A Multigrid-like Technique for Power Grid Analysis

Transcription

1 A Multirid-like Technique for Power Grid Analysis Joseph N. Kozhaya, Sani R. Nassif, and Farid N. Najm 1 Abstract Modern sub-micron VLSI desins include hue power rids that are required to distribute lare amounts of current, at increasinly lower voltaes. The resultin voltae drop on the rid reduces noise marin and increases ate delay, resultin in a serious performance impact. Checkin the interity of the supply voltae usin traditional circuit simulation is not practical, for reasons of time and memory complexity. We propose a novel multirid-like technique for the analysis of power rids. The rid is reduced to a coarser structure, and the solution is mapped back to the oriinal rid. Experimental results show that the proposed method is very efficient as well as suitable for both DC and transient analysis of power rids. Via Connection to power supply Level 3 Level 2 Level 1 I. Introduction In recent years, there has been an increased demand for hih performance and low power VLSI desins. Hih performance is achieved by technoloy scalin, increased functionality and competitive desins. On the other hand, a common technique used to obtain low power desins is to scale down the supply voltae. This stands to reason since the chip power P is proportional to the square of the supply voltae V DD. Thus, the demand for hih performance and low power has led to modern VLSI desins bein characterized by reduced feature size, increased functionality and lower supply voltae. Increased chip functionality results in the need for hue power distribution networks, also referred to as power rids since they typically have a rid structure. Lower supply voltae, on the other hand, makes the voltae variation across the power rids very critical since it may lead to chip failures. Voltae drops on the power rid reduce the supply voltae at loic ates and transistor cells to less than the ideal reference. This leads to reduced noise marins, hiher loic ate delays, and overall slower circuits. Reduced noise marins may lead to false switchin at certain loic ates and latches. Hiher loic ate delays, on the other hand, may slow down the circuit enouh so that timin requirements can not be met. Consequently, once voltae drops exceed certain desiner-specified thresholds, there is no uarantee that the circuit will operate properly [1], [2], [3]. Thus, it is clear that in modern VLSI circuits, power rids are becomin performance limitin factors. Consequently, efficient analysis of power rids [4], [5] is necessary for both (1) predictin the performance and (2) improvin the performance if necessary. Because of the lare dimensions of power rids in modern VLSI circuits, existin anal- J. N. Kozhaya is with IBM Corp., 1 River St., Essex Junction, VT He was with the ECE Department, University of Illinois, Urbana, IL, kozhaya@us.ibm.com S. R. Nassif is with IBM Austin Research Laboratory, Austin, TX, nassif@austin.ibm.com F. N. Najm is with the ECE Department, University of Toronto, Ontario, Canada, f.najm@utoronto.ca Contacts to devices Fi. 1. Power rid components. ysis methods are fallin behind. Thus, there is a need for new efficient, in terms of both execution time and memory, techniques for the analysis of power rids. In this paper, we propose an efficient analysis technique that follows the lines of thouht of multirid methods which are commonly used for the solution of smooth partial differential equations (PDEs). Specifically, our method is inspired by the alebraic multirid method, AMG, which is one variation of the multirid approach. Thus, section III describes the multirid approach with a specific emphasis on the alebraic multirid variation. After discussin the multirid technique, we present our proposed multiridlike approach for the efficient analysis of power rids in section IV. The efficiency of our proposed technique is verified by the experimental results iven in section IV-D. Finally, conclusions are provided in section V. II. Modelin and Analysis of Power Grids In this section, we discuss the basic modelin and analysis techniques of power rids. Specifically, section II-A discusses how to model typical power rids, sources, and drains, for efficient and accurate analysis. Section II-B, on the other hand, presents the basic analysis techniques and discusses some tricks to speedup the analysis. A. Modelin of Power Grids The connections from the power rid to the external supply voltae, V DD, are called the power sources since the current is supplied from the supply to the rid throuh these connections. The power drains, on the other hand, are those modules which draw current from the power rid. For instance, transistors, loic ates, latches, clock buffers, memory units, reister arrays, and I/O buffers are all considered power drains. The first step in power rid analysis involves modelin the rids as well as the power sources and drains [6]. Modelin of the rids, sources, and drains involves a tradeoff between accuracy and speed. The more complex the model

2 2 is, the more accurate the results of the analysis are but the more expensive the solution is (in terms of memory and CPU time). Typically, power distribution within an interated circuit is done from the top-level metal layer, which is connected to the packae, down throuh inter-layer vias and finally to the active devices, as illustrated in Fiure 1. We follow the same modelin approach as in [6], where the metal wires and vias are modeled as a linear, time-invariant and passive network consistin of resistive, capacitive and -rarelyinductive elements. For modern interated circuits such as microprocessors, such a network can easily include millions of nodes and tens of millions of elements. As for the power sources and drains, their models can be quite complex. The models for the power sources can be involved enouh to include sophisticated packae and board models. On the other hand, the models for the power drains can account for the complex interaction between the power rid, the underlyin non-linear circuit, and the time-varyin sinals propaatin across the chip. However, the hue size of the power rid makes it infeasible to include any but the simplest models for the power sources and drains. Hence, power sources are modeled as simple constant voltae sources and power drains are modeled as independent time-varyin current sources. Given the above, the complete