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Asymptotic Analysis for Overlap in Waveform Relaxation Methods for RC Type Circuits

Waveform relaxation (WR) methods are based on partitioning large circuits into sub-circuits which then are solved separately for multiple time steps in so-called time windows, and an iteration is used to converge to the global circuit solution in each time window. Classical WR converges quite slowly, especially when long time windows are used. To overcome this issue, optimized WR (OWR) was introduced which is based on optimized transmission conditions that transfer information between the sub-circuits more efficiently than classical WR. We study here for the first time the influence of overlapping sub-circuits in both WR and OWR applied to RC circuits. We give a circuit interpretation of the new transmission conditions in OWR and derive closed-form asymptotic expressions for the circuit elements representing the optimization parameter in OWR. Our analysis shows that the parameter is quite different in the overlapping case, compared to the nonoverlapping one. We then show numerically that our optimized choice performs well, also for cases not covered by our analysis. This paper provides a general methodology to derive optimized parameters and can be extended to other circuits or system of differential equations or space-time PDEs.

computer programs. Simulating circuits means developing a mathematical model to replicate the behavior of the real circuit, and then solve this model numerically. These simulations help the circuit designers to test their ideas, and optimize circuit parameters to achieve the desired output. Simulations save a lot of time and cost when designing circuits, and one can minimize the risk of unwanted hazards. Further, the number of electronic devices is continuously increasing tremendously which in turn increases the need for new design tools and techniques. Circuit solvers like the many different SPICE are no exception. Not only new approaches are desired but the tools need to have an ever increasing capacity to solve larger problems.
The specific class of problems considered in this work is a subset. We consider a subclass of circuits which are called RC circuits which are composed of resistors (R) and capacitors (C). Of course, the circuits will also include current and voltage sources. This circuit is also called RC filter since it is used to filter certain frequencies and let pass others. The most common are high pass filters and low pass filters. We study here the low pass RC filter circuit which is shown in Fig. 1. This circuit allows signals with frequencies lower than a certain cut off frequency to pass. Such circuits are often used in acoustics, optics and electronics and hence their study is of great importance. We formulate an effective mathematical model for its simulation using the well known and widely used MNA (Modified Nodal Analysis), which resembles the finite element assembly procedure in numerical partial differential equations (PDEs) [10,23]. This leads to a large system of ordinary differential equations (ODEs) where the unknowns are voltages at the nodes [24]. It is computational costly to solve such systems especially if their size is in the order of millions. Also, since 2004, the CPU frequency has been stagnating around 3 GHz and there is little hope that in the near future there will be any improvement [8]. Due to the availability of large numbers of processors, we are left with the option of using parallel computing in order to increase simulation speed. Over the years, various numerical tools have been developed to solve PDEs and large systems of equations effectively using parallel computing. One such tool is Waveform Relaxation.
Waveform Relaxation (WR) was developed in 1982 by Lelarasmee, Ruehli and Sangiovanni-Vincentelli [26], for analyzing electric circuits. In WR, the circuit or the system of differential equations is decomposed into many sub-circuits or subsystems which are then solved independently over an entire time window using standard time integration techniques. After each solve, information is transferred between neighboring subsystems using transmission conditions and the process is repeated until convergence is reached. The classical WR algorithm converges slowly when large time windows are used. To overcome this issue, smart transmission conditions were introduced, which led to optimized waveform relaxation (OWR) since different parameters in the transmission conditions need to be optimized. The convergence analysis of the classical WR methods for RC circuits was carried out by Ruehli et al [31]. Further, studies of WR and OWR algorithms applied to small RC circuits can be found in [2,12], and for large RC circuits, see [3,18]. There are also analyses at the discrete level for OWR applied to RC circuits [34]. WR and OWR were also studied for RLCG transmission lines [1,15,16,20,11]. A different approach for the interconnect circuits has been studied in [27], where the interconnect circuit is partitioned using Norton interfaces which depend upon resistors. However, in all these references, both WR and OWR were studied for the case of non-overlapping circuits. We show here that WR for circuits is related to domain decomposition for PDEs, in the case of the RC circuit the heat equation [13,9], and for RLCG circuits the wave equation [14], and domain decomposition methods often use overlap to improve convergence. In this article, we study the effect of overlap on the convergence factor of WR and OWR and present for the first time an analysis for the optimized parameter in the transmission conditions for RC circuits of infinite length, a result which was announced in the short proceedings paper [18].
In WR methods, only the space domain or system of differential equations is split into multiple sub-domains or subsystems. One can further combine these WR methods with certain other time parallel methods, where the time domain is divided into multiple time subdomains. On each time subdomain of the WR method, Parareal [17,19] or the pipeline Schwarz waveform relaxation method [30], or Waveform Relaxation with Adaptive Pipelining (WRAP) [25], or Revisionist Integral Deferred Correction (RIDC) [29,6] can be considered. In the pipeline Schwarz waveform relaxation method and the WRAP method, WR iterates are computed in a pipeline manner to provide speedup and stability, while RIDC produces high order solutions in the same time as the lower order methods [6]. Further, multigrid WR methods [33,22] can also be implemented for parabolic differential equations.
WR methods are becoming popular for the simulation of field-circuit coupled problems. Such problems arises if one wants to look closely into the device. In this situation, a lumped circuit element is replaced by a PDE model. Another perspective for such problems comes from the practical use by the engineer, where different parts of a domain are not equally important and hence some of its properties can be neglected by replacing a part of it by a simple electric circuit, while in other parts the space-time discretization of the PDE is kept. Recently WR method were used for such field-circuit problems at CERN to simulate the quench protection system [4,21]. Moreover, if the RC circuits in the field-circuit coupled problem is divided into multiple sub-circuits, then the results obtained in our manuscript can be directly implemented in them. Further, OWR methods are also applied for the coupled electromagnetic field and power-electronic simulations [7].
We explain in Section 2 how one can obtain the mathematical model for the RC circuit of infinite length using MNA, and we show that the RC circuit is an approximation to the time dependent heat equation. In Section 3 we present and study the classical WR algorithm with overlap for RC circuits, and identify its convergence problems for low frequencies. In Section 4 we introduce the OWR with overlap for RC circuits. We give a circuit interpretation of the new transmission conditions, and also show that they can be interpreted as Robin transmission conditions for the associated heat equation. We then optimize the transmission conditions of OWR in Section 5 using asymptotic techniques. In Section 6 we show how to generalize our results to the many sub-circuit case. We perform some numerical tests to support our theoretical results in Section 7, and present our conclusions in Section 8.

