A framework for synthesis of reduced order models

A framework for model reduction and synthesis is presented, which enables the re-use of reduced order models in circuit simulation. Especially when model reduction exploits structure preservation, we show that using the model as a current-driven element is possible, and allows for synthesis without controlled sources. Two synthesis techniques are considered: (1) by means of realizing the reduced transfer function into a netlist and (2) by unstamping the reduced system matrices into a circuit representation. The presented framework serves as a basis for reduction of large parasitic R/RC/RCL network

Model order reduction for multi-terminal circuits

Analysis of effects due to parasitics is of vital importance during the design of large-scale integrated circuits, since it gives insight into how circuit performance is affected by undesired parasitic effects. Due to the increasing amount of interconnect and metal layers, parasitic extraction and simulation may become very time consuming or even unfeasible. Developments are presented, for reducing systems describing R and RC netlists resulting from parasitic extraction. The methods exploit tools from graph theory to improve sparsity preservation especially for circuits with multi-terminals. Circuit synthesis is applied after model reduction, and the resulting reduced netlists are tested with industrial circuit simulators. With the novel RC reduction method SparseMA, experiments show reduction of 95% in the number of elements and 68x speed-up in simulation time

Atrasentan and renal events in patients with type 2 diabetes and chronic kidney disease (SONAR): a double-blind, randomised, placebo-controlled trial

Background: Short-term treatment for people with type 2 diabetes using a low dose of the selective endothelin A receptor antagonist atrasentan reduces albuminuria without causing significant sodium retention. We report the long-term effects of treatment with atrasentan on major renal outcomes. Methods: We did this double-blind, randomised, placebo-controlled trial at 689 sites in 41 countries. We enrolled adults aged 18–85 years with type 2 diabetes, estimated glomerular filtration rate (eGFR)25–75 mL/min per 1·73 m 2 of body surface area, and a urine albumin-to-creatinine ratio (UACR)of 300–5000 mg/g who had received maximum labelled or tolerated renin–angiotensin system inhibition for at least 4 weeks. Participants were given atrasentan 0·75 mg orally daily during an enrichment period before random group assignment. Those with a UACR decrease of at least 30% with no substantial fluid retention during the enrichment period (responders)were included in the double-blind treatment period. Responders were randomly assigned to receive either atrasentan 0·75 mg orally daily or placebo. All patients and investigators were masked to treatment assignment. The primary endpoint was a composite of doubling of serum creatinine (sustained for ≥30 days)or end-stage kidney disease (eGFR <15 mL/min per 1·73 m 2 sustained for ≥90 days, chronic dialysis for ≥90 days, kidney transplantation, or death from kidney failure)in the intention-to-treat population of all responders. Safety was assessed in all patients who received at least one dose of their assigned study treatment. The study is registered with ClinicalTrials.gov, number NCT01858532. Findings: Between May 17, 2013, and July 13, 2017, 11 087 patients were screened; 5117 entered the enrichment period, and 4711 completed the enrichment period. Of these, 2648 patients were responders and were randomly assigned to the atrasentan group (n=1325)or placebo group (n=1323). Median follow-up was 2·2 years (IQR 1·4–2·9). 79 (6·0%)of 1325 patients in the atrasentan group and 105 (7·9%)of 1323 in the placebo group had a primary composite renal endpoint event (hazard ratio [HR]0·65 [95% CI 0·49–0·88]; p=0·0047). Fluid retention and anaemia adverse events, which have been previously attributed to endothelin receptor antagonists, were more frequent in the atrasentan group than in the placebo group. Hospital admission for heart failure occurred in 47 (3·5%)of 1325 patients in the atrasentan group and 34 (2·6%)of 1323 patients in the placebo group (HR 1·33 [95% CI 0·85–2·07]; p=0·208). 58 (4·4%)patients in the atrasentan group and 52 (3·9%)in the placebo group died (HR 1·09 [95% CI 0·75–1·59]; p=0·65). Interpretation: Atrasentan reduced the risk of renal events in patients with diabetes and chronic kidney disease who were selected to optimise efficacy and safety. These data support a potential role for selective endothelin receptor antagonists in protecting renal function in patients with type 2 diabetes at high risk of developing end-stage kidney disease. Funding: AbbVie

Passivity preserving model reduction in the context of spectral zero interpolation

