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We present an iterative scheme for solving Poisson’s equation in 2D. Using finite differences, we discretize the equation into a Sylvester system,
*AU* +
*UB* =
*F*, involving tridiagonal matrices
*A* and
*B*. The iterations occur on this Sylvester system directly after introducing a deflation-type parameter that enables optimized convergence. Analytical bounds are obtained on the spectral radii of the iteration matrices. Our method is comparable to Successive Over-Relaxation (SOR) and amenable to compact programming via vector/array operations. It can also be implemented within a multigrid framework with considerable improvement in performance as shown herein.

Poisson’s equation ∇ 2 u = f , an elliptic partial differential equation [

The Poisson equation on a rectangular domain is given by

∇ 2 u = f in Ω = { x , y | 0 ≤ x ≤ a , 0 ≤ y ≤ b } (1)

where u = u ( x , y ) is to be solved in the 2D domain Ω , and f ( x , y ) is the forcing function. Typical boundary conditions for this equation are either Dirichlet, where the value of f ( x , y ) is specified on the boundary, or Neumann, where the value of the normal derivative is specified on the boundary. These are given mathematically as,

u = g D or ∂ u / ∂ n ≡ n ^ ⋅ ∇ u = g N on ∂ Ω , (2)

where n ^ is the outward unit normal along ∂ Ω and g D and g N are the function values specified by Dirichlet or Neumann boundary conditions. It is also possible to have mixed boundary conditions along the boundary, where some edges have Dirichlet and some have Neumann, so long as the problem is well-posed. Furthermore, edges could be subject to Robin boundary conditions of the form c 1 u + c 2 ∂ u / ∂ n = g R . Any numerical scheme designed to solve the Poisson equation should be robust in its ability to incorporate any form of boundary condition into the solver. A detailed discussion of boundary condition implementation is given in the Appendix. Discretizing (1) using central differences with equal grid size Δ x = Δ y = h leads to an M × N rectangular array of unknown U, such that U i , j ≈ u ( x i , y j ) (assuming that a and b are both integer multiples of h). This discretization leads to a linear system of the form A U + U B = F , the Sylvester equation, which can be solved either directly or iteratively. The direct method utilizes the Kronecker product approach [

K u = f where K = kron ( I , A ) + kron ( B T , I ) (3)

where u and f are appropriately ordered M N × 1 column vectors obtained from the M × N arrays U and F, and K is a sparse M N × M N matrix. The Kronecker product, kron ( P , Q ) , of any two matrices P and Q is a partitioned matrix whose ij^{th} partition contains matrix Q multiplied by component p i j of P. Due to the potentially large size of the system given in (3), direct solvers are not the preferred solution approach. Specifically addressed here is an iterative approach to solving the Sylvester equation,

A U int + U int B = F int , (4)

where U int is the m × n array of interior unknowns (not including the known boundary values when Dirichlet boundary conditions are given) with m = M − 2 and n = N − 2 . Operator matrices A and B are given by the O ( h 2 ) finite difference approximation to the second derivative,

1 h 2 [ − 2 1 1 − 2 1 ⋱ ⋱ ⋱ 1 − 2 1 1 − 2 ] . (5)

For an array of unknowns U int ∈ ℝ m × n , the operator matrices are of dimension A ∈ ℝ m × m , and B ∈ ℝ n × n . The matrix structure of U should be modified such that the first index i corresponds to the x-direction, and the second index j corresponds to the y-direction. With this orientation, multiplying U int by the matrix A on the left approximates the second derivative in the x-direction, and multiplying U int by the matrix B on the right approximates the second derivative in the y-direction.

Examining (4) it might seem natural to move one term to the right-hand side of the equation to achieve an iterative scheme such as:

A U + U B = F ⇒ A U = F − U B

⇒ U k + 1 = A − 1 F − A − 1 U k B , (6)

However, this scheme diverges, and an alternative approach is required to iterate on the Sylvester system. An appropriate method is to break up the iterative scheme into two “half-steps’’ as follows

1) First half-step: U k → U *

A U = F − U B ⇒ ( A − α I + α I ) U = F − U B

⇒ ( A − α I ) U * = F − U k ( B + α I ) (7)