Mathematical Model
We are interested in determining the voltages at the nodes of an RC circuit as in Fig. 1.
To develop a mathematical model of this circuit, we use MNA, originally described by Ho, Fig. 3 Element stamp for a capacitor C. Ruehli and Brennan [24] in 1975. This technique produces a system of ODEs of the forṁ v = Av + f, where the entries of the matrix A contain the elements of the circuit, v is the unknown vector of voltages at the nodes, and f contains the source terms. We now show how the matrix A is built using MNA, which is similar to the finite element assembly procedure [5].
In an electric circuit, any electric device with two terminals is called an element, and its terminals are called nodes. An electrical circuit is thus a system consisting of a set of elements and a set of nodes. The contribution of every element to the matrix equation is described by means of a template, called an element stamp [28]. Each element has a unique stamp which is given with the help of Kirchoff's Current Law (KCL) and Kirchoff's Voltage Law (KVL). For example, consider a resistor R placed between nodes with voltages v + and v − , as shown in Fig. 2 (left). Let i be the current passing through it. Then using KCL, we The corresponding element stamp is shown in Fig. 2 (right). The element stamp for a capacitor C is shown in Fig. 3.
To build the "descriptor system" form Mv = Kv +f, we start with a zero RHS vector f ∈ R N , and zero matrices M, K ∈ R N×N , where N is the number of nodes in the circuit. The vector v contains the unknown voltages at the nodes and remains unchanged throughout this process. We then read every element of the circuit one by one. As an element is read, its element stamps are added to the matrices M and K. Since the element stamp of a capacitor contains derivative terms, its element stamp is added to the matrix M, while the element stamp of a resistor is added in matrix the K. The independent voltage source term is stamped into the vectorf, see Fig. 4 for the element stamp of an independent voltage source. This process continues until all elements of the circuit are read and thus we obtain a system of differential equations in time of the formv = Av + f, where A := M −1 K and f := M −1f . For the infinite RC circuit in Fig. 1, this system looks like where and f = (I s (t)/C −∞ , 0, 0, ..., 0) T . This system of ODEs can also be viewed as semi-discretization by the method of lines of a heat equation in space-time: if we consider small resistors and capacitors, R i ≈ ∆ x and C i ≈ ∆ x, then each equation of the system (1) takes the form and as ∆ x → 0, we arrive at heat equation Hence we can consider the RC circuit of infinite length as an approximation for the one dimensional heat equation on the unbounded domain Ω x = (−∞, ∞). For a short vector v, one can solve the system of ODEs (1) by simple discretization and matrix manipulation. However if the size of the vector v is in the millions and for more complicated circuits, solving the system (1) is computationally costly, and one needs parallel methods. One such method is WR as we now show.