power rid model is composed of a linear network of RLC elements excited by constant voltae sources and time varyin current sources. Note that none of the network RLC elements are connected to round, all the voltae sources (sources) are between certain rid nodes and round, and all the current sources (drains) are between rid nodes and round. The behavior of such a system can be expressed followin the modified nodal analysis (MNA) [7] formulation as the followin ordinary differential equation: Gx + Cẋ = u(t) (1) where x is a vector of node voltaes, and source and inductor currents; G is the conductance matrix; C includes the capacitance and inductance terms, and u(t) includes the contributions from the sources and the drains. In fact, u(t) has three kinds of rows: i) rows with a positive V DD value, which correspond to nodes that are connected to power sources, ii) rows with a neative value of current (or sum of currents), which correspond to nodes that are connected to a power drain (or more than one), and iii) rows with, which correspond to all other nodes. In all what follows, we inore the effects of on-chip inductance and assume that the power rid is modeled as an RC network only. This is motivated by the fact that in today s technoloy, on-chip inductance in the power rid is too small to sinificantly affect the analysis results. Thus, in summary, the power rid is modeled as an RC network, the power sources are modeled as constant voltae sources, and the power drains are modeled as time-varyin current sources. Fiure 2 shows a 3 3 rid with one voltae source at node 1 (indicated by an X) and two current sources at Fi. 2. I (ma).25.5 Example of 3x3 rid. t (ns) 36 Fi. 3. Current drawn by an inverter. nodes 5 and 7 (indicated by a dot). This example shows the model for a two layer power rid with three wires in each layer, one voltae source at node 1, and two power drains, one drawin current from node 5 and the other from node 7. An example of a current source associated with an inverter power drain is shown in Fi. 3 and expressed mathematically as follows where t is the time in nanoseconds and I is the current in milliamps: { 41 I = 6 t t t t.51 9 B. Analysis of Power Grids Due to the lare size of typical power rids, eneral circuit simulators such as Spice [8] are not adequate for power rid analysis because of CPU time and memory limitation. The inefficiency of standard simulators comes about because (a) they require a lumped element approximation of the circuit which requires the translation of a reular eometrical structure to an expansive set of equivalent circuit elements, and (b) they use eneral purpose solution methods meant to be robust in the face of stiff systems of equations. By contrast, power rids are well behaved spatially (nearly reular) and temporally (damped). This motivates a special-purpose simulator for power rids which can make use of these properties. Solvin (1) requires the use of some numerical interation formula. Typically, Backward Euler (BE) interation formula is the formula of choice mostly due to its stability

3 3 G x 1 21 G x 2 32 G 3 36 x 3 41 G x G x G 6 69 x 74 G x G 8 89 x G 9 8 x 9 1 x 1 A x I 5 I 7 V DD Fi. 4. Linear system resultin from MNA. properties [7]. Applyin Backward Euler to (1) results in a set of linear equations: (G + C/h)x(t + h) = u(t + h) + x(t)c/h (2) which can be readily simplified to A x(t + h) = b with A = G + C/h and b = u(t + h) + x(t)c/h. The solution of (2) requires the inversion (factorization) of the matrix A = G + C/h which is independent of x, time-invariant, lare and sparse. We note, however, that if we hold the time step h constant, then only one initial factorization is required, with a forward/backward solve at each time step. Since, for lare matrices, a factorization is sinificantly more expensive than a forward/backward solve [9], the use of a constant time step results in lare savins. The time step needs to be kept small enouh to insure the accuracy of the solution. For application in the analysis of power rids of diital circuits, we find that usin 1 steps per clock cycle (i.e. h =.1 T period ) is sufficient. To illustrate, we simulate a simple rid of 33 wires in each of the horizontal and vertical directions, connected to a sinle voltae source at one of the corners, and loaded with 1 time-varyin current sources at random locations. The resultin electrical model has 189 nodes and a total of 19 equations. We perform the simulation for 1 time steps. Both, Spice and our simulator, produce the same results. However, Spice [8] takes 13.3 sec. of CPU time, whereas our simulator implementin the method above takes.73 sec. for a net speedup of about 18x. Due to the superlinear dependence of solve time on matrix size, the speedup will be even more dramatic for the much larer systems normally encountered when simulatin realistic power rids. To better explain the process of power rid analysis, modified nodal analysis is applied to the example iven in Fiure 2. This results in the linear system shown in Fiure 4. Let N be the number of nodes of the power rid. Then, N = 9 in the example rid shown in Fiure 2 since the rid is 3 3. Furthermore, there is one voltae source which means that the A matrix is a 1 1 matrix where the first 9 equations are the KCL equations at all 9 nodes and the last equation is the KVL equation at node 1 where the voltae source is located. As for the current sources at nodes 5 and 7, they appear in the riht hand side vector b. On the other hand, ij defines the conductance between b 1 G G 3 36 G G G G G G 9 A x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x V DD 21 V DD 41 V DD I 5 I 7 Fi. 5. Linear system after reformulation. the two neihborin nodes i and j. Thus, ij = ji which results in the A matrix bein symmetric. Furthermore, the diaonal entries of the A matrix are defined as follows: G i = ij + C i /h (3) j N i where C i is the capacitance at node i, h is the time step, and N i = {j ij } is the set of neihbors of node i. In eneral, the system matrix, A, of the linear system A x = b is symmetric but not necessarily positive definite [9]. However, the problem can be reformulated to result in a symmetric, positive definite matrix, A. Basically, the MNA formulation builds the linear system A x = b by assertin both the KCL and KVL equations at the power source nodes as well as the KCL equations at all the remainin rid nodes [7]. Then, the solution, x, of