The Floating Random Walk Method With Symmetric Multiple-Shooting Walks for Capacitance Extraction

A key factor affecting the computational time of floating random walk (FRW)-based capacitance extraction is the variance of underlying Monte Carlo (MC) sample of capacitance. For achieving a fixed accuracy of result, the number of walks executed is proportional to the variance of this underlying random variable. In this work, we study the way to reduce the variance of random variable in FRW method through some theoretical analysis. An FRW method with symmetric multiple-shooting (SMS) walks is proposed, which stems out <inline-formula> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> symmetric subwalk paths from a same sample point on Gaussian surface (with <inline-formula> <tex-math notation="LaTeX">$N_{s}$ </tex-math></inline-formula> being 2, 4, 8 or 16). Theoretical analysis reveals that the method with SMS walks could reduce the number of walks compared to the FRW method with important sampling (IS) approach under some assumption, and thus runs faster even considering the increase of hops within a walk. Its benefits also include the reduction of sampling points on Gaussian surface, which shows large benefit when the sampling on a complex Gaussian surface is very costly. Numerical experiments on the parallel-plate structure have validated the correctness of the theoretical analysis on the variances. With test cases from integrated circuit and flat panel display design, the proposed method with SMS walks is compared with the method with IS approach and the method with both IS and stratified sampling (SS) approach. The results show that the proposed method with SMS walks runs in similar speed or much faster than the FRW method using the IS+SS scheme, with up to <inline-formula> <tex-math notation="LaTeX">$10.1 \times $ </tex-math></inline-formula> speedup.

finite difference method (FDM), finite element method (FEM) and boundary element method (BEM).The FRW method, in contrast to the deterministic numerical methods, avoids the geometric discretization of simulation domain.It does not need to employ a matrix solver to deal with the linear equation system, but adopts the Monte Carlo (MC) method for numerical estimation.The FRW-based solvers are favored because of their scalability to large-scale problems (lowmemory costs), tunable accuracy and convenience for parallel computing.
In the past few years, the FRW method has been extended to efficiently handle structures with cylindrical inter-tier-vias in 3-D ICs [5], non-Manhattan conductors [6], general-shape floating metals [7], random process variation [8] and nonstratified dielectrics [9].Also, the FRW method has been modified to execute capacitance simulation in the design process of FPDs [10].For higher performance, it has GPU, FPGA and distributed parallel computing implementations [11], [12], [13], [14].Besides, it has been extended to handle stratified dielectrics more efficiently [15].To provide encryption service, random walk method can utilize the capacitance macromodel for protecting sensitive structural information from foundry or IP vendor [16].A reliable FDM technique was proposed recently to build capacitance macromodel with high accuracy [17].With new absorbing boundary conditions, the FRW method can be extended to simulate infinite large domain [18].The most recent work on FRW method is to improve the accuracy on extracting structures with nonstratified dielectrics with a deep-learning-driven approach [19].
The variance reduction technique is very important for accelerating the random walk-based capacitance extraction.As the number of executed walks increases, the sample variance of capacitance estimator which reflects the stochastic error of the capacitance is reduced.The FRW procedure terminates after the preset accuracy demand (depicted by the stochastic error) is met.The sample variance in the FRW procedure equals the variance of the underlying random variable (the capacitance estimate from an FRW path) divided by the number of walks.Therefore, effective reduction of the variance of underlying variable for an FRW path enables executing fewer walks and thus less computational time for achieving the same accuracy of result.In this work, a variance reduction technique via executing symmetric multiple-shooting (SMS) walks is proposed, which enables an FRW method with SMS walks for faster capacitance extraction.The method stems out multiple subwalk paths from a single point on Gaussian surface.Although the multiple subwalk paths include more hops for a walk, the reduction of variance (i.e., walks) and the sample points on Gaussian surface make larger benefit on reducing the overall computational time, especially for extracting structures with complicated multiblock master conductor.
The major contributions of this work are as follows.1) A rigorous variance analysis for the random variables in the FRW methods for capacitance extraction and related computation of electric potential and electric field intensity is conducted.It derives several theoretical formulas for variances of electric potential and field intensity for the parallel-plate structure, and we have validated their correctness with numerical experiments.2) In order to reduce the samples on Gaussian surface during the FRW method for capacitance extraction, the FRW methods with naive multiple-shooting (NMS) walks and SMS walks are proposed.We have conducted variance analysis on the proposed methods and found out that the methods with NMS walks have less variance than the existing method with importance sampling (IS) [4].Furthermore, the analysis reveals that the method with SMS walks has less variance than that with NMS walks, and has less computational time than the method with IS approach even considering the increase of hops within a walk, if assuming the first transition cube on top or bottom of Gaussian surface is always a single-dielectric one.3) Numerical experiments with IC and FPD structures have validated the benefit of the FRW method with SMS walks.Compared with the state-of-the-art variance reduction technique for FRW, i.e., the importance sampling plus stratified sampling (IS+SS) scheme [4], the FRW method with SMS walks shows similar or much less (up to 10.1× speedup) computational time while achieving the same accuracy of result.Besides, a speedup estimation formula is proposed, based on which we can automatically choose whether to use the FRW method with SMS walks or use the IS+SS scheme instead, for achieving the shortest computational time.The remainder of this article is organized as follows.In Section II, the FRW method for capacitance extraction is reviewed along with the existing techniques for variance reduction and sampling on the Gaussian surface.In Section III, we conduct theoretical analysis and derive the analytical expressions for the variances of underlying random variables within several FRW methods.In Section IV, we propose the FRW method with NMS walks and SMS walks, and discuss the performance of these FRW variants.In Section V, experimental results validating the theoretical analysis and the proposed methods are presented.Section VI gives the conclusions.

