA K67819 and by the European Union Social Fund (Grant Agreement No. TAMOP 4.2.1./B-09/1/KMR-2010-0003).
1
From the relations (2) and (3) we obtain the classical maximum/minimal principle for the harmonic functions directly: if u ∈ C 2 (Ω) and −∆u = 0 then the two-sided estimation of the form min g(s) ≤ u(x) ≤ max g(s). (4) s∈∂Ω
s∈∂Ω
This estimation shows that g ≥ 0 implies the relation u ≥ 0. Example 2 A modified example is the following: Find a function u ∈ C 2 (Ω) such that −∆u + c(x)u = f
in
Ω,
and
u=g
on
∂Ω,
(5)
If the function c ∈ C(Ω), and c ≥ c0 > 0 then for the classical solution of problem (5) the relation Then the following (a priori) two-sided estimates for the classical solution of problem (1) are valid for any x ∈ Ω: n o n o f (x) f (x) min 0, min , min g(s) ≤ u(x) ≤ max 0, max , max g(s) . x∈Ω c(x) s∈∂Ω x∈Ω c(x) s∈∂Ω
(6)
is valid for any x ∈ Ω. Example 3 Let us consider the heat distribution in the one-dimensional rod of the length L, where the temperature is fixed at both ends. If u(x, t) denotes the temperature at the point x ∈ [0, L] and at the time-instant t ≥ 0, then a simple continuous model can be described by the problem ∂u ∂2u = , x ∈ (0, L), t > 0, (7) ∂t ∂x2 u(x, 0) = u0 (x), u(0, t) = u(L, t) = 0, (8) where the given function u0 denotes the initial heat distribution. For the classical solution we know the following: n o n o min 0, min u0 (x) ≤ u(x, t) ≤ max 0, max u0 (x) . x∈(0,L)
x∈(0,L)
(9)
The estimation (9) implies the non-negativity principle: if u0 is non-negative, then the solution u(x, t) is also non-negative for all x and t. We can also show that the solution is decreasing in time, i.e., u(x, t1 ) ≥ u(x, t2 ) whenever t1 < t2 . Usually the above continuous problems cannot be solved analytically, therefore we apply some numerical method for its solution. The most typical methods are the finite difference method (FDM) and the finite element method (FEM). In these approaches the common point is that, by replacing the domain Ω by the sequence of the discrete meshes and defining on these meshes the sequence of the discrete problems, solving each discrete problem we get the sequence of the numerical solution. Our aim is to preserve the above qualitative properties to these solutions.
2
2
The discrete problem
The FDM and the FEM lead the above problem to the solution of the system of linear algebraic equation of the form Ay = f , (10) or, for the parabolic problem to the iteration of the form Ay(m+1) = By(m) + f ,
m = 0, 1, . . . .
(11)
In these problems the matrices A and B are given matrices from IRN ×N , where N is the number of the mesh-point, at which we search the numerical solution. These matrices have special partitioned form µ ¶ A1 A2 A= , (12) 0 I where submatrices A1 ∈ IRN1 ×N1 , A2 ∈ IRN1 ×N2 , the identity matrix I ∈ IRN2 ×N2 , and N = N1 + N2 . We note that the number N1 corresponds to the number of the inner mesh-point, where the discrete mesh-equation is defined. At the same time, N2 corresponds to the number of the boundary mesh-point, where the numerical solution is given from the boundary condition. Hence, the unknown vector y also can be partitioned as y = (y1 , y2 ) with y1 ∈ IRN1 and y2 ∈ IRN2 , and, in fact, the second component y2 is known. Remark 4 The problem (11) for a fixed iteration step (i.e., for a fixed m) can be rewritten in form of the linear system, similar to the form (10): µ ¶ µ (m+1) ¶ µ ¶ f A −B y = (13) 0 I y(m) y(m) The above motivations give the reason to the formulation and consideration of the following problem. We consider the block-matrix H ∈ IRk×k and the block-vector y ∈ IRk in the form µ ¶ µ ¶ H1 H2 y1 H= , y= , 0 I y2
(14)
where submatrices H1 ∈ IRk1 ×k1 , I ∈ IRk2 ×k2 , H2 ∈ IRk1 ×k2 , 0 ∈ IRk2 ×k1 , y1 ∈ IRk1 and y2 ∈ IRk2 with k = k1 + k2 . In the sequel, for arbitrary vectors v, w ∈ IRk , we will use the following notations: max{v} := max{v1 , v2 , . . . , vk }, max{0, v} := max{0, max{v}}, (15) max{v, w} := max{max{v}, max{w}}. Then the considered problem is: having some information on the sign of the elements of the vector Hy, we define some information for the location of the maximum component of the vector y. In the rest of the paper we deal with this problem in the general algebraic setting.