the resultin linear system ives the voltaes at all the rid nodes as well as the currents supplied by the different voltae sources. However, if we are only interested in the voltaes at all the rid nodes, then we can inore the KCL equations correspondin to the power source nodes. Furthermore, the voltae at a power source node is known to be exactly the supply voltae, V DD. Thus, we can reformulate the problem by substitutin the value of the supply, V DD, for the voltae of all the power source nodes and inorin the KCL equations for the power source nodes. We illustrate by applyin this reformulation to the 3 3 rid example iven above. Node 1 has a voltae source, so we replace the KCL equation at node 1 (equation 1) with the KVL equation at node 1 (equation 1). Furthermore, we substitute the value x 1 = V DD in the equations which depend on x 1. The resultin system is shown in Fiure 5. Observe that the riht hand side is also chaned to result in a new vector, b, where two extra terms 21 V DD and 41 V DD are added at indices 2 and 4 respectively. It is clear that the resultin matrix A is still symmetric. It can also be shown to be positive definite. For that, note first that i, ij = ij since ij = for j / N i. j i j N i This, toether with (3) results in the followin: G i = ij + C i /h = ij + C i /h ij i j N i j i j i (4) b

4 4 Equation (4) holds for every node i or equivalently for every row of the system matrix A. This shows that the modified matrix A is diaonally dominant and it was already pointed out that A is also symmetric. Consequently, A is a symmetric and positive definite matrix [9]. As a matter of fact, A can be shown to be an M-matrix [1] since it satisfies the followin: 1. a ii > i 2. a ij i j 3. a ii a ij i j i 4. a ii > a ij for at least one i j i where a ij is the entry of the A matrix at row i and column j. Thus, in the rest of the paper, we will use the fact that the A matrix is a non-sinular M-matrix. III. Multirid Method In a well desined power rid, the rid resistance is much smaller than the equivalent sink resistance since the power rid is required to deliver as constant a voltae as possible to all sinks. This causes local power disturbances, as would be caused by a lare localized sink, to be spread across an area much larer than that of the sink causin the disturbance. This spreadin leads to voltae distributions which are spatially smooth, and motivates solution methods which can make use of this smoothness to speed up the solution process. Furthermore, we note that the analysis of power rids results in a system of linear equations structurally identical to that of a finite element discretization of a twodimensional parabolic partial differential equation (PDE). This motivates us to consider the power rid problem as a discretization of a continuous PDE where the solution is needed at a spatially fixed set of points. Consequently, efficient methods for solvin PDEs are worth considerin as potential competitive solvers for the power rid problem. Recently, the multirid method (MG) has become the standard for solvin smooth PDEs [11], [12], [13]. Multirid involves two complementary steps: i) relaxation and ii) coarse rid correction. Relaxation involves runnin a few iterations of an iterative solver in order to smooth the error components; that is, reduce the hih frequency error components. Coarse rid correction, on the other hand, involves mappin the problem to a coarser rid, solvin the problem at the coarser rid, and then mappin the solution back to the oriinal rid. These two complementary steps work toether to provide an efficient technique for solvin PDEs. Thus, in this paper, we arue the suitability/efficiency of the multirid technique for power rid analysis. Initial interest in multirid resulted from a detailed analysis of iterative methods and the reasons for their slow converence. Historically, multirid methods faced slow acceptance durin their early staes until their practical efficiency was demonstrated by Brandt in 1973 [12], [14]. Then in 1975/1976 Hackbusch developed the fundamental principles of multirid without the knowlede of the existin literature. In his work, Hackbusch discussed a lot of theoretical and practical issues. He also presented a eneral converence theory of multirid methods [13]. While classical iterative methods suffer from slow converence as the rid dimension increases (equivalently rid spacin decreases), multirid performance doesn t deteriorate. As a matter of fact, multirid has been shown to have optimal performance in that a system of N equations can be solved with O(N) complexity. Not only is the multirid technique of optimal complexity but also the constant involved is small enouh to provide an advantae of multirid over other methods [13]. We should point out here that the multirid method falls under the cateory of iterative solvers like Jacobi and Gauss-Seidel as opposed to direct solvers like Gaussian elimination. We have already mentioned that the analysis of classical iterative methods has led to interest in multirid. So we will start with a brief analysis of classical iterative methods. Given a linear system Ax = b and an initial uess ˆx, an iterative scheme involves the followin: ˆx k+1 = ˆx k + M 1 (b Aˆx k ) (5) where k is the iteration index and ˆx k is an approximation of the exact solution x obtained at iteration k. As for M, it should be an easy-to-invert matrix defined such that M 1 A 1. Let e = x ˆx be the error defined as the difference between the exact solution x and the approximate solution, ˆx. It can be shown that the error can be expressed as a linear combination of low frequency and hih frequency Fourier modes [11]. Furthermore, the analysis of classical iterative methods leads to the followin observation [11], [15]: Classical iterative methods efficiently reduce the hih frequency error components but are inefficient in reducin the low frequency error components. In order to avoid the limitations of classical methods, multirid methods consist of two complementary components [11], [13]: 1. Relaxation (smoothin) which reduces the hih frequency error components. 2. Coarse rid correction which reduces the low frequency error components. Relaxation involves runnin a few iterations of a classical iterative solver. This follows from the observation that classical iterative methods act as ood smoothers as illustrated in Fiure 6 [15]. Coarse rid correction, on the other hand, involves mappin the problem to the coarser rid, solvin the mapped problem, and mappin the solution back to the fine rid. The key tools needed for communication between the two rids (fine and coarse) are the interrid transfer operators which are referred to as the restriction and prolonation operators. One intuitive motivation for coarse rid correction is that the solution at a coarse rid typically provides a ood initial uess for the iterative solver at the fine rid and thus results in rapid converence. Another motivation for this approach is that