II. FLOATING RANDOM WALK METHOD FOR
CAPACITANCE EXTRACTION The problem of capacitance extraction is to compute the capacitances of a master conductor i: C ij 's.Here, C ii is the total or self capacitance of conductor i, while C ij is the coupling capacitance between conductor i and j (i = j).The capacitances are the coefficients in the linear expression of the master conductor's electric charge with respect to all conductors' electric potential (voltage).The FRW method for capacitance extraction is based on the Gauss theorem expressing the electric charge of conductor i as where G is a closed surface surrounding the master (called Gaussian surface), and the formula of electric potential where S denotes the surface of a conductor-free cube (called transition cube) [3], [4].ε(r) and E n (r) denote the dielectric permittivity and normal electric field intensity at point r, respectively.For (2), r is usually fixed at the center of transition cube, while r 1 is a point on S. P(r, r 1 ) is the surface Green's function which can be regarded as a probability density function (PDF).The discretized form of P(r, r 1 ) is called green's function table (GFT), essentially a set of probabilities for describing the PDF, computed with FDM [4], [15].
The FRW method can be illustrated with Fig. 1.The MC integration method is applied to evaluate (1).Thus, a random r is sampled on G.To evaluate the integrand of (1), one can utilize (2) because electric field intensity E(r) is the negative gradient of potential φ(r), or where n(r) denotes the unit vector along the outer normal direction of G at point r.This derives where P 1 (r, r 1 ) denotes the surface Green's function for transition cube S 1 (see Fig. 1).The subscript r indicates that the gradient is with respect to r.
If one defines function w(r) = (1/g)E n (r), where g is a constant which forms a PDF for sampling G: g = 1/ G ε(r)ds, then ω(r, r 1 )P 1 (r, r 1 )φ(r 1 )ds (5) where and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Here, ω(r, r 1 ) is called weight value and is the sampling value (or estimate) of capacitance [4].The integral in (5) includes φ(r 1 ), which can be further expanded by substituting (2).This means to construct another transition cube centered at r 1 and make sampling on its surface S 2 (see Fig. 1).This operation is repeated, and results in a procedure like a FRW starting from Gaussian surface and consisting of a series of random hops.At each hop, the largest conductor-free transition cube is built, and its GFT determines how to randomly select the hop's destination.The random walk finally terminates at a conductor's surface, say conductor j's, and the weight value for this walk becomes an estimate of the capacitance C ij .With millions of such FRW paths, the averages of capacitance estimates well approximate the desired capacitances [4].
In order to efficiently execute the FRW procedure, the GFTs for the transition cubes with various dielectric configurations are calculated offline during a precharacterization stage.Similarly, the calculation of weight value leverages the precalculated weight value tables (WVTs) which store the three partial-derivative components of ∇ r P 1 (r, r 1 ), i.e., Here, (x, y, z) denotes the coordinates of r.The values of W (x) (r, r 1 ), W (y) (r, r 1 ) and W (z) (r, r 1 ) for different r 1 's on a unit-size transition cube are stored, with r being the cube's center.Moreover, special approaches can be applied to reduce the volume of GFTs and WVTs for multilayer-dielectric transition cubes [15].The FRW method is applicable to the computation of electric potential, based on (2), and the computation of electric field intensity, based on (4) [20].The theory of FRW method is built upon the MC method and the Markov random process, which reveals that the result converges to the accurate capacitance [21], [22].The contribution (i.e., weight value or 0) of an FRW path to a certain capacitance can be regarded as a sample of a random variable governed by the Markov random process with certain setting of absorbing condition.For example, the contribution to C ij corresponds to a random variable (say the distribution is F) governed by the Markov process with the absorbing condition of setting conductor j 1V and the others 0V.With this bias setting, Q i will equal C ij in value.Thus, the expectation of F equals C ij .After N walk walks, we will have N walk i.i.d.samples from F and the estimator Ĉij is obtained by averaging them.According to the central limit theorem (CLT) [23], Ĉij is approximately a normally distributed random variable, whose expectation equals that of F, i.e., the accurate capacitance C ij .Because statistical error is the major factor, the accuracy of capacitance is measured by the random variable's variance or standard deviation (STD).Assuming the STD of F is σ , the computed Ĉij has the STD error which decreases with the increase of N walk .Approximating σ with the capacitance estimates from the random walks, one can monitor the accuracy of capacitance with (9).Once it (or practically, its relative quantity) satisfies a preset criterion, the FRW procedure can be terminated [4].Equation ( 9) means that the convergence rate of FRW method depends on the variance (σ 2 ) of the random variable governed by the Markov process.If one can modify the random walk scheme to reduce the variance, fewer walks would be required to attain a given accuracy criterion.
For accelerating the FRW method for capacitance extraction, the variance reduction approaches based on importance sampling (IS) and stratified sampling (SS) have been proposed [4].The idea of IS is to convert (5) to g|∇ r P 1 (r, r 1 ) • n(r)| q(r, r 1 )φ(r 1 )ds (10) where and the new PDF for sampling S 1 is The discretized form of this probability function (denoted by PDF-IS table) is actually used, which can be obtained by calculating the linear combinations of WVT entries.With this scheme, the new weight value (capacitance estimate) becomes whose value roughly depends on whether r 1 is on the region outside Gaussian surface (S + 1 ) or inside Gaussian surface (S − 1 ) [4].And, computing this weight value can be facilitated by precalculating the sums of the absolute values of WVT entries for the x, y and z directions, respectively.They can be precomputed and stored in a table named SUM-WVT.
The idea of stratified sampling (SS) is to divide the surface S 1 into multiple regions, and then sample on them separately to compute the capacitance estimates and their STDs.Practically, two regions (i.e., S − 1 and S + 1 ) are usually set according to ω's sign.For attaining the same accuracy criterion, it has been demonstrated that N walk and the runtime of the FRW method can be reduced remarkably with the approach of IS and SS applied [4].The combined IS and SS can be seen as the state-of-the-art variance reduction technique for the FRWbased capacitance extraction.
A FRW starts from generating a sample on the Gaussian surface.The virtual Gaussian surface sampling (VGSS) method [24] enables this first step executed correctly, without actually constructing and sampling the single whole geometry of Gaussian surface for the actual conductor net composing of multiple conductor blocks.It is achieved by utilizing the rejection sampling (RS) technique and sampling points on the combination of the Gaussian surfaces for individual conductor blocks (called virtual Gaussian surface).In order to ensure that sampling on the virtual Gaussian surface is equivalent to that on the complicated single whole Gaussian surface, some sample points on the former (e.g., those inside other conductor blocks) must be rejected.The RS technique is also employed to ensure that the accepted samples follow desired probability distribution.Therefore, when the master net includes a lot of Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
blocks and is of complex geometry, this VGSS sampling step costs considerable time to obtain a valid sample on Gaussian surface.The total computational time of FRW method can be expressed as (14) where N hop denotes the average number of hops per walk, T walk and T hop denote the average time for executing an FRW walk and hop, respectively.T 1 here denotes the average time for generating a sample on Gaussian surface and calculating the weight value.If using R 1 to denote the ratio of we can further derive Usually R 1 > 1 and can be much larger if the master net has complicated geometry.Suppose that we can somehow reuse the sample point on Gaussian surface and launch multiple subwalks from it, which makes N walk reduced by a factor and N hop multiplied by the same factor.According to (16), this would lead to the reduction of T total , especially when R 1 is a large number.This inspires the idea of performing multipleshooting walks.In next sections, we will propose such an accelerated FRW method and prove its benefit on variance reduction.