The WR Algorithm
For a general WR algorithm, one decomposes the system (1) into many sub-systems, but to understand the key features of WR, we consider a decomposition into two sub-systems only. We decompose the infinite RC circuit from Fig. 1 at node 0 into two equal sub-circuits, and motivated by the PDE relation shown in (3) and the Schwarz WR algorithms studied in [13,9] which use overlap, we include an overlap of n nodes in the first sub-circuit. Let us denote the first sub-system unknowns by u(t) and the second sub-system unknowns by Fig.  5 shows the decomposition of the circuit when two nodes are in the overlap. We see that when we decompose the circuit, we need to add a voltage source u 3 in the first sub-circuit and w 0 in the second. At each iteration k, these voltage sources are given by transmission conditions which transfer information between the sub-circuits. The system of differential equations for the decomposed sub-circuits arė where the unknowns u k+1 n+1 (t) and w k+1 0 (t) are determined by the transmission conditions We see that at each iteration, these transmission conditions transfer voltages at the interface. Comparing this with Schwarz WR applied to PDEs in [13,9], these conditions can be interpreted as Dirichlet boundary conditions. To start the algorithm, we specify an initial guess for the solution w 0 n+1 (t) and u 0 0 (t), and then solve the sub-systems (4) for all time t ∈ (0, T ]. Note that the two sub-systems can be solved in parallel, since in the transmission conditions (5) the two sub-systems use both data from the previous iteration, like in a block Jacobi method [32] from linear algebra. One could also do the solves sequentially, and use the newest value available in the second transmission condition, w k+1 0 (t) = u k+1 0 (t), which would be more like a block Gauss Seidel iteration from linear algebra, but the convergence analysis is similar, so we will focus on the parallel version here.

Convergence Analysis of the classical WR Algorithm
The presence of different resistors R i and capacitors C i makes the convergence analysis difficult, so to simplify, we assume that all resistors and capacitors have the same value, i.e. R i := R and C i := C for all i ∈ Z which leads to a i = a, b i = −2a and c i = a for all i ∈ Z.
We study the convergence of the WR algorithm (4) with transmission conditions (5) using the Laplace transform: Definition 31. If f (t) is a real or complex valued function of the non negative real variable t, then the Laplace transform is defined by the integral Since the system (4) is linear, the error equations correspond to the homogeneous problem, f = 0, with zero initial conditions, u k+1 (0) = w k+1 (0) = 0. The Laplace transform allows us to show thatû k+1 andŵ k j are the Laplace transforms of u k j and w k j , and ρ n,cla (s) is the convergence factor. We can then use the Parseval equality for s = σ + iω, σ ≥ 0, to obtain a convergence estimate in the weighted Convergence in Laplace space with a convergence factor ρ n,cla (s) less than one implies convergence in the time domain in the weighted norm · σ , and for σ = 0, we obtain convergence in L 2 . We now find a closed form for the convergence factor ρ n,cla . The Laplace transform for s ∈ C of the WR algorithm (4) is given by where we have already included the transmission conditions (5); this shows the dependence ofû onŵ and vice-versa. Solving the sub-systems (7) requires solving a recurrence relation of the form In order to find the convergence factor of the WR algorithm, we need the following lemma.
Proof. The proof of this lemma can be found in [18].