5 5 Low Frequency Hih Frequency before relaxation after relaxation a. b. Fi. 6. Smoothin effect of iterative methods: a) Sliht reduction of the amplitude of low frequency components. b) Sinificant reduction of the amplitude of hih frequency components. the low frequency error components at the fine rid Ω h appear more oscillatory at the coarse rid Ω 2h as shown in Fiure 7 [11]. Then, relaxation at the coarser rid reduces those components. Multirid techniques work as follows [11]. Startin at the fine rid, a few relaxation steps (iterations) are applied to reduce the hih frequency modes of the error. Then the low frequency (smooth) modes of the error are well approximated by coarse rid correction. This leads to the efficient V-cycle multirid method sketched by the followin alorithm [11]. In the followin, R 2h h and Ph 2h correspond to the restriction and prolonation operators respectively. V h, on the other hand, identifies the call to the V cycle and h defines the level at which the V cycle is called. As for ν 1 and ν 2, these are constants that define the number of iterations to be performed. These constants are chosen empirically and typically have values of 2 or 3. ˆx h V h (ˆx h, b h ) 1. Relax ν 1 times on A h x h = b h with a iven initial uess ˆx h. If 2. Ω h = coarsest rid, then o to 4. Else b 2h R 2h h (bh A hˆx h ) ˆx 2h ˆx 2h V 2h (ˆx 2h, b 2h ) 3. Correct ˆx h ˆx h + P 2hˆx2h h 4. Relax ν 2 times on A h x h = b h with initial uess ˆx h. We conclude this section by notin that each of the components of multirid has its own advantaes and disadvantaes. However, the multirid method derives its power by combinin all these methods exploitin their advantaes while avoidin their disadvantaes. A. Alebraic Multirid The standard multirid methods, SMG, are focused on solvin a continuous problem with a known underlyin eometry. The process involves discretizin the operator on a sequence of increasinly refined rids and definin proper transfer operators between the rids. The coarsest rid is chosen so that the cost of solvin the residual problem at that rid is neliible. On the other hand, the finest rid is chosen to provide a desired deree of accuracy. Typically, SMG methods involve uniform coarsenin and linear interpolation to define the coarse rid and the rid transfer operators. Note that for certain classes of problems, it may be hard or even impossible to apply the standard multirid technique. One such interestin class of problems that relates directly to the power rid problem, is the class of oriinally discrete problems. For an oriinally discrete problem with unknown eometry, discretization at different resolution rids is impossible. Even if the eometry of the problem is known, discretization may still be hard especially if the eometry is irreular. This is because uniform coarsenin and linear interpolation can t be applied to a discrete problem defined on an irreular rid. As a matter of fact, it is hard to define what uniform coarsenin of an irreular rid means. For such problems, the alebraic multirid, AMG, provides an alternative to standard multirid, SMG, which attempts to solve a eneral system of equations usin the multirid principles. Of course, for alebraic multirid to be a competitive alternative, it has to maintain the efficiency advantae of the standard multirid methods. Both, SMG and AMG, involve relaxation and coarse rid correction. Furthermore, the efficiency of either method relies mostly on the choice of the multirid components: the relaxation operator and the inter-rid transfer operators. In SMG methods, uniform coarsenin and linear interpolation define the coarse rid and the rid transfer operators. Thus, the efficiency of SMG methods is decided by the choice of the relaxation operator which is chosen to reduce those error components not well approximated by coarse rid correction [16]. AMG methods, on the other hand, work the opposite way. That is, the choice of the relaxation operator is first fixed and then, the coarsenin procedure and interpolation technique are chosen to reduce those error components not well reduced by smoothin. Thus, the efficiency of AMG methods is decided by the choice of the

6 k = 4 wave on N = 12 rid k = 4 wave on N = 6 rid Fi. 7. Low frequency modes on fine rid Ω h appear more oscialltory on coarse rid Ω 2h. coarsenin procedure and the interpolation method [16]. Consequently, most of the followin analysis will taret coarse rid correction. Specifically, the coarse rid correction operator at level m, C m, is defined as follows [11], [16]: C m = I m P m m+1 (Am+1 ) 1 R m+1 m Am (6) where I m is the identity matrix at level m, A m is the system matrix at level m, A m+1 is the system matrix at level m+1 (the coarser rid), and R m+1 m and Pm+1 m are the inter-rid transfer operators. R m+1 m is the restriction operator used to map the problem to the coarser rid: b m+1 = R m+1 m bm. Pm+1 m is the prolonation operator used to map the solution back to the fine rid: x m = Pm+1 m xm+1. In AMG, the interrid transfer operators, as well as the coarse rid system matrix, A m+1, are completely defined once the interpolation (prolonation) operator, Pm+1 m is defined [11], [16]: R m+1 m = (Pm m+1 )T and A m+1 = R m+1 m Am P m m+1 (7) Analysis of AMG and the properties of the interrid transfer operators lead to the followin important result which indicates that the space of solution vectors at level m, R nm, can be decomposed into two subspaces, the rane space of Pm+1 m and the null space of Rm+1m Am as follows [11]: R nm = R(Pm+1 m ) N(Rm+1 m Am ) (8) It follows then that the error e m R nm can always be decomposed into two components e m = s m +t m where s m R(Pm+1) m and t m N(R m+1 m A m ). The effect of coarse rid correction on each of the error components is iven by the followin [11]: C m s m = (9) C m t m = t m (1) (9) clearly indicates that coarse