This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.

Introduction to the Finite Element Method

Nonlinear Programming: A Unified Approach

Practical cache replacement policies attempt to emulate optimal replacement by predicting the re-reference interval of a cache block. The commonly used LRU replacement policy always predicts a near-immediate re-reference interval on cache hits and misses. Applications that exhibit a distant re-reference interval perform badly under LRU. Such applications usually have a working-set larger than the cache or have frequent bursts of references to non-temporal data (called scans). To improve the performance of such workloads, this paper proposes cache replacement using Re-reference Interval Prediction (RRIP). We propose Static RRIP (SRRIP) that is scan-resistant and Dynamic RRIP (DRRIP) that is both scan-resistant and thrash-resistant. Both RRIP policies require only 2-bits per cache block and easily integrate into existing LRU approximations found in modern processors. Our evaluations using PC games, multimedia, server and SPEC CPU2006 workloads on a single-core processor with a 2MB last-level cache (LLC) show that both SRRIP and DRRIP outperform LRU replacement on the throughput metric by an average of 4% and 10% respectively. Our evaluations with over 1000 multi-programmed workloads on a 4-core CMP with an 8MB shared LLC show that SRRIP and DRRIP outperform LRU replacement on the throughput metric by an average of 7% and 9% respectively. We also show that RRIP outperforms LFU, the state-of the art scan-resistant replacement algorithm to-date. For the cache configurations under study, RRIP requires 2X less hardware than LRU and 2.5X less hardware than LFU.

High performance cache replacement using re-reference interval prediction (RRIP)

User-facing, latency-sensitive services, such as websearch, underutilize their computing resources during daily periods of low traffic. Reusing those resources for other tasks is rarely done in production services since the contention for shared resources can cause latency spikes that violate the service-level objectives of latency-sensitive tasks. The resulting under-utilization hurts both the affordability and energy-efficiency of large-scale datacenters. With technology scaling slowing down, it becomes important to address this opportunity. We present Heracles, a feedback-based controller that enables the safe colocation of best-effort tasks alongside a latency-critical service. Heracles dynamically manages multiple hardware and software isolation mechanisms, such as CPU, memory, and network isolation, to ensure that the latency-sensitive job meets latency targets while maximizing the resources given to best-effort tasks. We evaluate Heracles using production latency-critical and batch workloads from Google and demonstrate average server utilizations of 90% without latency violations across all the load and colocation scenarios that we evaluated.

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Heracles: improving resource efficiency at scale

Deep learning workloads are common in today's production clusters due to the proliferation of deep learning driven AI services (e.g., speech recognition, machine translation). A deep learning training job is resource-intensive and time-consuming. Efficient resource scheduling is the key to the maximal performance of a deep learning cluster. Existing cluster schedulers are largely not tailored to deep learning jobs, and typically specifying a fixed amount of resources for each job, prohibiting high resource efficiency and job performance. This paper proposes Optimus, a customized job scheduler for deep learning clusters, which minimizes job training time based on online resource-performance models. Optimus uses online fitting to predict model convergence during training, and sets up performance models to accurately estimate training speed as a function of allocated resources in each job. Based on the models, a simple yet effective method is designed and used for dynamically allocating resources and placing deep learning tasks to minimize job completion time. We implement Optimus on top of Kubernetes, a cluster manager for container orchestration, and experiment on a deep learning cluster with 7 CPU servers and 6 GPU servers, running 9 training jobs using the MXNet framework. Results show that Optimus outperforms representative cluster schedulers by about 139% and 63% in terms of job completion time and makespan, respectively.

Optimus: an efficient dynamic resource scheduler for deep learning clusters

The inherent temporal locality in memory accesses is filtered out by the L1 cache. As a consequence, an L2 cache with LRU replacement incurs significantly higher misses than the optimal replacement policy (OPT). We propose to narrow this gap through a novel replacement strategy that mimics the replacement decisions of OPT. The L2 cache is logically divided into two components, a Shepherd Cache (SC) with a simple FIFO replacement and a Main Cache (MC) with an emulation of optimal replacement. The SC plays the dual role of caching lines and guiding the replacement decisions in MC. Our pro- posed organization can cover 40% of the gap between OPT and LRU for a 2MB cache resulting in 7% overall speedup. Comparison with the dynamic insertion policy, a victim buffer, a V-Way cache and an LRU based fully associative cache demonstrates that our scheme performs better than all these strategies.

