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.

Hamilton-Jacobi Theory for Degenerate Lagrangian Systems with Holonomic and Nonholonomic Constraints

We extend Hamilton-Jacobi theory to Lagrange-Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton-Jacobi equation as the Dirac-Hamilton-Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton-Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac-Hamilton-Jacobi equation reduces to a variant of the nonholonomic Hamilton-Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac-Hamilton-Jacobi equation can be used to exactly integrate the equations of motion.Comment: 44 pages, 3 figure

Termodinamičke osnove termokemijskih energetskih sustava i gorivnih članaka

This research treats power optimization for energy converters, such as thermal, solar and electrochemical engines (fuel cells). A common methodology is developed for the assessment of power limits in thermal systems and fuel cells. Thermodynamic analyses lead to converter efficiency and limiting power. Steady and dynamic systems are investigated. Static optimization of steady systems applies the differential calculus or Lagrange multipliers, dynamic optimization of unsteady systems uses variational calculus and dynamic programming. The primary result of the first is the limiting value of power, whereas that of the second is a total generalized work potential. The generalizing quantity depends on the thermal coordinates and a dissipation index, h, i.e. the Hamiltonian of the problem of minimum entropy production. The advanced thermodynamics, of an irreversible nature, implies stronger bounds on work delivered or supplied than the classical reversible work. It is shown how various analytical developments can efficiently be synthesized to quantitatively evaluate power limits in thermal systems and fuel cells of a simple topology (without countercurrent flows).Ovo se istraživanje bavi optimizacijom snage sustava za pretvorbu energije poput termičkih, solarnih i elektrokemijskih (gorivni članci). U radu je razvijena jedinstvena metoda procjene granice snage u termičkim sustavima i gorivnim člancima. Termodinamičkim analizama dolazi se do učinkovitosti sustava za pretvorbu i granične snage. Istražuju se stacionarni i nestacionarni sustavi. Za statičku optimizaciju stacionarnih sistema primjenjuju se diferencijalni račun ili Lagrangeovi faktori; dinamička optimizacija nestacionarnih sustava koristi varijacijski račun i dinamičko programiranje. Rezultat prvog je ograničavajuća vrijednost snage dok je rezultat drugog ukupni poopćeni potencijal rada. Poopćenje ovisi o termičkim koordinatama i indeksu disipacije, h, npr. Hamiltonov operator problema minimalne entropije. Razvijena termodinamika nepovratnih sustava implicira čvršće granice na potrošeni ili predani rad nego što je to kod termodinamike povratnih procesa. Pokazano je kako različite analize mogu efikasno biti sintetizirane u svrhu kvantitativne procjene granica snage u termičkim sustavima i gorivnim člancima jednostavne topologije (bez protustrujnih tokova)

The Geometry of Monotone Operator Splitting Methods

We propose a geometric framework to describe and analyze a wide array of operator splitting methods for solving monotone inclusion problems. The initial inclusion problem, which typically involves several operators combined through monotonicity-preserving operations, is seldom solvable in its original form. We embed it in an auxiliary space, where it is associated with a surrogate monotone inclusion problem with a more tractable structure and which allows for easy recovery of solutions to the initial problem. The surrogate problem is solved by successive projections onto half-spaces containing its solution set. The outer approximation half-spaces are constructed by using the individual operators present in the model separately. This geometric framework is shown to encompass traditional methods as well as state-of-the-art asynchronous block-iterative algorithms, and its flexible structure provides a pattern to design new ones

Geometric Variational Models for Inverse Problems in Imaging

This dissertation develops geometric variational models for different inverse problems in imaging that are ill-posed, designing at the same time efficient numerical algorithms to compute their solutions. Variational methods solve inverse problems by the following two steps: formulation of a variational model as a minimization problem, and design of a minimization algorithm to solve it. This dissertation is organized in the same manner. It first formulates minimization problems associated with geometric models for different inverse problems in imaging, and it then designs efficient minimization algorithms to compute their solutions. The minimization problem summarizes both the data available from the measurements and the prior knowledge about the solution in its objective functional; this naturally leads to the combination of a measurement or data term and a prior term. Geometry can play a role in any of these terms, depending on the properties of the data acquisition system or the object being imaged. In this context, each chapter of this dissertation formulates a variational model that includes geometry in a different manner in the objective functional, depending on the inverse problem at hand. In the context of compressed sensing, the first chapter exploits the geometric properties of images to include an alignment term in the sparsity prior of compressed sensing; this additional prior term aligns the normal vectors of the level curves of the image with the reconstructed signal, and it improves the quality of reconstruction. A two-step recovery method is designed for that purpose: first, it estimates the normal vectors to the level curves of the image; second, it reconstructs an image matching the compressed sensing measurements, the geometric alignment of normals, and the sparsity constraint of compressed sensing. The proposed method is extended to non-local operators in graphs for the recovery of textures. The harmonic active contours of Chapter 2 make use of differential geometry to interpret the segmentation of an image as a minimal surface manifold. In this case, geometry is exploited in both the measurement term, by coupling the different image channels in a robust edge detector, and in the prior term, by imposing smoothness in the segmentation. The proposed technique generalizes existing active contours to higher dimensional spaces and non-flat images; in the plane, it improves the segmentation of images with inhomogeneities and weak edges. Shape-from-shading is investigated in Chapter 3 for the reconstruction of a silicon wafer from images of printed circuits taken with a scanning electron microscope. In this case, geometry plays a role in the image acquisition system, that is, in the measurement term of the objective functional. The prior term involves a smoothness constraint on the surface and a shape prior on the expected pattern in the circuit. The proposed reconstruction method also estimates a deformation field between the ideal pattern design and the reconstructed surface, substituting the model of shape variability necessary in shape priors with an elastic deformation field that quantifies deviations in the manufacturing process. Finally, the techniques used for the design of efficient numerical algorithms are explained with an example problem based on the level set method. To this purpose, Chapter 4 develops an efficient algorithm for the level set method when the level set function is constrained to remain a signed distance function. The distance function is preserved by the introduction of an explicit constraint in the minimization problem, the minimization algorithm is efficient by the adequate use of variable-splitting and augmented Lagrangian techniques. These techniques introduce additional variables, constraints, and Lagrange multipliers in the original minimization problem, and they decompose it into sub-optimization problems that are simple and can be efficiently solved. As a result, the proposed algorithm is five to six times faster than the original algorithm for the level set method