rid correction perfectly approximates and thus completely nullifies those error components in the rane space of interpolation. (1), on the other hand, shows that coarse rid correction has no effect on those error components in the null space of R m+1 m Am. The above analysis toether with the definition of alebraic smoothness provide a mechanism for choosin the interpolation operator. So we define alebraic smoothness next. An error is defined to be alebraicly smooth if it is characterized by the fact that the residual, r = Ae, is small compared to the error, r e [16]. Furthermore, it is expected that on averae r i a ii e i [16]. This observation proves useful in providin a ood approximation of the error in terms of its neihborin error values: = a ii e i r i + a ij e j a ii e i + a ij e j (11) j N i j N i where N i = {j i : a ij } denotes the neihborhood of i. Geometrically, N i denotes the rid nodes which are directly connected to node i. Furthermore, since the A matrix is an M-matrix, it can be shown that the error satisfies the followin inequality [16]: a ij (e i e j ) 2 a ii e 2 2 (12) i j i (12) states that the smooth error varies slowly in the direction of stron connections. That is, if aij a ii is relatively lare, then (e i e j ) has to be small to satisfy (12) and thus, variation in the error values between nodes i and j is small. (11) and (12) provide a mechanism for definin a ood interpolation operator. In this context, ood refers to an interpolation operator such that the error approximately lies in its rane space. Furthermore, most AMG rid reduction alorithms are uided by (11) and/or (12) [16]. IV. Proposed Multirid-like Power Grid Analysis Method In this section, we present the details of a novel approach for power rid analysis. The technique follows the eneral lines of thouht of the multirid theory. However, it uniquely tarets the specifics of the power rid problem to result in an efficient analysis method. Power rid analysis involves three steps:

7 7 1. Modelin the power rids, the power sources, and the power drains. 2. Formulatin the linear system Ax = b usin modified nodal analysis (MNA). 3. Solvin the linear system Ax = b to obtain the voltaes at all nodes of the power rid. The necessity of efficient modelin of power rids, sources, and drains for efficient analysis has already been discussed. Furthermore, the formulation of the linear system Ax = b by applyin modified nodal analysis (MNA) has been illustrated earlier. Thus, it remains to discuss how to efficiently solve the resultin linear system Ax = b. As explained in section III-A, the power rid problem is an oriinally discrete problem thus motivatin a solution usin the alebraic multirid technique. However, successful application of alebraic multirid requires the definition of a ood interpolation operator. This imposes a rid reduction mechanism that satisfies the followin requirement [16]: For each removed node, i, every node j which is stronly connected to i should be either kept, or stronly connected to at least one node k, where k is both kept and stronly connected to i. Satisfyin the above requirement may lead to inefficient rid reduction. Specifically, the resultin reduced rid may not be coarse enouh; that is, the rid reduction removes only a small number of nodes. This translates to an expensive solution, in terms of CPU time and memory, of the reduced rid. Thus, to avoid the limitation of AMG, we propose a rid reduction alorithm similar to the reduction usin standard multirid. However, since typical power rids may be irreular, our alorithm is desined to efficiently handle the irreularities of the power rid and produce a sinificantly reduced coarser rid at every iteration. The details of the alorithm are iven in section IV-A. Once the rid is reduced, our proposed approach defines the interpolation operator so as to maintain the requirements of the alebraic multirid method. That is, the interpolation operator is defined such that the error components which are not well-reduced by smoothin lie in its rane space. Followin (11), assume that the interpolation operator is defined such that: e h i = e h i = eh i if i is kept (13) j N i a ij and a ii e H j if i is removed (14) where h and H denote the fine and coarse rids respectively. This definition uarantees that the error lies in the rane space of the interpolation operator since e h = PH h eh. Note that if aij a ii is small, then the effect of node j is neliible and thus, can be inored. That is, it is enouh to interpolate the voltae at a removed node i from those nodes that are stronly connected to i. Based on this, we interpolate the voltae at a removed node, m, from the voltaes of those kept nodes which are stronly connected to m. Since the eometry of the rid is available, the kept nodes which are stronly connected to m can be easily identified. Typically, these would be the eometric neihbors of m, and/or their correspondin neihbors. The exact definition of the interpolation operator is discussed and illustrated in section IV-B. Thus, the proposed method combines the advantaes of the multirid techniques, the standard and the alebraic, while avoidin their limitations. Furthermore, there is one other sinificant advantae of the proposed approach over reular multirid. As noted earlier, multirid consists of two major components: relaxation (or smoothin) and coarse rid correction. Relaxation basically smoothes the error components and coarse rid correction approximates those smooth error components. In power rids, however, the error components are typically smooth since the power rids are desined so as to have smooth voltae variation over the rid. Consequently, in solvin the power rid problem, it is possible to inore the relaxation step of the multirid and concentrate only on the coarse rid correction step. That is exactly what is done by our proposed approach and the results in section IV-D show that this assumption leads to sinificant speed-ups while incurrin very small errors. Assumin smooth voltae variation and inorin the relaxation step, our proposed method falls under the cateory of direct solvers as opposed to the multirid technique which is an iterative solver. This promises even more speedups when performin transient analysis. Given an initial rid Ω h with the associated linear system A h x h = b h, the proposed approach can be summarized as follows: 1. Apply the rid reduction alorithm described