Pengwen Chen

ePlace: Electrostatics Based Placement Using Nesterov's Method

ePlace is a generalized analytic algorithm to handle large-scale standard-cell and mixed-size placement. We use a novel density function based on electrostatics to remove overlap and Nesterovs method to minimize the nonlinear cost. Steplength is estimated as the inverse of Lipschitz constant, which is determined by our dynamic prediction and backtracking method. An approximated preconditioner is proposed to resolve the difference between large macros and standard cells, while an annealing engine is devised to handle macro legalization followed by placement of standard cells. The above innovations are integrated into our placement prototype ePlace, which outperforms the leading-edge placers on respective standard-cell and mixed-size benchmark suites. Specifically, ePlace produces 2.83%, 4.59% and 7.13% shorter wirelength while runs 3.05×, 2.84× and 1.05× faster than BonnPlace, MAPLE and NTUplace3-unified in average of ISPD 2005, ISPD 2006 and MMS circuits, respectively.

ePlace-MS: Electrostatics-Based Placement for Mixed-Size Circuits

We propose an electrostatics-based placement algorithm for large-scale mixed-size circuits (ePlace-MS). ePlace-MS is generalized, flat, analytic and nonlinear. The density modeling method eDensity is extended to handle the mixed-size placement. We conduct detailed analysis on the correctness of the gradient formulation and the numerical solution, as well as the rationale of dc removal and the advantages over prior density functions. Nesterovs method is used as the nonlinear solver, which shows high yet stable performance over mixed-size circuits. The steplength is set as the inverse of Lipschitz constant of the gradient function, while we develop a backtracking method to prevent overestimation. An approximated nonlinear preconditioner is developed to minimize the topological and physical differences between large macros and standard cells. Besides, we devise a simulated annealer to legalize the layout of macros and use a second-phase global placement to reoptimize the standard cell layout. All the above innovations are integrated into our mixed-size placement prototype ePlace-MS, which outperforms all the related works in literature with better quality and efficiency. Compared to the leading-edge mixed-size placer NTUplace3, ePlace-MS produces up to 22.98% and on average 8.22% shorter wirelength over all the 16 modern mixed-size benchmark circuits with the same runtime.

Fourier Phase Retrieval with a Single Mask by Douglas-Rachford Algorithm

The Fourier-domain Douglas-Rachford (FDR) algorithm is analyzed for phase retrieval with a single random mask. Since the uniqueness of phase retrieval solution requires more than a single oversampled coded diffraction pattern, the extra information is imposed in either of the following forms: 1) the sector condition on the object; 2) another oversampled diffraction pattern, coded or uncoded. For both settings, the uniqueness of projected fixed point is proved and for setting 2) the local, geometric convergence is derived with a rate given by a spectral gap condition. Numerical experiments demonstrate global, power-law convergence of FDR from arbitrary initialization for both settings as well as for 3 or more coded diffraction patterns without oversampling. In practice, the geometric convergence can be recovered from the power-law regime by a simple projection trick, resulting in highly accurate reconstruction from generic initialization.

ePlace: Electrostatics-Based Placement Using Fast Fourier Transform and Nesterov's Method

We develop a flat, analytic, and nonlinear placement algorithm, ePlace, which is more effective, generalized, simpler, and faster than previous works. Based on the analogy between placement instance and electrostatic system, we develop a novel placement density function eDensity, which models every object as positive charge and the density cost as the potential energy of the electrostatic system. The electric potential and field distribution are coupled with density using a well-defined Poissons equation, which is numerically solved by spectral methods based on fast Fourier transform (FFT). Instead of using the conjugate gradient (CG) nonlinear solver in previous placers, we propose to use Nesterovs method which achieves faster convergence. The efficiency bottleneck on line search is resolved by predicting the steplength using a closed-form equation of Lipschitz constant. The placement performance is validated through experiments on the ISPD 2005 and ISPD 2006 benchmark suites, where ePlace outperforms all state-of-the-art placers (Capo10.5, FastPlace3.0, RQL, MAPLE, ComPLx, BonnPlace, POLAR, APlace3, NTUPlace3, mPL6) with much shorter wirelength and shorter or comparable runtime. On average, of all the ISPD 2005 benchmarks, ePlace outperforms the leading placer BonnPlace with 2.83% shorter wirelength and runs 3.05× faster; and on average, of all the ISPD 2006 benchmarks, ePlace outperforms the leading placer MAPLE with 4.59% shorter wirelength and runs 2.84× faster.