III. VARIANCE ANALYSIS FOR THE FRW METHOD
In this section, we conduct the variance analysis on the random variables in the computation of capacitances and electric potential with the aforementioned FRW method.Each random walk path in the FRW method includes a Markov process.Without loss of generality, we consider the Markov process with the absorbing condition setting the conductor j 1V and the others 0V.j can equal i (the identity of master conductor).Therefore, the analysis is valid for the self capacitance C ii and any coupling capacitance C ij .

A. Notations
Under the context of the FRW method presented in Section II, we denote the Markov chain for the basic FRW method (i.e., without IS or SS) by {X t , t ≥ 0}, with a continuous state space ⊂ R 3 .The initial distribution and the one-step transition probability density are as follows: Define In other words, X T is the point where the Markov chain reaches the boundary ∂ for the first time.In the scenario of FRW, ∂ consists of the surfaces of all conductors.By sampling the Markov chain that starts from r and terminates on the absorbing boundary, one can draw a sample of φ(X T ).Note that X T is always with a known potential and φ(X T ) only takes binary values.We call φ(X T ) a "return value" of an FRW path for calculating electric potential.For calculating electric field intensity or induced charge, the return value is further multiplied by the weight value ω in ( 6), whose value depends on the first two points X 0 and X 1 .
We use three random variables to denote the "return value" of an FRW path for calculating electric potential, electric field intensity (w(r) is actually considered for simplicity) and electric charge, respectively They are unbiased as proven in Appendix A Under the setting of bias voltage, Q i equals C ij .Thus, cohering with the notations in Section II (see the analysis of ( 9)), we have If the IS approach [4] is used, w(r) will be evaluated with a different weight function ω (see ( 10)-( 13)).The transition probability density of Markov chain must be modified accordingly.We denote the IS-based Markov chain by { Xt , t ≥ 0}.The only distinction is that the first-step transition probability density is changed to The other transitions are identical to those in the basic FRW method.Likewise, we define Notations for the IS scheme are marked with tildes.The unbiasedness still holds, as discussed in Appendix A B. Variance of X φ (r) and X Q In this section, we first give the following theorem, which presents an important statement on the variance of X φ (r).Notice that there is only one FRW method for computing the electric potential at a point, whereas different FRW variants exist for computing capacitance and electric field intensity.Then, we analyze the second moment of X w (r).Finally, we derive the variance of X Q for two different FRW variants: 1) the vanilla FRW and 2) the method with IS [4].
Theorem 1: Suppose the FRW method is used to calculate the electric potential φ(r), under the condition that one conductor is at 1V and the other conductors 0V.X φ (r) denotes the random variable representing the return value for calculating φ(r) from one FRW path.Then and the variance Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Proof: The proof of E[X φ (r)] = φ(r) has been given in Appendix A. As X φ (r) takes binary values 0 or 1, its secondorder moment equals its expectation Thus, (25) has been proven.The variance of X φ (r) is then Theorem 1 reveals that the variance of X φ (r) only depends on the actual electric potential at point r.Under our problem setting, φ(r) has the value ranging from 0 to 1. So, the FRW method calculating φ(r) converges at different rate for different r.It converges the fastest for the position r close to or far away from the master conductor at voltage 1V.
The random variable X w (r) represents the value for w(r) from one FRW path.It depends on the outer normal direction n(r) of Gaussian surface at r. Below we take n(r) along the x-axis as an example to analyze the second moment of X w (r), which is useful for deriving Based on (5), we see that X w (r) depends on ω(r, r 1 )X φ (r 1 ) according to the PDF P 1 (r, r 1 ).So, with the Markov transition scheme and the conditional probability theory we derive If n(r) is along y-or z-axis, similar expression of E[X 2 w (r)] can be derived where only W (x) (r, r 1 ) is replaced with W (y) (r, r 1 ) or W (z) (r, r 1 ).
If the IS approach is applied, we have where Now, let us consider the variance of X Q .By the law of total expectation, we see Based on (29), it can be transformed to where G x , G y and G z are the parts of G whose normal direction is along x-, y-and z-axis, respectively.And, S 1x , S 1y and S 1z denote different transition cubes.The variance of X Q is then Similarly, according to (30), the second moment of XQ produced by the FRW method with IS approach is We denote XQ 's variance by D (IS) , which equals