in section IV-A to produce a coarser rid Ω 2h. 2. Define the interpolation operator, P2h h as explained in section IV-B. 3. If Ω 2h is not coarse enouh: (a) Copy Ω 2h to Ω h. (b) Update the interpolation operator, P2h h. (c) Go to step 1. If, on the other hand, Ω 2h is coarse enouh: (a) Map problem to coarse rid: A 2h = ( P2h) h T A h P2h h and b 2h = ( P2h) h T b h. (b) Solve problem at coarse rid, A 2h x 2h = b 2h. (c) Map solution back to fine rid: x h = P2h h x2h and exit. Note that the criterion for the reduced rid to be coarse enouh is user-specified. The criterion for a coarse-enouh rid involves a tradeoff between accuracy and speed. The coarser the rid is, the faster the solution is but the less accurate. Typically, a coarse-enouh rid is a rid where the size of the problem is small enouh to be solved efficiently. In our implementation, four levels of rid reduction (chosen empirically) prove sufficient to result in a reduced system which can be solved efficiently. In the rest of this section, we will be mostly concerned with the coarse rid correction process which is completely defined once the coarsenin stratey is chosen and the interpolation operator defined. In section IV-A we define

8 8 K K K K H R K K H R K K V R V R N R V R N R V R K K H R K K Fi. 8. Multiple resolution power rids. StatusF la N K H V R Indication No fla (default) Kept Visited Horizontally Visited Vertically Removed TABLE I Meanin of status flas. and discuss the rid reduction alorithm. Then in section IV-B, we illustrate how the interpolation operator is defined. Finally, in section IV-C, the advantaes of usin the proposed multirid-like technique for transient analysis are discussed. A. Grid Reduction A natural method for efficient rid reduction, inspired by SMG, is to skip every other wire, resultin in a situation as in Fi. 8. However, typical power rids may be irreular, i.e. different edes may have different lenths and different separation distances. Thus, the reduction alorithm should present a systematic mechanism for reducin any eneral rid. Furthermore, the alorithm should maintain the structure of the oriinal rid so that it can be recursively applied until a coarse enouh rid is obtained. The major objective of our reduction alorithm is to remove as many nodes as possible while maintainin the ability to estimate voltaes at the removed nodes by interpolation. The alorithm takes as input a fine rid Ω h and a list of nodes to be kept and produces as output a reduced rid Ω 2h with a smaller number of nodes. The list of kept nodes should consist of specific nodes of interest to the user, but our technique automatically enerates a default list containin the corner nodes and nodes where voltae sources are located. The alorithm makes use of certain status flas, which are explained in Table I, to decide whether a node is kept or removed. Furthermore, these flas indicate how to interpolate the voltae at a removed node from its kept neihbors. The rid reduction alorithm makes repeated use of a socalled node update operation, which is defined as follows: Startin from that node, o alon a horizontal (vertical) direction and fla all visited nodes with H (V). Fla extremities as kept. A node which is visited both horizontally and vertically (flaed with both H and V), is flaed as kept. The alorithm consists of three passes described as V K N R V K K K H R K K H R K K Fi. 9. Reduction of an irreular rid. follows: 1. First Pass: Update every kept node. 2. Second Pass: For each H (V) node, fla it as removed (R). Fla its neihbors alon the same row (column) as kept and update those neihbors. If a node is not flaed (N), fla it as removed (R), fla its diaonal neihbors as kept, and update those nodes. 3. Third Pass (defines interpolation): Voltae of a kept node is the same as that computed at the coarser rid. Voltae of an H (V) node which is then flaed as R is interpolated from its row (column) neihbors voltaes. Voltae of an N node which is then flaed as R is interpolated from its diaonal neihbors which are kept. The diaonal neihbors of a node X are defined as those nodes reached by oin 2 steps from X first horizontally and then vertically or first vertically and then horizontally. For example, if node Y is the upper neihbor of node X, then the left and riht neihbors of Y are diaonal neihbors of X. The alorithm is illustrated by the irreular rid Ω h shown in Fiure 9 which will be reduced to result in the rid Ω 2h. In our implementation of the rid reduction alorithm, rid nodes are ordered from top to bottom, left to riht. However, note that this is not a limitation of the alorithm which is robust enouh to handle any orderin of the nodes. Initially, all nodes have the default status of N except for the nodes which should be kept. In this example, these would be all the corner nodes of the rid (dashed nodes in Fi. 9). A ta consistin of two fields is associated with every node of the rid. The left field indicates the status of the node after the first pass and the riht field indicates the status of the node after the second pass. As shown in Fi. 9, after the first pass, an ede (row or column) consistin of at least one kept node, has its extremities flaed as kept. The remainin nodes on that ede are flaed with H or V based on whether the ede is horizontal or vertical. Note that some nodes still have a status fla of N which indicates that these nodes have not

9 9 A m B n r p Fi. 1. Basic Multirid operator. been visited durin the first pass. Then after the second pass, nodes with a K fla are kept while those with an R fla are removed thus resultin in the coarser rid Ω 2h. Finally, we point out that if the oriinal rid is reular, then the alorithm is optimal. That is, it results in maximal reduction in the number of nodes as illustrated in Fiure 1. In that case, every rid reduction results in a linear system with approximately 4x fewer unknowns and consequently 8x smaller CPU time for solution by direct sparse matrix methods. B. Interpolation AMG interpolation is uided by (11) and (12). Thus, the interpolation operator should be chosen to relate the voltae of a removed node, i, to the voltaes of those kept nodes which are stronly connected to i. Typically, AMG considers a connection between two nodes, i and j, to be stron when a ij / max l i a il θ, where θ 1 (θ is typically chosen to be.25 in practice [16]). With such a choice of the interpolation operator, the coarse rid