Theorem 1
The convergence factor ρ n,cla of the classical WR algorithm (7) with n nodes overlap for an RC circuit of infinite length is given by Proof. The detailed proof of this theorem can be found in [18], but for completeness, we present a brief sketch of the proof. Solving the recurrence relation ay j−1 + (b − s)y j + ay j+1 = 0 for y j =û k+1 j givesû k+1 j = A k+1 λ j 1 for j = (. . ., n − 2, n − 1, n), and for y j =ŵ k+1 j we getŵ k+1 j = D k+1 λ j 2 for j ∈ N, with λ 1 and λ 2 defined in Lemma 1. The transmission conditions (5) determine the constants A k+1 and B k+1 and we find The above expression together with Vieta's formulas From Lemma 1, we see that if |b| = 2a and s = 0, the convergence factor |ρ n,cla (0)| = 1, while for b = −(2 + ε)a with ε > 0, we have |ρ n,cla (s)| < 1. It is therefore interesting to study the case ε > 0 and then analyze the limit as ε → 0. Infact, in this limit, the rate of convergence deteriorates at ω = 0, as we show on the left in Fig. 6. We also show on the  Fig. 6 that increasing the overlap increases the rate of convergence, but the effect is very small for ω close to zero when s = iω.
For circuits, the introduction of ε leads to the addition of a resistorR = R/ε at each node of the circuit (see Fig. 7). The inclusion of the resistorsR presents a practical case where there is a small leakage of current in the dielectric current. The limit ε → 0 leads to the limitR → ∞, which means that no current passes through this resistor. Thus considering b = −(2 + ε)a and taking the limit ε → 0 states that the circuit in Fig. 7 is a good approximation to circuit in Fig. 5. Adding ε > 0 also corresponds to convergence in a weighted norm, with choosing σ = ε > 0 in s. This can be seen be substituting w(t) = u(t)e εt .

OWR Algorithm
The main drawback of the classical WR algorithm is that it is very slow especially when large time windows are used. Large time windows in real space correspond to small frequencies ω in s = σ + iω in Laplace space. When ε → 0 with b = −(2 + ε)a, we observe that |ρ n,cla (0)| → 1, which means the convergence rate slows down for small ω, and increasing the overlap does mostly improve the convergence of higher frequencies ω, as one can see in Fig. 6 on the right. The Dirichlet transmission conditions (5) which exchange just voltages at the interfaces are the main reason for this slow convergence. We thus search for better transmission conditions to exchange information between the sub-circuits. One way is to use optimized transmission conditions, which are defined for this RC circuit in [18] as where α, β ∈ R and k is the iteration index. These transmission conditions exchange both voltages and currents at the interface, which can be seen by dividing the first equation by α and the second by β . The term can be viewed as a current and u k+1 n+1 is a voltage. These transmission conditions are called optimized transmission conditions since we need to find the best (optimized) values for α and β such that the convergence factor is as small as possible.
For circuits, the introduction of the new transmission conditions (10) means two voltage sources need to be added at the interface of each sub-circuit. From Fig. 8, we see that the resistors R α := R(1 + α) and R β := R(1 − β ), which depend on the parameters α and β , are added. Similar to the WR algorithm, at each iteration the voltage sources w 3 , w 2 , u 0 , u 1 are transferred between the sub-circuits.
We can also interpret the new transmission conditions (10) as Robin transmission conditions for the discretized heat equation (2): if we divide the first equation of (10) byᾱ := α p for some p > 0 and the second equation byβ := −β p , we obtain If we consider nowᾱ ≈ ∆ x andβ ≈ ∆ x, then the fractions inside the above equations represent discretization for the derivative ∂ u ∂ x and ∂ w ∂ x , and we thus obtain in the limit Robin transmission conditions,

Convergence of OWR
As for the classical WR algorithm, we analyze the convergence of the OWR algorithm in Laplace space. We first rearrange the optimized transmission conditions as where all the voltages u i and w i depend on time t. We substitute these rearranged transmission conditions into (4) and take the Laplace transform to arrive at We need to give initial guessesŵ 0 n+1 ,ŵ 0 n ,û 0 0 andû 0 1 to start this algorithm.