C. Special Case With Parallel-Plate Structure
In the case of parallel-plate capacitors, between the two plates there is uniform electric field.The coupling capacitance between the two plates has the analytical solution where A is the area of the plate, d is the distance between the two plates, and ε is the permittivity of the homogeneous dielectric filling between the two plates.Constructing a parallel-plate structure with d much smaller than the edge size of plate, we can apply the FRW method to compute the electric potential, electric field intensity and coupling capacitance, which well match the above theoretical results.Fig. 2 illustrates the parallel-plate structure.Setting the z-axis along downward direction as in Fig. 2, we have the analytical solution of electric potential and field intensity Applying Theorem 1 to the parallel-plate structure, we see The MC variance reaches the maximum at z = (d/2), i.e., r is at just the middle between the two plates.The constant g Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. in (7) for this case is (1/εA).So, for r on the Gaussian surface G between the two plates Suppose XE (r) denotes the random variable for computing E n (r) with the FRW method with IS approach.We see XE (r) = g Xw (r).So, based on (30), it derives Here, we consider that n(r) is along the z-axis.Therefore, with the first transition cube and the corresponding single-dielectric WVT we can compute the variance of XE (r) directly from (41), without performing the FRW method.
IV. FLOATING RANDOM WALK ALGORITHM WITH SYMMETRIC MULTIPLE-SHOOTING WALKS In this section, we propose a novel FRW scheme with SMS walks, which stems out multiple subwalk paths from a point on the Gaussian surface.We prove that, with this kind of SMS walks the variance of FRW procedure for capacitance extraction can be remarkably reduced.Therefore, fewer samples on the Gaussian surface are needed for attaining certain accuracy, which benefits resolving the problem caused by costly Gaussian surface sampling.

A. Idea of Performing Multiple-Shooting Walks
We consider a structure with complex Gaussian surface which encloses the master conductor consisting of numerous conductor blocks.With the approach of VGSS [24], a quite large effort has to be paid to obtain a valid sample point on the Gaussian surface.If we can perform a different random walk scheme that leads to faster convergence of the FRW procedure, the number of walks (i.e., the times of sampling Gaussian surface) and the total runtime of the FRW method could be reduced.
For reducing the number of walks for attaining given accuracy, a straight-forward idea is to stem out several subwalks paths from the same starting point on Gaussian surface.We call it a multiple-shooting walk.The subwalks in a multipleshooting walk resemble multiple normal walk paths.Since the weight values from the subwalks are merged, the new estimator is expected to have smaller variance.Below we present some naive schemes of performing multiple-shooting walks, before proposing the approach with SMS walks in next section.
Fig. 3 shows two naive schemes of performing multipleshooting walks called NMS walks of the first kind (denoted by NMS-1st) and the second kind (denoted by NMS-2nd), respectively.In the naive schemes, suppose N s paths of random and independent subwalks are launched.As an example, here we assume N s = 2, and the subwalks are plotted in different colors.For the approach with NMS-1st, random walk path branches at the first point which is sampled on the Gaussian surface.For the approach with NMS-2nd, we let the subwalks share the second point r 1 .As shown in Fig. 3, the points r 1 and r 2 are randomly and independently selected on S 1 (for NMS-1st) or S 2 (for NMS-2nd).Afterwards, the hops are performed as in the normal FRW method to generate two subwalk paths.Notice that for correctly computing the capacitances, the weight value for each subwalk path should be the original weight value divided by N s .
Assuming that the IS approach is also applied and that N s = 2, we analyze the variance for this FRW method with NMS walks.We use random variable X Q to denote the return value of an NMS-1st walk, and X w (r) to denote the variable for w(r) accordingly.The second moment of Again, we first take n(r) along the x-axis as an example to do analysis.Then, like (30), we derive If n(r) is along y-or z-axis, similar expressions of E[ X 2 w (r)] to (43) can be derived.Based on these formulas and (42), it is straight-forward to write the formula of Similarly, with X Q and X w (r) denoting the random variables for the NMS-2nd walk, we can derive for the case with n(r) along the x-axis.Also, we can write the expression of E[ X 2 Q ] based on (44).Now, we can compare IS) .First, based on the Cauchy-Schwarz inequality, it can be proved that Considering that the electric potential φ(r 1 ) is a quantity between 0V and 1V, we see Finally, we arrive at the conclusion that which means that using NMS brings the effect of variance reduction but the reduction is less than 50% compared to the variance of the IS method.Naturally, the NMS-3rd, NMS-4th and etc. can be developed likewise.For instance, in the NMS-3rd method a walk path should branch at the third point.Similar inequalities to (45) hold for other kinds of NMS.More specifically, The above proposition can be generalized to cases with any N s other than 2, that is, variance reduction is ensured with the NMS scheme, but the variance is always higher than 1/N s of the original.The number of walks required for the same accuracy is reduced but still higher than 1/N s of the original.However, under the NMS scheme, the computational cost per walk increases almost N s times.Therefore, the method with NMS cannot outperform the IS method.
The FRW method using NMS walks can be further accelerated with the application of stratified sampling (SS).Just as in the normal FRW method, random walks are categorized into two types according to the sign of weight value, which bisects the probability space.Since the subwalks of one NMS (except NMS-1st) walk share the same weight value, the variance in the subspace tends to be significantly lower than in the whole space, which explains why the method with NMS-2nd+SS usually achieves the best performance among the approaches of NMS family.This will be validated by the numerical experiments in Section V-B.