2) Second half-step: U * → U k + 1

U B = F − A U ⇒ U ( B − β I + β I ) = F − A U

⇒ U k + 1 ( B − β I ) = F − ( A + β I ) U * (8)

where U * is some intermediate solution between steps k and k + 1 . Rearranging, this leads to

( I − 1 α A ) U * = U k ( I + 1 α B ) − 1 α F , U k + 1 ( I − 1 β B ) = ( I + 1 β A ) U * − 1 β F . (9)

The iterative scheme (9) is similar to the Alternating Direction Implicit (ADI) formulation [

d U d t = A U + U B − F , (10)

which achieves the solution to Equation (4) when it reaches steady-state. This method is separated into two half-steps, the first time step going from time k → k + 1 / 2 treating the x-direction implicitly and the y-direction explicitly. The second time step then goes from time k + 1 / 2 → k + 1 , treating the y-direction implicitly and the x-direction explicitly. The two half-steps are,

U k + 1 / 2 − U k Δ t / 2 = 1 h 2 [ A U k + 1 / 2 + U k B ] , U k + 1 − U k + 1 / 2 Δ t / 2 = 1 h 2 [ A U k + 1 / 2 + U k + 1 B ] , (11)

which leads to

( I − Δ t 2 A ) U k + 1 / 2 = U k ( I + Δ t 2 B ) − Δ t 2 F , U k + 1 ( I − Δ t 2 B ) = ( I + Δ t 2 A ) U k + 1 / 2 − Δ t 2 F . (12)

This iteration procedure looks nearly identical to our Sylvester iterations given in (9) with Δ t / 2 replaced by the unknown parameters 1/α and 1/β. However in our formulation, there is no pseudo-time dependency introduced. Instead, the eigenvalues of our operator matrices A and B are deflated to produce an iterative scheme that optimally converges, and finding the values of the parameters α and β becomes an optimization problem.

ConvergenceAfter the Sylvester Equation (4) is modified into the iterative system (9), the iterative scheme can be written as a single step by substituting the expression for the intermediate solution U * into the second step of the iterative process; this yields the single update equation for U k + 1 given by

U k + 1 = ( I + 1 β A ) ( I − 1 α A ) − 1 U k ( I + 1 α B ) ( I − 1 β B ) − 1 − [ 1 α ( I + 1 β A ) ( I − 1 α A ) − 1 − 1 β I ] F ( I − 1 β B ) − 1 . (13)

Assuming that an exact solution U exact exists that exactly satisfies the linear system (4), i.e. A U exact + U exact B = F , we define the error between the k^{th} iteration and the exact solution as

E k ≡ U k − U exact . (14)

Finding an update equation for the error is done by subtracting the error at the k^{th} step from the error at the ( k + 1 ) s t step, noting that the expressions involving the forcing F disappear, we arrive at

E k + 1 = P E k Q , (15)

where the matrices P and Q are given by

P = ( I + 1 β A ) ( I − 1 α A ) − 1 , Q = ( I + 1 α B ) ( I − 1 β B ) − 1 . (16)

Denoting the m eigenvalues of A ∈ ℝ m × m by λ k A , and the n eigenvalues of B ∈ ℝ n × n by λ k B , the corresponding deflated eigenvalues of the iteration matrices P and Q are

λ k P = 1 + ( λ k A / β ) 1 − ( λ k A / α ) = α β ( β + λ k A α − λ k A ) , k = 1 , 2 , ⋯ , m , λ k Q = 1 + ( λ k B / α ) 1 − ( λ k B / β ) = β α ( α + λ k B β − λ k B ) , k = 1 , 2 , ⋯ , n . (17)

A sufficient condition for convergence of the iterative process is achieved if the spectral radii of both iteration matrices P and Q are less than one,

ρ ( P ) ≡ max | λ k P | < 1 and ρ ( Q ) ≡ max | λ k Q | < 1. (18)

The error at each consecutive iteration is decreased by the product of ρ ( P ) and ρ ( Q ) ,

E k + 1 = P E k Q , ‖ E k + 1 ‖ = ρ ( P ) ρ ( Q ) ‖ E k ‖ , ‖ E k + 1 ‖ = ( ρ ( P ) ρ ( Q ) ) k ‖ E 0 ‖ , (19)

where E 0 is the initial error. Often in practical applications, the exact solution is not known, so the error E k cannot be computed directly. In this case, the preferred measure in iterative schemes is given by the residual, which measures the difference of the left and right hand sides of the linear system being solved. This will be further discussed in the Results section.