Theorem 2
The convergence factor ρ n (s, α, β ) of the OWR algorithm (12) for an RC circuit of infinite length is given by Proof. To find the convergence factor, we proceed as in the proof of Theorem 1 to arrive atû k+1 j = A k+1 λ j 1 for j = (. . . , n − 1, n), andŵ k+1 j = D k+1 j λ j 2 for j ∈ N. To determine the constants A k+1 and D k+1 , we use the optimized transmission conditions (11) and get Using λ 1 λ 2 = 1 and equations (14), we arrive at where the convergence factor ρ n (s, α, β ) is given by (13). Similarly, we can also show that w k+1 j = ρ n (s, α, β )ŵ k−1 j , for j ∈ N, and this completes the proof.
We observe from (13) that the effect of overlap on the convergence factor given by 1 λ 2 1 n is the same for both the WR and the OWR algorithm. This means increasing the overlap increases the rate of convergence also for OWR. The convergence factor is also the same for all the circuit nodes irrespective of which sub-circuit they belong to. Further, for fast convergence, we would like to have the convergence factor |ρ n (s, α, β )| as small as possible. The parameters a, b represent circuit elements and cannot be changed, but we can choose α and β such that |ρ n (s, α, β )| becomes as small as possible. So what is the best possible choice for the parameters α and β ?
One can see that this is the best choice, since the solution in each sub-system depends on all the source terms f j and during the first iteration, each sub-system uses only the local f j to compute the approximation. It is only for the second iteration that information is transferred. Therefore, convergence cannot be achieved in less than 2 iterations.
Since λ 1 is a complicated function of s ∈ C, its inverse Laplace transform leads to non local operators in time for α opt and β opt . These non local operators are expensive to use since they require convolution operations. It is therefore of interest to approximate α opt and β opt by a polynomial in s. In this paper, we will focus on approximation of α opt and β opt by a constant.
Since s ∈ C with s = σ + iω, the above optimization problem is in four variables and hence already very difficult to solve. We simplify the problem further using some assumptions and the following lemmas.
Proof. The detailed proof of this lemma can be found in [18]. The idea is to show that ρ n (s, α, β ) is analytic in the right half of the complex plane and then to use the maximum modulus principle for analytic functions.
From Theorem 3, we observe that α opt and β opt are related to each other via the relation β opt = −α opt , which suggests the natural assumption β = −α. In our RC circuit of infinite length, this would mean that at the interface (where the circuit is split into two), the current flowing in both sub-circuits is equal but into opposite direction. This interpretation is easy to see for the non-overlapping case n = 0. Recall that the terms we can estimate ω min = π T and ω max = π ∆t . Note that ω min > 0, but to further simplify our analysis, we consider a wider range for ω, that is, ω ∈ [0, ω max ]. Therefore, our min-max problem (16) reduces to We observe numerically, see Figure 9, that the solution for the min-max problem (17) is given by equi-oscillation. However the behavior of equi-oscillation is different for the nonoverlapping and overlapping case. We first analyze the non-overlapping case, that is, n = 0, where the equi-oscillation occurs for ω = 0 and ω = ω max → ∞ (see the left plot of Fig. 9). This means the optimized α denoted by α * 0 satisfies where we have dropped the parameter −α for simplicity, ρ n (ω, α) := ρ n (ω, α, −α).
The analysis is different for the overlapping case n > 0. Numerically, we observe that the solution of the min-max problem (17) is also given by equi-oscillation, see the right plot of Fig. 9. But in this case, equi-oscillation for |ρ n (ω, α)| occurs for ω = 0 and ω =ω, wherẽ ω → 0 as ε → 0. The dependence of the optimized α * andω on ε for n = 1 can be seen in the right plot of Fig. 10 and in Fig. 11. We therefore use the ansatzω := C ω ε η , for some η > 0. Solving the problem (17) is equivalent to solving the system of equations |ρ n (0, α * n )| = |ρ n (ω, α * n )| and where α * n is the optimized α for an overlap of size n. We first solve the equation ∂ ∂ ω |ρ(ω, α * n )| = 0 and find a relation betweenω and α * n . Substituting this relation into the first equation of (23) then gives an explicit expression for α * n .
The analysis to find the expression for |ρ n (ω, α)| is similar. Substitutingω = C ω ε δ in to the expressions for A(ω, α) and B(ω, α) leads to Dividing A(ω, α) by B(ω, α) and using the Taylor expansion 1 1+z = 1−z + z 2 2 +O(z 3 ) gives us Similarly, we obtain the expression for D(ω), Multiplying the above expression by the expansion for A(ω) B(ω) , we find, which completes the proof.