B. Approach With Symmetric Multiple-Shooting Walks
In this section, we propose the FRW method with SMS walks for capacitance extraction.The motivation is that the weight value has opposite signs for two r 1 's on different sides of the Gaussian surface [4], [15].When the IS approach is applied, the weight value is almost always negative for r 1 outside G (i.e., on S + 1 ) and positive for r 1 inside G (i.e., on S − 1 ), according to (13).So, if within a walk we launch two subwalks passing through both sides of G and they finally reach the same conductor, the two weight values will cancel out each other.This makes zero overall contribution to capacitance estimation.If a lot more zero samples are obtained, the sample variance can be considerably reduced.A scheme for obtaining the two r 1 's is randomly choosing a point on S + 1 and then choosing its symmetric point with respect to G. Afterwards, two sequences of random hops are performed individually in the same manner as in normal FRW method.We call this kind of walk, including two symmetric r 1 's a SMS walk.
In Fig. 4, we illustrate two SMS walks.For the first transition cube whose center is on the part of G with normal directions along x-or y-axis, two r 1 's symmetric to each other with respect to G are shown.They are denoted by r + and r − , with r + outside G and r − inside G. Now we construct the unbiased estimator under the SMS scheme.In the approach with SMS walks, w(r) should be evaluated in a different manner.Take the case with x-axis normal direction as an example.We convert (5) into where q c (r, r + ) is a new PDF for sampling r + on S + 1 .The equality holds because S + 1 and S − 1 are of same shape and r − is a point symmetric to r + .Based on the idea of IS, we define Obviously S + 1 q c (r, r + )ds = 1 and sampling can be easily realized based on the PDF-IS table.Denote the Markov chain here by {X + t , t ≥ 0}, whose only distinction to {X t , t ≥ 0} is the first-step transition probability density changed to The symmetric or mirror chain {X − t , t ≥ 0} is defined as: X − 0 = X + 0 , X − 1 is symmetric to X + 1 w.r.t.G and other transitions are according to Green's function as in the original FRW process.Denote the new estimator for w(r) by X * w (r).By inspection of (46), we define Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.For an SMS walk starting from r and splitting at r + and r − , a pair of weight values with opposite signs will be obtained and should be merged with factors , respectively, to become unbiased.Note that the factors can be computed with the precalculated PDF-IS table, and in many situations they both equal 1/2.If n(r) is along the y or z-axis, we can similarly construct the SMS scheme for evaluating w(r).Denote the estimator for Q i via SMS walks by X * Q .Similar to other FRW variants, it is defined as The unbiasedness of the FRW method with SMS walks is stated in Theorem 2. This approach can also be extended to symmetrically shoot more subwalks, i.e., making N s > 2. We can have half of the N s r 1 's inside G and the other half outside, and the sampling is made on 1/N s surface of S 1 .In addition to the sample point, the other N s − 1 points on S 1 can be obtained via some symmetric transformations.For practicality concerns, we let N s be 4, 8 or 16 (the choice of N s and the corresponding mapping are further discussed in Appendix B), and define sampling subsurfaces as shown in Fig. 5, so that these N s points share the same or opposite WVT value due to symmetry in many cases [15].The sampling function is similar to (47), and the estimators follows the forms of ( 49) and (50) but involve N s Markov chains.We describe the algorithm as Algorithm 1, where Steps 10-17 are for consecutively performing the N s subwalks.
Theorem 2: The estimator in the FRW method with SMS walks, as defined in (50), is unbiased for electric charge Q i .For the SMS schemes developed with N s = 4, 8, 16, the unbiasedness also holds.
Proof: The proof is given in Appendix C.
It is important to clarify that the unbiasedness of the proposed SMS estimation does not rely on the symmetry of N s points' WVT values, such as Although the symmetry of WVTs inspires the SMS idea and simplifies the variance derivation (in the next section), it is not a prerequisite for the correctness of Algorithm 1.

C. Variance Analysis on the FRW With SMS Walks
In this section, we analyze the variance of the FRW method with SMS walks, and check if using the SMS walks reduces