Finding α and β is an optimization problem for achieving the fastest convergence rate of the Sylvester iterative scheme (9). Given the operator matrices A and B and their respective eigenvalues λ k A and λ k B , it seems feasible to find optimal values of α and β to minimize the spectral radii of the iteration matrices P and Q given in Equations (17). From (15) the error E k + 1 is found by multiplying by P on the left, and Q on the right, thus the convergence is governed by the spectral radii of both P and Q.

the high frequency eigenvalues of P and Q, in magnitude, will be greater than one, thus convergence condition (18) will not be satisfied. This provides the restriction for convergence that,

α = β . (20)

This optimal value of α = β will henceforth be called α * . It is important to note that the operator A ∈ ℝ m × m or B ∈ ℝ n × n with the larger dimension

l ≡ m a x ( m , n ) , (21)

has a larger range of eigenvalues.

λ k P = α * + λ k A α * − λ k A , k = 1 , 2 , ⋯ , m , λ k Q = α * + λ k B α * − λ k B , k = 1 , 2 , ⋯ , n . (22)

Finding the optimal parameter α * is done by considering the error reduction of Sylvester iterations on an arbitrary initial condition U_{0}. Assume that U_{0} can be decomposed into its constituent error (Fourier) modes, ranging from low frequency (smooth) to high frequency (oscillatory) modes. Given that U_{0} contains error modes of all frequencies, the most conservative method would be to choose α * such that the spectral radii ρ ( P ) and ρ ( Q ) are minimized over the full range of frequencies. This ensures that all modes of error are efficiently relaxed, and convergence is governed by the product of spectral radii.

Referring to the lower curve in

determining α * would be to set L min * = L max * and according to (22),

− ( α * + λ min A α * − λ min A ) = ( α * + λ max A α * − λ max A ) . (23)

Noting that eigenvalues for all dimensions collapse onto the curves shown in

α * = | min ( λ min A , λ min B ) | × | max ( λ max A , λ max B ) | , (24)

where absolute values are introduced as a reminder that λ A , λ B are negative. This value of the parameter α * most uniformly smooths all frequencies for any arbitrary U 0 containing all frequency modes of error. It can be seen that the spectral radii of P and Q shown in

The Sylvester multigrid formulation is based on the philosophy that most iterative schemes, including Sylvester iterations, relax high frequency modes fastest, leaving low frequency components relatively unchanged [

α mg ∗ = | m i n ( λ min A , λ min B ) | × | m i n ( λ mid A , λ mid B ) | , (25)

where, if m ≠ n , the minimum (i.e., most negative) middle eigenvalue λ mid ≡ ( λ min + λ max ) / 2 is chosen to shrink the optimal smoothing region R smooth mg such that high frequencies are smoothed most effectively. This choice of optimal parameter can be observed in

To find analytical expressions for α * and α mg ∗ , it is necessary to have values for λ A and λ B . For Dirichlet boundary conditions, analytical expressions for λ A and λ B are derived below, but for Neumann boundary conditions numerical approaches are necessary to find λ A and λ B . The operator matrices A and B are each of tridiagonal form,

[ d 0 d 1 d 1 d 0 d 1 ⋱ ⋱ ⋱ d 1 d 0 d 1 d 1 d 0 ] ∈ ℝ p × p . (26)

Tridiagonal matrices with constant diagonals, such as A and B for Dirichlet boundary conditions, have analytical expressions for their eigenvalues given by

λ k = d 0 + 2 d 1 cos ( k π p + 1 ) , k = 1 , 2 , ⋯ , p (27)

where p is the arbitrary dimension of the matrix [

λ k = 2 h 2 ( − 1 + cos ( k π p + 1 ) ) , k = 1 , 2 , ⋯ , p , (28)

which achieves minimum and maximum values given by

λ min = 2 h 2 ( − 1 + cos ( p π p + 1 ) ) and λ max = 2 h 2 ( − 1 + cos ( π p + 1 ) ) , (29)

respectively. Using (24), (25), and (29) the analytic expressions for optimal parameters for both conservative and multigrid approaches are given by