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3
Matrix maximum principles
According to Ciarlet’s and Stoyan’s works (see [2], [8], [9]) we introduce the following definitions. Definition 5 We say that a matrix H satisfies the Ciarlet matrix maximum principle (CMMP) if for arbitrary vectors y1 ∈ IRk1 and y2 ∈ IRk2 , such that H1 y1 + H2 y2 ≤ 0, the inequality max{y1 } ≤ max{0, y2 } holds. Definition 6 We say that a matrix H satisfies the Stoyan matrix maximum principle (SMMP) if for arbitrary vectors y1 ∈ IRk1 and y2 ∈ IRk2 , such that H1 y1 + H2 y2 = 0 and y2 ≥ 0, the inequality max{y1 } ≤ max{y2 } holds. Remark 7 We note that in [8] the SMMP was originally formulated for general, un-partitioned matrices as follows: a quadratic matrix H is said to satisfy the maximum principle if the relation Hy ≥ 0 implies that y ≥ 0, moreover, when max{y} = yi0 , (i.e., the maximum is taken on the i0 -th component) then (Hy)i0 > 0. An application of this principle to the structured matrix H of the form (14) yields the definition of the SMMP in Definition 6. The above definitions give information about the location of the maximum components of the unknown vector y ∈ IRk , using some a priori information: for the CMMP the non-negative maximum, while for the SMMP the maximum is taken over the last indices i = k1 + 1, k1 + 2, . . . , k, i.e., on the sub-vector y2 . Remark 8 If a matrix H satisfies the CM M P , then it is necessarily regular. To show this, it is enough to prove that the relation H1 y1 = 0 implies that y1 = 0. By the choice y2 = 0, the application of the CMMP implies the inequality y1 ≤ 0. Repeating this argument for −y1 , we obtain −y1 ≤ 0, which shows the validity of the required equality. The similar statement holds for the SMMP, too. The CMMP and SMMP properties can be guaranteed in the following way. Theorem 9 ([2]) The matrix H satisfies the CMMP if and only if the following two matrix conditions hold: (C1): H is monotone, i.e.,
µ H
−1
=
H−1 −H−1 1 1 H2 0 I
¶ ≥ 0;
(16)
(C2): as before, using the notation ekm ∈ IRkm (m = 1, 2) for the vectors with all coordinates equal to one, we have −H−1 (17) 1 H2 ek2 ≤ ek1 . The condition (C2) can be relaxed by the following sufficient condition (C2’): the row sums of the matrix H are all non-negative, i.e., H1 ek1 + H2 ek2 ≥ 0. The following statement gives an equivalent condition for the CMMP. 4
(18)
Lemma 10 A matrix H satisfies the CMMP if and only if the implications H1 y1 + H2 y2 ≤ 0, and y2 ≤ 0
⇒
max{y1 } ≤ 0;
(19)
H1 y1 + H2 y2 ≤ 0, and y2 ≥ 0
⇒
max{y1 } ≤ max{y2 }
(20)
are valid. Proof. It is obvious that CMMP implies both (19) and (20). Therefore have to show only the converse implication. From the assumption (19) it follows that the vector Hy is also nonpositive for any non-positive y. This yields the monotonicity of H, i.e., H−1 ≥ 0 is valid. On the other side, let us choose y1 = −H−1 1 H2 ek2 and y2 = ek2 . Then H1 y1 + H2 y2 = 0 and y2 ≥ 0. For these vectors we can use (20) and obtain the relation max{−H−1 1 H2 ek2 } ≤ max{ek2 } = 1. Hence, −H−1 H e ≤ e . Hence, according to the Theorem 9, we have showed the CMMP 2 k k 2 1 1 property for the matrix H. Remark 11 The CMMP obviously implies the SMMP. Therefore, the above conditions also guarantee the SMMP property. It is worth mentioning that for the un-partitioned matrix H the monotonicity and the condition He ≥ 0 (which is, in fact, the analogue of the condition (18)) are necessary conditions for validity of the SMMP. When H is an M-matrix∗ , then these conditions are necessary and sufficient ([8]). We can combine the CMMP and the SMPP as follows: we require that under the CMMP condition the implication in the SMMP is true, i.e., we introduce Definition 12 We say that a matrix H satisfies the Ciarlet-Stoyan matrix maximum principle (CSMMP) if for arbitrary vectors y1 ∈ IRk1 and y2 ∈ IRk2 , such that H1 y1 + H2 y2 ≤ 0, the relation max{y1 } ≤ max{y2 } holds. Obviously, the CSMMP implies both the CMMP and SMMP properties. This property can be guaranteed by the following statement. Lemma 13 Assume that H is monotone and the condition H1 ek1 + H2 ek2 = 0
(21)
holds. Then H has the CSMMP property. Proof. Let y1 ∈ IRk1 and y2 ∈ IRk2 be arbitrary vectors with the property H1 y1 +H2 y2 ≤ 0. −1 Since H is monotone, therefore H−1 1 ≥ 0 and −H1 H2 ≥ 0 (cf. (16)). Therefore we have −1 −1 y1 ≤ −H−1 1 H2 y2 ≤ −H1 H2 (max{y2 }ek2 ) = −(max{y2 })H1 H2 ek2 .
(22)
Due to the assumption (21), the relation (22) implies that y1 ≤ (max{y2 })ek1 ,
(23)
which proves the statement. In the next statement, we show that the conditions in the Lemma 13 are not only sufficient but they are necessary, too. ∗
A Z-matrix (a real square matrix with all off-diagonal entries are less than or equal to zero) A is called an M-matrix if the relationAv ≥ 0 implies that v ≥ 0. There are many equivalent definitions, see e.g., [1].
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Lemma 14 Assume that H has the CSMMP property. Then H is monotone and the relation H1 ek1 + H2 ek2 = 0
(24)
holds. Proof. Since the CSMMP property implies the CMMP property, therefore, due to the Theorem 9, H is monotone. In order to show the second condition, first, let us put y1 = −H−1 1 H2 ek2 and y2 = ek2 (as in the proof of the Lemma 10). Since for this choice the CSMMP is applicable, we get the estimation −1 max{−H−1 1 H2 ek2 } ≤ max{ek2 } = 1. Let us put now y1 = H1 H2 ek2 and y2 = −ek2 . The CSMMP is again applicable and we get the estimation max{H−1 1 H2 ek2 } ≤ max{−ek2 } = −1. The above two estimations clearly result in the equality −H−1 1 H2 ek2 = ek1 , which yields the required (24).
4
Conclusions
The matrix maximum principles, defined and analysed in Section 3, can be successfully applied to the construction of the qualitatively adequate discrete models for the continuous problems, given in Section 1. In fact, for the concrete schemes we have to give those conditions which guaranty the monotonicity of the matrices given in (12) and (13), respectively. (These conditions for the FDM and FEM can be defined and typically they yield stronger condition than the condition of the convergence, see [3, 4].) For the analyzing of the nonnegativity property of the discrete model to the problem, given in Examples 1 and 2, we recall [2] and also [3]. For the properties of the monotonically increase, discrete maximum principle and the nonnegativity preservation in the discrete parabolic models, which correspond to the Exmple 1 (even for the problem in more general form), we cite the papers [4] and [5]. For the discrete maximum principle in the elliptic problem of Example 2 with general boundary condition we refer our recent work [6].
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