Algorithm 1 FRW-Based Capacitance Extraction With SMS Walks
Input: A 3-D structure of conductors (the master conductor is indexed by i); a set of pre-characterized GFT, PDF-IS and SUM-WVT tables for dielectric environment; preset accuracy criterion for capacitance C ij ; N s = {2, 4, 8, 16}.Output: Master conductor's self-capacitance and coupling capacitances C ij (j = 1, 2, . . .). 1: Load the precharacterized GFT, PDF-IS and SUM-WVT tables; 2: Set up the Gaussian surface enclosing the conductor i; 3: C ij := 0, ∀j; N walk := 0; 4: while the computed capacitance results do not satisfy the preset accuracy criterion do 5: N walk := N walk + 1; 6: Generate a valid sample point r on the Gaussian surface; 7: Generate the largest conductor-free transition cube T whose center is r; Obtain the sampling probabilities for q c (r, r + ) from the PDF-IS table corresponding to T's dielectric profile, and randomly choose a point r + on 1/N s surface of T; Obtain a set R including N s points on T's surface, through performing symmetric transformations on r + (making them symmetrically distributed with respect to Gaussian surface); 10: Calculate weight value ω * using the SUM-WVT table entry corresponding to r 1 and the PDF-IS entries corresponding to the points in R (following a formula similar to (49)); 12: while r 1 does not fall on any conductor do 13: Generate the largest conductor-free transition cube T whose center is r 1 ; 14: Obtain the sampling probabilities from the GFT table according to T 's dielectric profile, and randomly choose a point on T 's surface to update r 1 ; 15: end while 16: C ij := C ij + ω * // Suppose r 1 falls on conductor j; 17: end for 18: end while 19: C ij := C ij /N walk , ∀j the variance and thus brings faster convergence.We assume the IS approach is applied and N s = 2.
Using the notations of X * w (r) and X * Q , we can write Again, we first take n(r) along the x-axis as an example to do analysis.Based on definition in (49) and using the property that W (x) (r, r + ) and W (x) (r, r − ) have opposite signs which is very similar to derivation in (70).The last term in ( 52) because the random walk paths for calculating φ(r + ) and φ(r − ) are independent.Then, based on Theorem 1 Due to the symmetric relationship of r + and r − , an important property (proved in [15]) that W (x) (r, r + ) = −W (x) (r, r − ) holds in the environment with stratified dielectrics.Therefore, (53) is finally transformed into Comparing it with (30), we see that Theorem 3: Consider the FRW method for capacitance extraction performed within multilayer-dielectric environments.We use XQ to denote the random variable which is the return value for capacitance C ij from a random walk path in the IS-based FRW method, and X * Q to denote that from an SMS walk when the approach with symmetric 2-shooting walks is performed.Further we denote ) on all first transition cubes with z-axis normal direction, where r + and r − are symmetric points, then Proof: We have previously proved (55) in the case of x-axis normal direction.In multilayer-dielectric environments, the exact proof of (55) also holds for y-axis normal direction.Since it is assumed that W (z) (r, r + ) = −W (z) (r, r − ), we can assert the inequality in (55) for z-axis normal direction as well.Together with (32) and (51) indicating the relation between the second moments, we can then derive , the inequality between the variance is also true Theorem 3 means that the variance in the proposed FRW method with the SMS walks is less than half of that in the FRW method with the IS approach.Because the variance of FRW method is inversely proportional to the number of walks (see (9)), the original FRW method needs at least twice the walks as the proposed method to achieve the same accuracy.Even if we consider that each SMS walk includes two subwalk paths, the method with SMS walks should be advantageous in terms of computational time.
Now we justify the assumptions in Theorem 3. Equations stand when S 1 encloses layered dielectrics.If S 1 involves more complex environment, such as conformal dielectrics, most FRW implementations will use a layered-dielectric model to approximate its WVT [15].So, the symmetry of W (x) or W (y) is usually guaranteed.Equation W (z) (r, r + ) = −W (z) (r, r − ) stands only when S 1 encloses single dielectric.In actual scenarios, it may only approximately hold.Therefore, Theorem 3 just provides partial support to the efficiency of the proposed SMS scheme.The variance reduction needs to be validated by numerical experiments.Please see Section V-B.
For generalized SMS schemes with N s > 2, the assertion like Theorem 3, i.e., D (SMS) < (1/N s )D (IS) , cannot be proven theoretically.However, the reduction of variance has also been observed in experiments like those in Section V-B.More discussions are given in Appendix B.
The state-of-the-art variance reduction is IS+SS [4], where stratification is based on both the sign of ω and the face index of Gaussian surface.This SS is not compatible with SMS schemes because: 1) the sign of ω(r, r + ) is fixed and 2) stratification based on Gaussian surface is difficult to realize due to VGSS.Moreover, the relationship between D (IS+SS) and D (SMS) is uncertain in theory, so the choice between them may depend on the input structures of conductors case by case.However, for the structures with complex Gaussian surface, the approach with SMS walks requires much fewer sample points on Gaussian surface, and therefore may exhibit better-overall runtime benefit than the approach using SS.
In previous analysis on capacitance estimators and their variances, the normalization coefficient of sampling function on Gaussian surface, denoted by g, is treated as a known constant (see (13)).In real problems, however, g −1 is difficult to compute analytically and will be estimated based on all samples on the virtual Gaussian surface [24].According to Theorem 3, much fewer samples would be obtained in the SMS schemes, thus increasing the variance of the statistical ĝ−1 .Therefore, to avoid the potential impact on accuracy, we redefine the result variance to include the variance of both ĝ−1 and the original estimator Ĉ as follows: In the experiments, we set the termination criterion of the FRW method to control this merged variance, ensuring the accuracy of capacitance results.As we noticed that ĝ−1 converges so fast that it has minor impact on the merged variance, previous derivations and propositions remain valid.
V. NUMERICAL EXPERIMENTS We first verify the variance analysis in Sections III and IV using the results on a parallel-plate structure.In this experiment, we extend the RWCap3 program for 3-D capacitance extraction [25] to compute electric potential and electric field intensity.Then, we validate the efficiency of the FRW method with SMS walks (Algorithm 1) proposed in Section IV-B, which is also implemented based on RWCap3.Finally, we discuss in what case using FRW via SMS walks can be advantageous, and explore methods for estimating the speedup ratio of the SMS schemes over IS+SS and making automatic choice.
The test structures for capacitance extraction include IC and FPD cases from industry designs.All experiments are carried out on a Linux server with Intel Xeon 8375C CPUs at 2.9 GHz.

A. Validating the Variance Analysis With Parallel-Plate Structure
A structure with two opposite parallel metal plates is constructed, where the size of plate is much larger than the distance between them (d).The experiment setup is shown in Fig. 6.Thus, a uniform electric field is formed.Setting point r at different positions between the two plates, we evaluate its potential φ(r) with the FRW method, and electric field intensity E n (r) with the IS approach and SMS walks, respectively.N s is set to 2. For each quantity we execute 10 7 walks to get sufficiently accurate estimations.Based on the FRW samples, we can estimate the variances of underlying random variables X φ (r), XE (r) and X * E (r).On the other hand, these variances of random variables have been theoretically derived.D[X φ (r)] can be computed with Theorem 1, i.e., (39).D[ XE (r)] can be calculated with (41).D[X * E (r)] can be derived based on (54).Specifically The integral can be evaluated with the first transition cube and without performing the FRW method, similarly to the evaluation of (41).Therefore, we obtain the theoretical values for the three variances.In Table I, we list the theoretical values and computed values for the three variances, with r being nine equally spaced points between the plates.The electric potentials of these points are also listed.From the results, we see that the theoretical values well match those obtained from executing 10 7 random walks, with maximum discrepancy less than 0.3%.This validates the corresponding variance formulas derived in Sections III and IV, and partially validates the theoretical analysis revealing the variance reduction effect of the FRW method with SMS walks.