α * = 2 h 2 ( − 1 + cos ( l π l + 1 ) ) ( − 1 + cos ( π l + 1 ) ) , α mg ∗ = 2 h 2 ( − 1 + cos ( l π l + 1 ) ) ( − 2 + cos ( π l + 1 ) + cos ( l π l + 1 ) ) , (30)

where again l ≡ m a x ( m , n ) . Having expressions for α * and α mg ∗ allows λ P and λ Q to be found analytically using (22) which subsequently allows the spectral radii of the iteration matrices P and Q to be calculated. Knowing the spectral radii of the iteration matrices P and Q is highly advantageous, as it allows for an analysis of the Sylvester iterative scheme.

The analysis of standard Sylvester iterations can be performed and describes the error reduction with each consecutive iteration using (19). Having the optimal parameters given by (30) and eigenvalues of P and Q in (22), the spectral radii can be calculated to be

ρ ( P ) = − 1 + cos ( m π m + 1 ) + ( − 1 + cos ( l π l + 1 ) ) ( − 1 + cos ( π l + 1 ) ) − 1 + cos ( m π m + 1 ) − ( − 1 + cos ( l π l + 1 ) ) ( − 1 + cos ( π l + 1 ) ) , ρ ( Q ) = − 1 + cos ( n π n + 1 ) + ( − 1 + cos ( l π l + 1 ) ) ( − 1 + cos ( π l + 1 ) ) − 1 + cos ( n π n + 1 ) − ( − 1 + cos ( l π l + 1 ) ) ( − 1 + cos ( π l + 1 ) ) . (31)

Rewriting the last expression of (19), we see that

‖ E k ‖ ‖ E 0 ‖ ~ ( ρ ( P ) ρ ( Q ) ) k . (32)

If we want to reduce our error to ‖ E k ‖ ~ ϵ ‖ E 0 ‖ and we wish to know how many iterations it will take to achieve this error reduction, using (32) we set ( ρ ( P ) ρ ( Q ) ) k ~ ϵ , and solving for k, we find it will take

k ~ l o g ( ϵ ) l o g ( ρ ( P ) ρ ( Q ) ) (33)

iterations to reduce the error by ϵ . Here log can be with respect to any base, as long as the same one is used in both the numerator and denominator; e.g., the natural log can be used. Recall that the exact solution U exact of (4) is only an approximate solution of the differential Equation (1) we are actually solving. Due to this, we can only expect accuracy of the truncation error of the approximation. With an O ( h 2 ) method, U i , j exact differs from U ( x i , y j ) on the order of h 2 so we cannot achieve better accuracy than this no matter how well we solve the linear system. Thus, it is practical to take ϵ to be something proportional to the expected global error, e.g. ϵ = C h 2 for some fixed C [

To calculate the order of work required asymptotically as h → 0 , (i.e. m → ∞ ) using (33) and our choice for ϵ , we see that

k ~ l o g ( C ) + 2 l o g ( h ) l o g ( ρ ( P ) ρ ( Q ) ) . (34)

The expressions for ρ ( P ) and ρ ( Q ) in (31) contain several cosine terms which can be Taylor expanded about different values. Cosines with arguments like π x can be expanded about x = 1 or x = 0 depending on the form of x, namely

c o s ( π x ) ~ − 1 + π 2 2 ( x − 1 ) 2 + O ( ( x − 1 ) 3 ) for x ≈ 1, c o s ( π x ) ~ 1 − π 2 2 x 2 + O ( x 3 ) for x ≈ 0, (35)

where, from (31), the form of x is something like m / ( m + 1 ) or 1 / ( m + 1 ) , which clearly approach one or zero, respectively, in the limit that m → ∞ . Using these expansions, along with the fact that 1 / ( 1 − x ) ~ 1 + x + O ( x 2 ) for x ≪ 1 to simplify the spectral radii, we arrive at the following