correction would efficiently reduce the error. In our reduction alorithm, the status flas indicate which neihbors of a removed node are to be used for interpolation, based on the fact that they are kept and stronly connected to a removed node m. As for the interpolation weihts, these are obtained by considerin the values of conductances between the nodes. Thus, if the voltae at a removed node m is interpolated from the voltaes at nodes A and B, then the (linear) interpolation function INT () is defined as: V (m) = INT (V (A), V (B)) = a V (A) + a 1 V (B) (15) ma where a = ma+ mb and a 1 = mb ma+ mb. Here, ma it is the conductance between nodes m and A, and mb is the conductance between nodes m and B. Note that our technique for choosin the interpolation weihts is inspired by the technique used in AMG. To illustrate, consider a removed node m whose voltae will be interpolated from the voltaes at the kept nodes A and B. AMG uses the followin interpolation scheme [16]: V (m) = a ma a mm V (A) + a mb a mm V (B) (16) where a ma is the entry of the A matrix relatin nodes m and A, and a mb is the entry of the A matrix relatin nodes m and B. As for a mm, one AMG approach is to define it Fi. 11. C D Interpolation from reduced rid nodes. as the diaonal entry of the A matrix correspondin to node m. Another common AMG method defines a mm as: a mm = a ma + a mb. For the power rid problem, a ma = ma and a mb = mb, which shows that our interpolation technique is motivated by AMG. However, this is not the full story. Recall that our rid reduction alorithm differs from AMG rid reduction methods; it is actually based on SMG reduction - it uses only eometric information and removes as many nodes as possible. Hence, it is possible to come across cases where a removed node i has all the nodes that are stronly connected to it removed as well. To illustrate this, consider Fi. 11, where a filled node indicates a removed node and a blank node indicates a node that is kept. In this example, we assume that every horizontal or vertical link represents a stron connection but two nodes that are separated by two or more links are not stronly connected. This situation is typical of power rids. Thus, r is stronly connected to m and m is stronly connected to B, but r and B are not stronly connected. Nodes such as B that are separated by two stron links from r, but which are themselves not stronly connected to r, are said to be two-level stronly connected to r. Our reduction would remove node r, as well as all the nodes that are stronly connected to it, m, n, p, and q. However, it can be shown that our alorithm uarantees that, if a node i is removed, either some nodes that are stronly connected to i are kept or some nodes that are two-level stronly connected to i are kept. Therefore, in our interpolation technique, if all stronly connected neihbors of a node have been removed alon with it, we use its two-level stronly kept neihbors for interpolation. This is clearly illustrated in Fi. 11 where the voltae at node r is interpolated from those nodes which are two-level stronly connected to r; specifically, nodes A, B, C, D, and E. Note that this approach maintains the advantae of efficient rid reduction as well as meets the requirement of a ood interpolation operator. q E

10 1 C. Time Domain Analysis In section II-B we pointed out that the fixed time step BE interation method offers lare efficiency ains because it requires only one matrix inversion for all time steps. However, this efficiency comes at the cost of requirin the use of a direct solver. Since our proposed approach uses a direct solve, it is clear that it promises sinificant speed-ups when transient analysis is performed. In addition to bein advantaeous over iterative solvers, the proposed multirid-like technique offers sinificant advantaes over direct solvers as well. Basically, iven an N N linear system Ax = b, the proposed technique solves this system by mappin the problem to a reduced N H N H linear system A H x H = b H where N H N, solvin the reduced system, and mappin the solution back to the oriinal system. Note that most of the cost for solvin the oriinal system usin the proposed technique lies in solvin the reduced system A H x H = b H. Furthermore, both A and A H are sparse matrices and thus a factorization of either matrix is sinificantly more expensive than a forward/backward solve [9]. However, since N H N, then factorizin A is sinificantly more expensive than factorizin A H. The same observation holds true for performin a forward/backward solve. Consequently, the proposed technique promises more speed-ups when transient analysis is performed since transient analysis involves several forward/backward solves. It is clear that direct solvers offer sinificant speed-ups over iterative solvers when transient analysis is performed. However, the major problem with direct solvers is their hih memory demand which proves to be a limitin factor for many applications [1]. As a matter of fact, if the dimension of a linear system is very lare, it may be impossible to solve such a system usin a direct solver. In such cases, an iterative solver has to be used and the problem is seriously aravated when performin transient analysis because an iterative solver has to be used at every time step thus losin the speed-up advantae of direct solvers. However, our proposed technique offers an efficient solution to this problem because it uses a direct solver to solve a reduced system of a much smaller dimension. That is, our technique avoids the memory limitation of direct solvers while maintainin their speed-up advantae. It avoids the memory limitation by solvin a reduced system of a much smaller dimension. On the other hand, it maintains the speed-up advantae because the reduced system matrix is factorized only once with several forward/backward solves performed for transient analysis. Of course, the advantaes of the proposed technique come at a sliht cost in the accuracy of the solution since the relaxation step is inored. However, the results in the next section show that this error is small enouh to maintain the efficiency and suitability of the proposed technique. D. Experimental Results The proposed multirid method has been implemented and interated into a linear simulator written in C++. All experimental results reported in this section were obtained by runnin the simulations on a 4MHz ULTRA 2 Sun workstation