B. Validating the FRW Method With SMS Walks
In this section, we validate the performance of the FRW method with SMS walks for capacitance extraction.The baselines include the FRW method with IS approach, the FRW method with NMS walks, and the FRW method with IS+SS scheme.There are eight test structures from diverse scenarios, denoted by cases 1-8.Cases 1 and 2 are simple structures  under 180-and 45-nm technology nodes, obtained from [4].Cases 3 and 4 consist of tens of parallel wires located within the middle metal layer, sandwiched between two large parallel panels.Case 5 is from the design of a static random-access memory (SRAM).Cases 6-8 are from FPD and touchscreen design, containing millions of conductor blocks.A local view of case 6 is shown in Fig. 7, which is a typical structure of touch sensor.All test cases in our experiments involve multidielectric environments, and cases 1, 2, 5, and 6 include nonstratified dielectrics.For each case, one wire is set as the master conductor.Statistics about the cases are listed in Table II, where #blocks and #blocks m mean the number of all conductor blocks and the number of master-conductor blocks, respectively.Among the cases, cases 5-8 include relatively complicated master conductors formed by a lot of conductor blocks.The termination criterion for all the FRW variants is set to 1-σ error on self-capacitance being 0.5% of the mean value.Experiments are conducted with serial computing on cases 1-5, and in parallel with eight threads on cases 6-8, as running serial-computing program costs too much time.First, we run the FRW method with NMS walks and compare it with the method with IS approach.cases 1-4 are of relatively simple structure.Without complex Gaussian surface, the sampling of the start point of random walk path becomes inexpensive.Therefore, the variance is the dominant factor affecting the runtime of the FRW method.The results are listed in Table III, with N s = 2 for the FRW method with NMS walks.From the table, we see that the method with NMS walks indeed reduces the #walks.Since #walks is directly proportional to the variance, the data in Table III well corroborate the inequality in (45).However, an increase is observed in the runtime after applying the NMS walk.The reason is that an NMS walk is about twice expensive as a regular walk.The FRW method via NMS walks is also tested with varied N s , as demonstrated in Fig. 8.We see that larger N s won't bring runtime benefits for these simple cases, which is consistent with the theoretical analysis.
Next, we evaluate the performance of the FRW method with SMS walks.It has four variants denoted as SMS (2), SMS (4), SMS (8) and SMS (16), where the numbers refer to the corresponding values of N s .As the FRW methods with the IS approach and the NMS walks run relatively slow, we consider the FRW method accelerated by the SS technique for comparison, e.g., the IS+SS scheme and the NMS-2nd+SS scheme.The IS+SS is a combined technique first proposed in [4] and can be seen as the state-of-the-art variance reduction technique for FRW-based capacitance extraction.The NMS-2nd subwalks share the same weight value, so the SS technique can be applied and reduce the variance.The NMS-2nd+SS scheme is generally the most efficient among the NMS variants of FRW method.As for other variants of FRW method, they failed to converge in a acceptable time when extracting the large cases.The computational results are listed in Table IV, including the results on time and self-capacitance.For the first four cases, the results can be compared with those in Table III, which reveals that the SS technique remarkably accelerates the convergence.In Table IV, the last column shows the maximum speedup of the four SMS variants over the IS+SS scheme.From it we see that the proposed FRW with SMS walks runs the fastest for most of cases, especially for large cases with complex master nets.While for the small simple case, the runtimes of IS+SS scheme and the proposed FRW with SMS walks are comparable.The speedup of FRW with SMS over IS+SS is up to 10.1× (for case 8).
In Table IV, the reference values of capacitance from Raphael [26], which is an FDM-based field solver, are also listed.They validate the accuracy of the FRW methods.Although Raphael failed to solve cases 5-8 due to prohibitive memory cost, for cases 1-4 the errors of FRW methods are all within the range of statistical error.
Finally, we apply the proposed FRW method with SMS walks to extract the capacitances within an analog circuit design.The case is a voltage-controlled oscillator (VCO) design at a 65nm technology, where there are 38 signal nets and totally 377K conductor blocks.Our methods are compared with a commercial FRW solver, all run with serial computing to extract the capacitances of the 38 nets.The termination criterion is always set to 1-σ error on self capacitance being 1% of the mean value.The runtimes are 857 s, 344 s and 493 s for the FRW method with IS+SS, the FRW method with SMS (4) and the commercial solver, respectively.This means the proposed SMS scheme enables faster computational speed.The discrepancy between the extracted capacitance matrices with our FRW methods and that with the commercial solver is shown in Fig. 9, where the off-diagonal entry is calculated by the difference on coupling capacitance divided by the self capacitance.The figure validates the accuracy of the proposed SMS scheme (with discrepancy to the commercial solver less than 3%).While comparing the coupling capacitances more carefully, we see the discrepancies are small on large coupling capacitances.For example, the self capacitance and two largest coupling capacitances of net "VDD" are 7.18/2.18/0.598pF from the SMS (4) scheme, and 7.28/2.22/0.615pF from the commercial solver.