ρ ( P ) ~ ρ ( Q ) ~ 1 − π l + 1 + 1 4 ( π l + 1 ) 2 , (36)

when m , n ≫ 1 . Since h = 1 / ( m + 1 ) , (34) combined with (36) gives the following order of work needed for convergence to within ϵ ~ C h 2 :

k ~ − 2 l o g ( m + 1 ) 2 l o g ( 1 − π l + 1 ) ~ l π l o g ( m ) , (37)

where only linear terms are used from (36), and the latter simplified expression can be deduced by using the property that l o g ( 1 + x ) ~ x + O ( x 2 ) for x ≪ 1 . Note that when m = n , the order of work for Sylvester iterations is k ~ ( m / π ) l o g ( m ) , which is comparable to the work necessary for the Successive Over-Relaxation (SOR) algorithm to solve Poisson’s equation [

Problems solved by Sylvester iterations can, in general, be written shorthand as L U = F , where L is a linear operator. In the case of Poisson’s equation, L is the Laplacian operator. As an error measure, the discrete ‖ ⋅ ‖ 2 norm of the residual, r ≡ F − L U , can be measured at each iteration. This number provides the stopping criterion for our iterative schemes, namely the iterations are run until

‖ r ( k ) ‖ = ‖ F − L U ( k ) ‖ < tol × ‖ r ( 0 ) ‖ , (38)

where r ( 0 ) is the initial residual, and tol is the tolerance. The tolerance is set to machine precision tol ~ 10 − 16 to illustrate the asymptotic convergence rate,

q ( k ) = ‖ r ( k ) ‖ ‖ r ( k − 1 ) ‖ , (39)

however, in practice, the discretization error O ( h 2 ) is the best accuracy that can be expected. These numerical results were run using MATLAB on a 1.5 GHz Mac PowerPC G4. The model problem that is solved is given by

∇ 2 u = − 2 [ y 2 ( 1 − 6 x 2 ) ( 1 − y 2 ) + x 2 ( 1 − 6 y 2 ) ( 1 − x 2 ) ] in Ω , u = 0 on ∂ Ω , (40)

where Ω = { x , y | 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 } , and whose exact solution

u exact ( x , y ) = ( x 2 − x 4 ) ( y 4 − y 2 ) , (41)

is known so errors can be computed [

For comparison, standard Sylvester iterations were tested against Successive Over-Relaxation (SOR) with Chebyshev acceleration (see e.g., [

In multigrid Sylvester iterations, the performance of the V ( ν 1 , ν 2 ) -cycle using Sylvester iterations is compared to that using the traditional Gauss-Seidel (GS)

System Size | Sylvester | SOR | Sylvester | SOR | ||
---|---|---|---|---|---|---|

M × N | Iterations | Iterations | q sylvester | q sor | CPU Time | CPU Time |

16 × 16 | 84 | 93 | 0.638 | 0.674 | 0.059 | 0.072 |

32 × 32 | 173 | 185 | 0.806 | 0.819 | 0.474 | 0.612 |

64 × 64 | 352 | 368 | 0.899 | 0.905 | 4.591 | 5.082 |

128 × 128 | 709 | 735 | 0.949 | 0.951 | 43.13 | 48.28 |

256 × 256 | 1473 | 1469 | 0.975 | 0.975 | 744.5 | 786.7 |

System Size | Sylvester | GS | Sylvester | GS | ||
---|---|---|---|---|---|---|

M × N | V(2,1)-cycles | V(2,1)-cycles | q sylvester mg | q gs mg | CPU Time | CPU Time |

16 × 16 | 12 | 14 | 0.055 | 0.067 | 1.53 | 1.66 |

32 × 32 | 13 | 15 | 0.062 | 0.078 | 2.07 | 2.16 |

64 × 64 | 13 | 15 | 0.066 | 0.082 | 3.29 | 2.94 |

128 × 128 | 13 | 15 | 0.068 | 0.083 | 6.34 | 4.87 |

256 × 256 | 13 | 15 | 0.068 | 0.083 | 24.1 | 15.4 |

512 × 512 | 13 | 15 | 0.069 | 0.083 | 151.8 | 102.3 |

iterations. The parameter ν 1 represents the number of smoothing iterations done on each level of the downward branch of the V-cycle, while ν 2 represents the number done on the upward branch. In practice, common choices are ν = ν 1 + ν 2 ≤ 3 , so our performance is based on the V ( 2,1 ) -cycle [

Sylvester iterations provide an alternative iterative scheme to solve Poisson’s equation that is comparable to SOR in the number of iterations necessary to converge, namely converging to discretization accuracy within k ~ ( m / π ) l o g ( m ) iterations. The true benefit of the Sylvester iterations, however, comes from its adaptive ability to smooth any range of error frequencies, thus being a perfect candidate for smoothing in a multigrid framework. Multigrid V ( 2,1 ) -cycles using Sylvester smoothing have an asymptotic convergence rate of q sylvester mg = 0.069 (versus q gs mg = 0.083 for Gauss-Seidel smoothing) and indicate significant improvement in efficiency over standard Sylvester iterations.