with 2GB of RAM and runnin the SunOS 5.7 operatin system. The practicality and efficiency of the proposed technique are illustrated by applyin it for the analysis of the power rids of two real industrial ASIC desins. We will refer to these desins as C 1 and C 2. Both desins, C 1 and C 2 are.18µ CMOS desins and have a supply voltae of 1.8 V. Given the power rid, the technique requires as input the currents associated with the different power drains on the chip. Different current measures can be used for the analysis dependin on the application of interest. For instance, while peak current is a ood representative measure for IR drop, averae current is a better measure for electromiration analysis. On the other hand, a current waveform is the suitable current measure for transient analysis. A straiht-forward technique for obtainin any current measure of interest is to simulate the power drains under nominal loads and realistic switchin factors. This is how the current measures we use for our analysis are obtained. In all our experiments for DC analysis, we use the peak current drawn by the power drains as our current measure. As for transient analysis, the current measure used is the current waveform associated with the different power drains. The irreular power rids of the two desins, C 1 and C 2, were simulated. Several rid reductions are applied and the problem accordinly mapped to the coarser rids (as explained earlier, the reduction is repeated until the rid is coarse enouh as specified by the user). Specifyin four levels of reduction, Table II shows the number of nodes of the rid at every level. Table II also shows the CPU times for solvin the iven linear system usin both a reular direct solver (shown in column 4) as well as the proposed multirid-like technique (shown in column 5). It is clear that the proposed technique is almost 16 to 2 faster than traditional simulation. Note that the same direct solver is used for solvin both the oriinal system as well as the reduced system which verifies that the speedup is not due to an advantae of one solver over another. In order to verify the accuracy of the results provided by the proposed technique, the exact solution is compared to the estimated solution returned by the proposed technique. The historams of percentae errors in the voltaes of the different nodes of the power rids correspondin to the two desins, C 1 and C 2, are shown in Fiures 12 and 13 respectively. For the desin C 1, the distribution of the node voltae errors has a mean of.77% and a standard deviation of.333%. As for the desin C 2, the error distribution has a mean of.26% and a standard deviation of.167%. Furthermore, Fiures 12 and 13 also show that the errors at all the power rid nodes of both desins lie in the 1.% to 1.% rane. In fact, desin C 1 has errors that rane from.93% to.66% while desin C 2 has errors that rane from.3% to 1.%. Thus, it is clear that the proposed technique provides an accurate solution to the power rid

11 11 Desin name Level Number of nodes Exact solve time (sec) MG solve time (sec) C C TABLE II Grid reduction and CPU times usin exact solve as well as multirid-like technique Number of nodes Percentae error in node voltaes Fi. 12. Error in nodes voltaes for the C 1 desin Number of nodes Percentae error in node voltaes Fi. 13. Error in nodes voltaes for the C 2 desin.

12 12 Desin name Exact transient solve time MG transient solve time C seconds seconds C hours seconds TABLE III Grid reduction and CPU times for transient analysis Exact Waveform Estimated Waveform 1.82 Exact Waveform Estimated Waveform Node Voltae (Volts) Node Voltae (Volts) Time (nano seconds) Time (nano seconds) Fi. 14. Error in the voltae waveform at a power rid node in the C 1 desin. Fi. 15. Error in the voltae waveform at a power rid node in the C 2 desin. problem at a sinificant speed-up over reular solvers. As explained earlier, the proposed multirid-like technique is even more advantaeous when applied for transient analysis. This is illustrated by Table III which shows the time required to run a transient simulation of the power rids of the two desins usin a reular solver and the proposed technique. The power rids are simulated for a duration of 4ns with.4ns time steps. The speedup advantae is clear in both cases. However, it is more sinificant in the case of the C 2 desin and the reason is that desin C 2 is simulated usin an iterative solver due to memory limitations. Thus, for each time step, the power rid is simulated usin the iterative solver requirin a total of 14.3 hours. The proposed technique, on the other hand, uses a direct solver to solve the problem at the reduced rid. Thus, only one initial factorization is needed and only forward/backward solves are needed at the remainin time steps. The total time required for transient analysis usin the proposed multirid-like technique is seconds, representin a speed-up of 6 for transient analysis. Other methods to speed-up power rid analysis have been proposed [5]. In [5], a hierarchical power rid analysis technique is proposed which ives speed-ups between 2 and 5 for DC analysis. The authors also propose utiliz- in parallelism which increases the speed-ups to the rane 1 to 23 [5]. However, their proposed method offers no speed-ups when transient analysis is applied in serial mode. Smaller speed-ups between 1.8 and 5.1 can still be observed when parallel execution is used for transient analysis. Note that the speed-up comparison is a function of the linear solvers bein used as well as the size of the problems bein solved. However, experimental results show that our method promises more sinificant speed-ups at a minimal cost in accuracy. Furthermore, these speed-ups are evident for both DC as well as transient analysis. Finally, it remains to verify the accuracy of the resultin approximate solution. This is illustrated by Fiures 14 and 15 which show the voltae waveform at some node of the power rids of desins C 1 and C 2 respectively. It is clear that the multirid-like technique accurately tracks the exact voltae waveform at the iven node. Fiures 16 and 17 show the estimation error in the voltae drop (as opposed to voltae), showin a worst-case error of about 16%. Thus, the multirid-like technique provides relatively ood accuracy for both DC as well as transient analysis of the power rids with the added advantae of sinificant speed-up over reular analysis techniques.