C. More Discussions and Estimating FRW-SMS' Speedup
In order to analyze the breakdown of the FRW's runtime and discuss what kind of structure is more suitable for the proposed Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.FRW with SMS method, we collect more detailed results and listed them in Table V. #walks represents the number of executed walks and #hops is the average number of hops per walk.Notice that for different FRW variants to attain the same accuracy on the same task, the required #walks is directly proportional to the variance of the underlying random variable, due to (9).So, from the table we see that the #walks (i.e., the variance) of SMS ( 2) is approximately half of that of IS+SS, while the #hops is double.The variance reduction of SMS (2) over IS+SS is at least 1.55× and at most 10.3×.For SMS (4), SMS (8), and SMS (16), the times of variance reduction over IS+SS becomes larger as N s increases.Moreover, comparing #walks in Table III and that in Table V, we can verify the claimed variance reduction in Theorem 3.
Regarding the overall runtime (16), parameter R 1 reveals the effectiveness of RS on the Gaussian surface sampling.Its value is also listed in Table V. R 1 will be large when the Gaussian surface of the master conductor is highly complicated and the RS is costly.Since with the SMS walk we can reuse the samples on the Gaussian surface, thus save runtime on those cases with R 1 .on (16), we can also estimate the speedup of SMS over IS+SS by Sp = #walks (IS+SS) R 1 + #hops (IS+SS)   #walks (SMS) R 1 + #hops (SMS)  (61) where the superscript indicates the quantities from the IS+SS or SMS scheme.For the SMS scheme which runs the fastest, the estimated " Sp" and the actual speedup "Sp" are listed in the last two columns of Table V.Their consistency validates the above analysis and motivation of proposed SMS scheme.
Finally, we present a practical strategy for automatic selection between the proposed SMS scheme and the state-ofthe-art IS+SS.Based on (14) and considering that #walks is proportional to the variance, we have (i.e., the average runtime of a walk), and the variances of the both schemes (see context of ( 9)).Then, we can use (62) to estimate the speedup ratio of SMS over IS+SS, whose results are listed in Table V and are consistent with actual speedups.With the either scheme, running 1000 walks costs no more than 0.1 s.Combining some heuristic rule, this enables an automatic choice between the IS+SS and SMS schemes with negligible overhead, and the acceleration of the FRW-based capacitance extraction for the structures with complicated master conductor.
VI. CONCLUSION In this work, we have studied the variances of underlying variables in the FRW method for capacitance extraction analytically, and proposed a novel FRW method with SMS walks which largely reduces the variance.The proposed method reuses the sample point on the Gaussian surface, and launches N s subwalks (N s = 2, 4, 8 or 16) from it with the starting points of subwalks symmetrically distributed on the first transition cube.We have proved that the proposed method converges with the number of SMS walks fewer than 1/N s of number of walks used in the method with IS approach, under some assumption.Experimental results have validated our theoretical analysis via simulating a parallel-plate structure, and showed that the proposed method runs up to 10.1× faster than the FRW method with the state-of-the-art IS+SS scheme.Finally, we have discussed which kind of structure is more suitable for using the proposed method and proposed an automatic mechanism to choose between the proposed method and the IS+SS scheme for best accelerating the capacitance extraction.
Again by the law of total expectation and based on (70) The unbiasedness of symmetric 2-shooting method has been proven.For the method with N s = 4, 8, 16, the first-step sampling function is similar to (47).The estimator for w(r) is similar to (49) but involves N s Markov chains.Its expectation can be derived following the approach in (70), also equaling w(r).The estimator for Q i is defined the same as (50).Therefore, all proposed SMS schemes have been proven to yield unbiased results for capacitance extraction.

Fig. 1 .
Fig. 1.Illustration of the FRW method with a structure, including four conductors (2-D cross-section view).

Fig. 2 .
Fig. 2. Electric potential and electric field intensity in the parallel-plate structure (2-D side view).

Fig. 3 .
Fig. 3. Illustration of the FRW method with NMS walks of (a) the first kind and (b) the second kind.

Fig. 4 .
Fig. 4. Illustration of the FRW method with SMS walks (2-D crosssection view).On the right the two examples of the first transition cube are zoomed in.

Fig. 5 . 1 /
Fig.5.1/N s surface of the first transition cube S 1 , on which point r + is randomly sampled.The sampling surface includes the faces with a black edge, not those wholly outlined by blue gray lines.The Gaussian surface's normal direction n at transition cube's center r is assumed along the x-axis.

Fig. 6 .
Fig. 6.Illustration of the tested parallel-plate structure, where a path of symmetric 2-shooting walk is sketched.

Fig. 9 .
Fig. 9. Relative discrepancy between the capacitance results of our programs and a commercial FRW solver.
the extraction task, we can first run 1000 walks with the SMS scheme and the IS+SS scheme, respectively.With them we can estimate T

TABLE I THEORETICAL
AND COMPUTED VALUES OF THE VARIANCES OF THREE RANDOM VARIABLES IN THE APPLICATION OF FRW METHODS TO THE PARALLEL-PLATE STRUCTURE.THE QUANTITIES ARE FOR NINE POSITIONS EQUALLY SPACED BETWEEN THE TWO PLATES Fig. 7. Local view (top view) of case 6 (left) is zoomed in (right).The coloring differentiates metal layers.

TABLE II STATISTICS
FOR THE EIGHT TEST CASES

TABLE III COMPARISON
OF THE FRW METHODS WITH IS AND NMS WALKS Fig. 8. Computational time of the two NMS-based methods with varied N s , tested on cases 1-4.

TABLE IV COMPUTATIONAL
RESULTS OF SIX FRW VARIANTS.THE SPEEDUP IS THE MAXIMUM SPEEDUP RATIO OF THE PROPOSED FRW METHOD WITH SMS WALKS OVER THE FRW METHOD WITH IS+SS SCHEME TABLE V MORE DETAILED RESULTS OF THE FRW METHODS AND THE PREDICTED SPEEDUP OF THE PROPOSED METHOD OVER THE FRW METHOD WITH IS+SS SCHEME