Franklin, M.B. and Nadim, A. (2018) A Poisson Solver Based on Iterations on a Sylvester System. Applied Mathematics, 9, 749-763. https://doi.org/10.4236/am.2018.96052

1) Boundary condition implementation

Solving the Poisson equation using Sylvester iterations lends itself nicely to boundary condition implementation. Dirichlet boundary conditions of the form

u ( 0 , y ) = u 1 ( y ) , u ( a , y ) = u 2 ( y ) , u ( x , 0 ) = u 3 ( x ) , u ( x , b ) = u 4 ( x ) , (1)

where u 1 , u 2 , u 3 and u 4 are functions describing the edges of U, can be implemented as follows. The unknown values in the Sylvester system given in Equation (4) are the m × n array of interior values, where m = M − 2 , n = N − 2 . It is possible to incorporate Dirichlet boundary conditions directly into this interior system by examining the partitioned matrix product, for example AU, given by

with UB taking an analogous partitioned form. Multiplying through by h 2 associated with the operator matrices A and B, the partitioned Sylvester system for internal unknowns gives

A L ⋅ u 1 T + ( A int ) ( U int ) + A R ⋅ u 2 T + u 3 ⋅ B T T + ( U int ) ( B int ) + u 4 ⋅ B B T = ( h 2 ) ( F int ) , (3)

where all matrix-vector products are m × n outer products. Note that the product AU incorporates Dirichlet boundary conditions in the x-direction, and UB incorporates Dirichlet boundary conditions in the y-direction. Combining the partitioned systems incorporating both A and B matrix multiplications and boundary conditions yields

( A int ) ( U int ) = ( h 2 ) ( F int ) − ( A L ⋅ u 1 T + A R ⋅ u 2 T ) − ( u 3 ⋅ B T T + u 4 ⋅ B B T ) , (4)

which is an m × n linear system for U int .

For Neumann boundary conditions, the edge at which the condition is imposed becomes part of the internal unknowns in the Sylvester system. As an example, consider a Neumann boundary condition given by

∂ u ∂ x = g ( y ) on x = 0. (5)

Staying within the finite difference formulation of derivatives and letting g ( y j ) ≡ g j , this condition can be discretized and approximated with the O ( h 2 ) central difference approximation, which yields

( ∂ u ∂ x ) i , j ≈ U i + 1 , j − U i − 1 , j 2 h = g j for i = 0 , 0 ≤ j ≤ N . (6)

For a Neumann condition along the edge x = 0 , the row vector u 1 T described in (2) becomes a part of the internal array of unknowns U int . In order to implement this finite difference on the edge, we need to introduce a ghost layer with index i = − 1 , and pair Equation (6) to the second derivative operator AU for i = 0 . This gives

U 1 , j − U − 1 , j 2 h = g j ⇒ U − 1 , j = U 1 , j − 2 h g j , ( ∂ 2 u ∂ x 2 ) 0 , j ≈ ( U 1 , j − 2 h g j ) − 2 U 0 , j + U 1 , j h 2 = 2 U 1 , j − 2 U 0 , j h 2 − 2 g j h , (7)

which leads to the following partitioned form of AU,

where the additional term − 2 g j / h is taken to the right hand side of the Sylvester system such that F 0 , j int = F 0 , j int + 2 g j / h for 0 < j < N . Comparing (8) to (2), the size of U int changes from m × n to ( m + 1 ) × n , and A 0,1 is changed from 1 to 2 (shown boldface in (8)), and the right hand side is slightly modified along that edge. Similarly, any edge with a Neumann condition can be handled in this fashion. It is clear that both Dirichlet and Neumann boundary conditions are very simple to implement in the Sylvester iteration method, and only slightly modify the structure of the arrays involved.