This thesis presents a new passivity preserving model reduction method for circuit simulation, based on interpolation of dominant spectral zeros. Implemented as an eigenvalue approximation problem, the dominant spectral zero method (dominant SZM) is based on the subspace accelerated dominant pole algorithm (SADPA), which computes dominant spectral zeros automatically. The application of dominant SZM is extended beyond its interpolatory nature, proposing solutions for several problems in passive reduction. In particular, better approximation is achieved when combined with partial realization for descriptor systems, a framework for SISO reduction of the voltage transfer function in transmission lines is presented, and the implementation of MIMO dominant SZM is developed. Dominant SZM reduces automatically passive circuits irrespective of how the system equations are formulated, transmission line models with controlled sources, or circuits containing susceptance elements. Results show that dominant SZM gives comparable and often more accurate reduced models than state of the art techniques

Model order reduction for multi-terminals systems : with applications to circuit simulation

Ever since its beginnings in the 1950’s, the integrated circuit (IC) has profoundly changed our lives. The way we work, travel, communicate, or address medical problems today has been facilitated by advances in microelectronics, which permit more functionality to be built on the same silicon area, at decreasing cost. As the feature size of devices on a chip shrink and circuits operate at increasing frequencies, the electromagnetic coupling effects between different IC components can no longer be ignored. To understand their impact on chip performance, these so called parasitic effects must be simulated. Parasitic networks are often so large, that state of the art simulation tools are insufficient to handle them: the simulations are either too lengthy, or cannot be carried out at all. The mathematical reason behind this is that the underlying systems are too large to be solved with the numerical algorithms implemented in simulation software. Model order reduction (MOR) provides one avenue for enabling faster simulations at little accuracy loss. However, when the systems have many input/output nodes, i.e., they have many terminals, performing the reduction itself becomes even more challenging. In this thesis model reduction methods are developed for multi-terminal systems arising in industrial problems. To solve these effeciently, the methods rely jointly on concepts from numerical linear algebra, electrical engineering, and computer science. This thesis begins with an overview of MOR in Chapter 1 and places the reduction of multi-terminal systems in the context of existing approaches. Aside from being efficient and accurate, multi-terminal MOR methods should also ensure that the reduced model is easily inserted in the simulation environment in place of the original, and that it is indeed cheaper to simulate. Although all reduction methods are expected to satisfy these properties, this thesis shows that such expectations are rarely met when traditional approaches are applied to very large electrical networks with many terminals which arise in industrial problems. Hence, an improved framework for multi-terminal model reduction and synthesis is proposed. The methods developed in this thesis address three global problems: (1) the efficient and accurate reduction of multi-terminal circuits, (2) the appropriate synthesis of the reduced model into a netlist equivalent with the same terminal nodes, and (3) the re-simulation of the reduced circuit (instead of the original) with emphasis on accuracy and simulation time. In Chapter 2, a basic framework is developed for the reduction of multi-terminal networks and the synthesis of reduced multi-terminal models. Chapter 2 shows that, if the circuit equations are prepared appropriately, a multi-terminal RLC network can be reduced so that the synthesis step is also greatly simplified. In particular, from the reduced mathematical construction, an equivalent circuit containing only RLC elements can be obtained, without introducing unintended circuit elements such as controlled sources. In addition, the reduced circuit has the same terminal nodes as the original and is coupled easily to other circuit blocks in the simulation setup. The framework establishes the mathematical principle which allows voltage sources, non-linear devices or other parts of a larger network to be de-coupled from to specific linear part to be reduced. It also ensures that these elements can be re-coupled in the simulation phase to the reduced circuit via its terminals. In Chapters 3 and 4, new methods for reducing large, multi-terminal R, and RC networks are derived, with emphasis on accuracy, efficiency and sparsity. It is shown that, if the projection which reduces an R/RC network performs a Schur-complement operation on the original conductance matrix, the resulting reduced network will have only positive resistors, which may be important for certain circuit simulators. This projection is also shown to exactly preserve the path resistance between terminals, and for RC circuits, the slope of the response in addition (in system theoretic terms, two multi-port admittance moments at DC are matched). The efficiency and sparsity considerations are dealt with especially in Chapter 4. These become critical for circuits with terminal numbers exceeding thousands, and node numbers exceeding hundreds of thousands. Reducing them by traditional means is either inefficient, or results in dense reduced models which are more expensive to simulate than the originals. Chapter 4 however develops a new method which is able to reduce efficiently such challenging RC netlists, while ensuring that the reduced models are sparse and fast to simulate. The key principles of the approach are graph partitioning, fill-reducing node re-orderings, and a reducing projection which is constructed accordingly. This preserves sparsity and also maintains accuracy by moment matching irrespective of how the circuit is partitioned. Based on the result of Chapter 2, the reduced models thus obtained are synthesized without controlled sources, have the same terminal nodes as the original ones, and are therefore inserted easily in the desired simulation flow. With the decrease of transistor feature sizes and increase of operating frequencies, it becomes important to also investigate the effects of parasitic inductances (e.g., skin or proximity effects) on chip performance. This requires time-consuming simulations of very large multi-terminal RLC(K) networks, and thus motivates the need for appropriate reduction methods. Chapter 5 identifies the main challenges of multi-terminal RLC reduction, and presents a skeleton for approaching them based on partitioning principles similar to those of Chapter 4. Important problems pertaining to RLC reduction are also identified in Chapter 5, which are otherwise rarely explicitly addressed in the literature. These include matching the response at DC when the underlying conductance matrix is singular, or the presence of singularities in the reduced conductance matrix which in turn negatively affect the simulation of the reduced model. For the latter problem especially, a solution is proposed in Chapter 5 . Two constraints which are known to limit the applicability of traditional reduction methods to multi-terminal systems are as follows: (a) the underlying matrix pencils are often singular and (b) the reducing projections may destroy the structure of the input/output matrices and with that, the physical interpretation of terminal nodes. The advanced methods of Chapters 3, 4 and 5 automatically by-pass these limitations due to the special way in which the reducing projection is formed. Chapter 6 brings an additional contribution by showing how, despite the known limitations, more general reduction methods (e.g., the Loewner approach) are also able to handle multi-terminal systems. The new reduction-synthesis framework of Chapter 6 eliminates the pencil singularity using a simple pre-processing of the original circuit, and recovers the connectivity at all terminal nodes using a post-processing of the reduced model. Among the main results of this thesis is the improvement in reduced model sparsity and reduction efficiency, achieved with the help of graph partitioning. State-of-theart partitioning algorithms such as nested dissection or Mondriaan already served this purpose, nevertheless Chapter 7 shows that the results could be further strengthened with the help of partitioning criteria designed especially for multi-terminal MOR. A high level description of the desired objectives is also derived there