FFTPL: An analytic placement algorithm using fast fourier transform for density equalization

We propose a flat nonlinear placement algorithm FFTPL using fast Fourier transform for density equalization. The placement instance is modeled as an electrostatic system with the analogy of density cost to the potential energy. A well-defined Poissons equation is proposed for gradient and cost computation. Our placer outperforms state-of-the-art placers with better solution quality and efficiency.

Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Alternating projection (AP) of various forms, including the parallel AP (PAP), real-constrained AP (RAP) and the serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed to produce good-quality initial guess. Numerical experiments show that the null initialization is more accurate than the spectral initialization and that AP converges faster to the true object than other iterative schemes such as the Wirtinger flow (WF). In numerical experiments AP with the null initialization converges globally to the true object.

Linear convergence analysis of the use of gradient projection methods on total variation problems

Optimization problems using total variation frequently appear in image analysis models, in which the sharp edges of images are preserved. Direct gradient descent methods usually yield very slow convergence when used for such optimization problems. Recently, many duality-based gradient projection methods have been proposed to accelerate the speed of convergence. In this dual formulation, the cost function of the optimization problem is singular, and the constraint set is not a polyhedral set. In this paper, we establish two inequalities related to projected gradients and show that, under some non-degeneracy conditions, the rate of convergence is linear.

From Circuit Theory, Simulation to SPICE Diego : A Matrix Exponential Approach for Time-Domain Analysis of Large-Scale Circuits

SPICE (Simulation Program with Integrated Circuit Emphasis) is a widely used circuit simulation framework for integrated circuit designs. The basic skeleton of SPICE time domain simulation was derived from the versions developed in UC Berkeley during the 1970s. In this paper, we report most recent numerical integration methods to improve traditional SPICE time integration schemes, which are based on linear multi-step and low order approximation for the circuit differential equation system. Recently, matrix exponential based time domain simulation algorithms are being developed to address long-term issues in the standard numerical integration methods. We review the related techniques in matrix exponential based approaches and state several distinguished features in challenging simulation problems, such as linear power network analysis and nonlinear circuit system simulation (SPICEDiego). We believe that the matrix exponential approaches can shed new light on the research and development of future circuit simulation algorithmic systems.

Phase retrieval by linear algebra

The null vector method, based on a simple linear algebraic concept, is proposed as an initialization method for nonconvex approaches to the phase retrieval problem. For the stylized measurement with random complex Gaussian matrices, a nonasymptotic error bound is derived, stronger than that of the spectral vector method. Numerical experiments show that the null vector method also has a superior performance for the realistic measurement of coded diffraction patterns in coherent diffractive imaging.

ePlace-3D: Electrostatics based Placement for 3D-ICs

We propose a flat, analytic, mixed-size placement algorithm ePlace-3D for three-dimension integrated circuits (3D-ICs) using nonlinear optimization. Our contributions are (1) electrostatics based 3D density function with globally uniform smoothness (2) 3D numerical solution with improved spectral formulation (3) 3D nonlinear pre-conditioner for convergence acceleration (4) interleaved 2D-3D placement for efficiency enhancement. Our placer outperforms the leading work mPL6-3D and NTUplace3-3D with 6.44% and 37.15% shorter wirelength, 9.11% and 10.27% fewer 3D vertical interconnects (VI) on average of IBM-PLACE circuits. Validation on the large-scale modern mixed-size (MMS) 3D circuits shows high performance and scalability.