Optimization techniques for high-performance digital circuits

The relentless push for high performance in custom dig-ital circuits has led to renewed emphasis on circuit opti-mization or tuning. The parameters of the optimization are typically transistor and interconnect sizes. The de-sign metrics are not just delay, transition times, power and area, but also signal integrity and manufacturability. This tutorial paper discusses some of the recently pro-posed methods of circuit optimization, with an emphasis on practical application and methodology impact. Circuit optimization techniques fall into three broad categories. The rst is dynamic tuning, based on time-domain simulation of the underlying circuit, typically combined with adjoint sensitivity computation. These methods are accurate but require the specication of in-put signals, and are best applied to small data- ow cir-cuits and \cross-sections " of larger circuits. Ecient sensitivity computation renders feasible the tuning of cir-cuits with a few thousand transistors. Second, static tuners employ static timing analysis to evaluate the per-formance of the circuit. All paths through the logic are simultaneously tuned, and no input vectors are required. Large control macros are best tuned by these methods. However, in the context of deep submicron custom de-sign, the inaccuracy of the delay models employed by these methods often limits their utility. Aggressive dy-namic or static tuning can push a circuit into a precip-itous corner of the manufacturing process space, which is a problem addressed by the third class of circuit op-timization tools, statistical tuners. Statistical techniques are used to enhance manufacturability or maximize yield. In addition to surveying the above techniques, topics such as the use of state-of-the-art nonlinear optimization methods and special considerations for interconnect siz-ing, clock tree optimization and noise-aware tuning will be brie y considered.

An Efficient Integrated Circuit Simulator And Time Domain Adjoint Sensitivity Analysis

In this paper, we revisit time-domain adjoint sensitivity with a circuit theoretic approach and an efficient solution is clearly stated in terms of device level. Key is the linearization of the energy storage elements (e.g., capacitance and inductance) and nonlinear memoryless elements (e.g., MOS, BJT DC characteristics) at each time step. Due to the finite precision of computation, numerical errors that accumulate across timesteps can arise in nonlinear elements

Differential-Algebraic Equations

Differential-Algebraic Equations (DAE) are today an independent field of research, which is gaining in importance and becoming of increasing interest for applications and mathematics itself. This workshop has drawn the balance after about 25 years investigations of DAEs and the research aims of the future were intensively discussed

Development and application of 2D and 3D transient electromagnetic inverse solutions based on adjoint Green functions: A feasibility study for the spatial reconstruction of conductivity distributions by means of sensitivities

To enhance interpretation capabilities of transient electromagnetic (TEM) methods, a multidimensional inverse solution is introduced, which allows for a explicit sensitivity calculation with reduced computational effort. The main conservation of computational load is obtained by solving Maxwell's equations directly in time domain. This is achieved by means of a high efficient Krylov-subspace technique that is particularly developed for the fast computation of EM fields in the diffusive regime. Traditional modeling procedures for Maxwell's equations yields solutions independently for every frequency or, in the time domain, at a given time through explicit time stepping. Because of this, frequency domain methods are rendered extremely time consuming for multi-frequency simulations. Likewise the stability conditions required by explicit time stepping techniques often result in highly inefficient calculations for large diffusion times and conductivity contrasts. The computation of sensitivities is carried out using the adjoint Green functions approach. For time domain applications, it is realized by convolution of the background electrical field information, originating from the primary signal, with the impulse response of the receiver acting as secondary source. In principle, the adjoint formulation may be extended allowing for a fast gradient calculation without calculating and storing the whole sensitivity matrix but just the gradient of the data residual. This technique, which is also known as migration, is widely used for seismic and, to some extend, for EM methods as well. However, the sensitivity matrix, which is not easily given by migration techniques, plays a central role in resolution analysis and would therefore be discarded. But, since it allows one to discriminate features in the a posteriori model which are data or regularization driven, it would therefore be very likely additional information to have. The additional cost of its storage and explicit computation is comparable low disbursement to the gain of a posteriori model resolution analysis. Inversion of TEM data arising from various types of sources is approached by two different methods. Both methods reconstruct the subsurface electrical conductivity properties directly in the time domain. A principal difference is given by the space dimensions of the inversion problems to be solved and the type of the optimization procedure. For two-dimensional (2D) models, the ill-posed and non-linear inverse problem is solved by means of a regularized Gauss-Newton type of optimization. For three-dimensional (3D) problems, due to the increase of complexity, a simpler, gradient based minimization scheme is presented. The 2D inversion is successfully applied to a long offset (LO)TEM survey conducted in the Arava basin (Jordan), where the joint interpretation of 168 transient soundings support the same subsurface conductivity structure as the one derived by inversion of a Magnetotelluric (MT) experiment. The 3D application to synthetic data demonstrates, that the spatial conductivity distribution can be reconstructed either by deep or shallow TEM sounding methods