Model order reduction for multi-terminal circuits

Analysis of effects due to parasitics is of vital importance during the design of large-scale integrated circuits, since it gives insight into how circuit performance is affected by undesired parasitic effects. Due to the increasing amount of interconnect and metal layers, parasitic extraction and simulation may become very time consuming or even unfeasible. Developments are presented, for reducing systems describing R and RC netlists resulting from parasitic extraction. The methods exploit tools from graph theory to improve sparsity preservation especially for circuits with multi-terminals. Circuit synthesis is applied after model reduction, and the resulting reduced netlists are tested with industrial circuit simulators. With the novel RC reduction method SparseMA, experiments show reduction of 95% in the number of elements and 68x speed-up in simulation time

Passivity-preserving model reduction using dominant spectral-zero interpolation

In this paper, the dominant spectral-zero method (dominant SZM) is presented, a new passivity-preserving model-reduction method for circuit simulation. Passivity is guaranteed via spectral-zero interpolation, and a dominance criterion is proposed for selecting spectral zeros. Dominant SZM is implemented as an iterative eigenvalue-approximation problem using the subspace-accelerated dominant-pole algorithm. Passive circuits are reduced automatically irrespective of how the original system equations are formulated (e.g., circuit models containing controlled sources or susceptance elements). Dominant SZM gives comparable and often more accurate reduced models than known techniques such as PRIMA, modal approximation, or positive real balanced truncation

SparseRC : sparsity preserving model reduction for RC circuits with many terminals

A novel model order reduction (MOR) method for multi-terminal RC circuits is proposed: SparseRC. Specifically tailored to systems with many terminals, SparseRC employs graph-partitioning and fill-in reducing orderings to improve sparsity during model reduction, while maintaining accuracy via moment matching. The reduced models are easily converted to their circuit representation. These contain much fewer nodes and circuit elements than otherwise obtained with conventional MOR techniques, allowing faster simulations at little accuracy loss

A new approach to passivity preserving model reduction : the dominant spectral zero method

A new model reduction method for circuit simulation is presented, which preserves passivity by interpolating dominant spectral zeros. These are computed as poles of an associated Hamiltonian system, using an iterative solver: the subspace accelerated dominant pole algorithm (SADPA). Based on a dominance criterion, SADPA finds relevant spectral zeros and the associated invariant subspaces, which are used to construct the passivity preserving projection. RLC netlist equivalents for the reduced models are provided