Multiple Sub-circuits
We studied so far only the decomposition of the system of equations (1) into two subsystems. However, for the practical use on parallel computers, we need to split the system into many sub-systems and apply the WR or OWR algorithm to them. We thus split the RC circuit of infinite length in Fig. 1 into N s sub-circuits which are denoted by v r , r = 1, 2, . . . , N s . Assume that each sub-circuit v r contains M r nodes. Applying the Optimized Waveform Relaxation algorithm (with α, β as the optimization parameters) with n nodes overlap leads tȯ v k+1 where r = 2, 3, . . . , N s − 1 and we have considered the source term f = 0 for simplicity. For α = ∞, β = −∞, we obtain the classical Waveform Relaxation algorithm.
In the case of OWR for two sub-domains, we saw that with the optimal transmission conditions (15), convergence is achieved in 2 iterations and the parameters α, β represent non-local operators in time. In the case of N s sub-circuits, with the use of optimal transmission conditions, one can show that convergence can be achieved in N s iterations [12], and this result still holds in the overlapping case. Again these parameters α, β are nonlocal operators in time and the analysis becomes very complicated. Hence we will use α * 0 and α * n given in Theorem 4 and Theorem 5 from our two sub-circuit analysis and test their performance numerically for the multiple sub-circuit case.

Numerical Results
We consider an RC circuit with R = 0.5kΩ, C = 0.63pF , a = 1 RC and b = −(2 + a)ε with ε > 0, ε → 0. Our analysis for both WR and OWR was performed assuming that the length of the circuit is infinite. For our numerical experiments, we consider a circuit of length N = 200, a homogeneous source, zero initial conditions, and apply the backward Euler scheme for time integration with ∆t = 0.1 and T = 2000. We use random initial guesses to start both the WR and OWR algorithm. The left plot of Fig. 12 clearly shows the important influence of our optimized transmission conditions (10) on the convergence of the algorithm. We used the optimized α for OWR given in Theorem 4 and Theorem 5. We also illustrate in the same figure the influence of overlap on convergence. Increasing the overlap increases the convergence rate in both the WR and OWR algorithm, but the impact of the optimized transmission conditions is much more important for performance than the influence of the overlap.  We have seen in (2) that for ε → 0, the RC circuit of infinite length is an approximation to the one dimensional heat equation. We now apply the WR and OWR algorithms with different overlaps n = 0 and n = 2 to the heat equation (2). For OWR we need the values of α * 0 and α * 2 , which we obtain both by solving the min-max problem numerically, and by using the asymptotic expression in Theorem 4 and Theorem 5 respectively from our circuit analysis for small ε = 10 −4 . From the right plot of Fig. 12, we observe that OWR converges much faster than WR. Also, the numerically calculated α * 0 and α * 2 give us the best convergence. For overlap n = 2, the convergence of OWR using our asymptotic expression for α * 2 for ε = 10 −4 needs only 4 more iterations, than the best possible choice computed numerically. This is very little compared to the iterations and time required to compute α * 2 numerically. This shows that our expressions in Theorem 4 and Theorem 5 can also be used when OWR is applied to the heat equation.
We next compare the numerically and asymptotically optimized values of α * n . For the non-overlapping case n = 0 and ε = 10 −4 , we show in Fig. 13 on the left the error reduction for different values of α, together with the numerically and asymptotically optimized value. We see that our asymptotic formula underestimates the optimal choice a bit. However, the right plot in Fig. 13 shows that for the two different values of the optimized α, the corresponding convergence factors |ρ 0 | in Laplace space are very close to each other, and we also show the improvement in the overall convergence for OWR compared to WR. In Fig. 14 we show the corresponding results for the overlapping case n > 0, and the observations are  similar as for n = 0. Finally, we consider the case when we split our RC circuit with size N = 1000 into multiple sub-circuits. We choose ε = 10 −5 and T = 1000, with zero initial condition, homogeneous source and random initial guesses. We use our optimized α in in Theorem 5 from the two sub-circuit analysis. We first consider N s = 5 and then N s = 50 sub-circuits. We see in Fig. 15 that our asymptotically optimized parameters for two sub-circuits work extremely well for the many sub-circuit case, and make the OWR solver into a rather effective method for solving many sub-circuit problems.

Conclusion
We presented a first analysis for the influence of overlap on the convergence factor for both the WR and OWR methods applied to RC circuits. We found that increasing the overlap increases the convergence rate for both WR and OWR, but the impact of optimized transmission conditions is far more important than the impact of overlap. The overlap however changes the optimized parameters, and we provided closed form asymptotic formulas for them. Using OWR with these parameters leads to much lower iteration counts, also when OWR is used for many sub-circuits. We also showed that our optimized parameters can be used when solving heat equations with waveform relaxation techniques. A final important contribution is our interpretation of these optimized transmission conditions as circuit elements, which should help circuit designers to better understand and embrace this new technology.