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Emulating Optimal Replacement with a Shepherd Cache

Effective sharing of the last level cache has a significant influence on the overall performance of a multicore system. We observe that existing solutions control cache occupancy at a coarser granularity, do not scale well to large core counts and in some cases lack the flexibility to support a variety of performance goals. In this paper, we propose Probabilistic Shared Cache Management (PriSM), a framework to manage the cache occupancy of different cores at cache block granularity by controlling their eviction probabilities. The proposed framework requires only simple hardware changes to implement, can scale to larger core count and is flexible enough to support a variety of performance goals. We demonstrate the flexibility of PriSM, by computing the eviction probabilities needed to achieve goals like hit-maximization, fairness and QOS. PriSM-HitMax improves performance by 18.7% over LRU and 11.8% over previously proposed schemes in a sixteen core machine. PriSM-Fairness improves fairness over existing solutions by 23.3% along with a performance improvement of 19.0%. PriSM-QOS successfully achieves the desired QOS targets.

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Probabilistic shared cache management (PriSM)

Query Optimization focuses on finding the best query execution plan, given fixed hardware resources. In BigData settings, both pay-as-you-go clouds and on-prem shared clusters, a complementary challenge emerges: Resource Optimization: find the best hardware resources, given an execution plan. In this world, provisioning is almost instantaneous and time-varying resources can be acquired on a per-query basis. This allows us to optimize allocations for completion time, resource usage, dollar cost, etc. These optimizations have a huge impact on performance and cost, and pivot around a core challenge: faithful resource-to-performance models for arbitrary BigData queries. This task is challenging for users and tools alike due to lack of good statistics (high-velocity, unstructured data), frequent use of UDFs, impact on performance of different hardware types and a lack of understanding of parallel execution at such a scale. We address this with PerfOrator, a novel approach to resource-to-performance modeling. PerfOrator employs nonlinear regression on profile runs to model arbitrary UDFs, calibration queries to generalize across hardware platforms, and analytical framework models to account for parallelism. The resulting estimates are orders of magnitude more accurate than existing approaches (e.g, Hive's optimizer), and have been successfully employed in two resource optimization scenarios: 1) optimize provisioning of clusters in cloud settings---with decisions within 1% of optimal, 2) reserve skyline of resources for SLA jobs---with accuracies over 10x better than human experts.

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PerfOrator: eloquent performance models for Resource Optimization

The effectiveness of the last-level shared cache is crucial to the performance of a multi-core system. In this paper, we observe and make use of the DelinquentPC — Next-Use characteristic to improve shared cache performance. We propose a new PC-centric cache organization, NUcache, for the shared last level cache of multi-cores. NUcache logically partitions the associative ways of a cache set into MainWays and DeliWays. While all lines have access to the MainWays, only lines brought in by a subset of delinquent PCs, selected by a PC selection mechanism, are allowed to enter the DeliWays. The PC selection mechanism is an intelligent cost-benefit analysis based algorithm that utilizes Next-Use information to select the set of PCs that can maximize the hits experienced in DeliWays. Performance evaluation reveals that NUcache improves the performance over a baseline design by 9.6%, 30% and 33% respectively for dual, quad and eight core workloads comprised of SPEC benchmarks. We also show that NUcache is more effective than other well-known cache-partitioning algorithms.

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NUcache: An efficient multicore cache organization based on Next-Use distance

A Projection Error Propagation-based Regularization (PEPR) method is proposed and the reconstructed image quality is improved in Electrical Impedance Tomography (EIT). A projection error is produced due to the misfit of the calculated and measured data in the reconstruction process. The variation of the projection error is integrated with response matrix in each iterations and the reconstruction is carried out in EIDORS. The PEPR method is studied with the simulated boundary data for different inhomogeneity geometries. Simulated results demonstrate that the PEPR technique improves image reconstruction precision in EIDORS and hence it can be successfully implemented to increase the reconstruction accuracy in EIT.> doi:10.5617/jeb.158 J Electr Bioimp, vol. 2, pp. 2-12, 2011

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Improving Image Quality in Electrical Impedance Tomography (EIT) Using Projection Error Propagation-Based Regularization (PEPR) Technique: A Simulation Study

Kaushik Rajan

Papers

Emulating Optimal Replacement with a Shepherd Cache

Probabilistic shared cache management (PriSM)

PerfOrator: eloquent performance models for Resource Optimization

NUcache: An efficient multicore cache organization based on Next-Use distance

Improving Image Quality in Electrical Impedance Tomography (EIT) Using Projection Error Propagation-Based Regularization (PEPR) Technique: A Simulation Study