Efficient smart monte carlo based SSTA on graphics processing units with improved resource utilization

Static timing analysis (STA) is an essential yet time-consuming task during the circuit design flow to ensure the correctness and performance of the design. Thanks to the advancement of general-purpose computing on graphics processing units (GPUs), new possibilities and challenges have arisen for boosting the performance of STA. In this work, we present an efficient and holistic GPU-accelerated STA engine. We accelerate major STA tasks, including levelization, delay computation, graph propagation, and multicorner analysis, by developing high-performance GPU kernels and data structures. By dividing the STA workloads into CPU–GPU concurrent tasks with managed dependencies, our acceleration framework supports versatile incremental updates. Furthermore, we have extended our approach to multicorner analysis by exploring a large amount of corner-level data parallelism using GPU computing. Our implementation based on the open-source STA engine OpenTimer has achieved up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$4.07\times $ </tex-math></inline-formula> speed-up on single corner analysis, and up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$25.67\times $ </tex-math></inline-formula> speed-up on multicorner analysis on TAU 2015 contest designs and a 14-nm technology.

General-purpose computing on graphics processing unit (GPGPU) enables the possibility of parallel computing for Static Timing Analysis (STA) of VLSI designs. However, memory access and synchronization between massively many cores become challenges to parallelizing STA. In this work, we developed a fast CUDA-Accelerated STA engine (named CASTA) that incorporates four novel techniques including Table-Index Remapping (TIR), Texture-Accelerated Rendering (TAR), Cell Levelization & Type Sorting (CLTS) and Timing-Table Restructuring(TTR) to enable high parallelism. Cell Levelization & Type Sorting (CLTS) levelizes cells and sort their types in order to efficiently access the same timing library. Timing-Table Restructuring (TTR) modifies the data structure for timing signals of cells to increase memory throughput. Table-Index Remapping (TIR) re-maps the axes of timing tables to retrieve data more efficiently while Texture-Accelerated Rendering (TAR) expands look-up tables (LUTs) to avoid extrapolation and stores LUTs in the texture for speed. As a result, our experimental result indicates that CASTA successfully enables high parallelism and outperforms a commercial tool by a three-order speedup on average over several benchmark circuits.

Static timing analysis (STA) is an integral part in the overall design flow because it verifies the expected timing behaviors of a circuit. However, as the circuit complexity continues to enlarge, there is an increasing need for enhancing the performance of existing STA algorithms using emerging heterogeneous parallelism that comprises manycore central processing units (CPUs) and graphics processing units (GPUs). In this paper, we introduce several state-of-the-art STA techniques, including task-based parallelism, task graph partition, and GPU kernel algorithms, all of which have brought significant performance benefits to STA applications. Motivated by these successful results, we will introduce a task-parallel programming system to generalize our solutions to benefit broader scientific computing applications.

In this paper, we explore the implementation of Monte Carlo based statistical static timing analysis (SSTA) on a Graphics Processing Unit (GPU). SSTA via Monte Carlo simulations is a computationally expensive, but important step required to achieve design timing closure. It provides an accurate estimate of delay variations and their impact on design yield. The large number of threads that can be computed in parallel on a GPU suggests a natural fit for the problem of Monte Carlo based SSTA to the GPU platform. Our implementation performs multiple delay simulations at a single gate in parallel. A parallel implementation of the Mersenne Twister pseudo-random number generator on the GPU, followed by Box-Muller transformations (also implemented on the GPU) is used for generating gate delay numbers from a normal distribution. The μ and σ of the pin-to-output delay distributions for all inputs and for every gate, are obtained using a memory lookup, which benefits from the large memory bandwidth of the GPU. Threads which execute in parallel have no data/control dependencies on each other. All threads compute identical instructions, but on different data, as required by the Single Instruction Multiple Data (SIMD) programming semantics of the GPU. Our approach is implemented on a NVIDIA GeForce GTX 8800 GPU card. Our results indicate that our approach can obtain an average speedup of about 260x as compared to a serial CPU implementation. With the recently announced quad 8800 GPU cards, we estimate that our approach would attain a speedup of over 785x. The correctness of the Monte Carlo based SSTA implemented on a GPU has been verified by comparing its results with a CPU based implementation.

Purpose: For microdosimetric calculation event-by-event Monte Carlo (MC) simulation is considered the most accurate, but it is very time-consuming. In this work we present an event-by-event MC simulation of low energy electron and proton for accelerated microdosimetric MC simulations on a graphic processing unit (GPU). Methods: The MC simulation of particle was implemented in C and executed on a multi-core CPU, and a commercially available general purpose GPU using the compute unified device architecture (CUDA). Additionally, a hybrid implementation scheme was realized by employing OpenMP and CUDA so that both GPU and multi-core CPU were utilized simultaneously. The two implementation schemes have been tested and compared with the sequential single threaded MC simulation on the CPU. A calculation time comparison was established on the speed-up for a set of benchmarking cases of electron and proton. Results: Dosimetric results were obtained with both the parallel and serial MC codes. A GPU over CPU speed-up of 67.2 and 19.2 times was achieved for 300 eV and 2 keV electron tracks respectively. For proton tracks, the GPU-based code was approximately 5 times faster than the CPU-single-thread code. By incorporating a multi-core CPU and running the MC code simultaneously on the GPU and CPU, an increase of 2%-7% and 20% in the speedup was noticed for electrons and protons tracks respectively. A good dosimetric agreement between the CPU- and GPU-based MC methods was observed for both electrons and proton. Conclusions: A GPU-based MC method for microdosimetric calculations of low energy electron and proton has been presented. The results indicate the capability of our GPU-based implementation for accelerated MC simulations of both electron and proton without loss of accuracy. Lastly, the potential of a hybrid approach by utilizing simultaneously a GPU and a multi-core CPU for further acceleration of MC microdosimetric calculations has been demonstrated.

The assessment of flow-mediated dilation (FMD) is widely used to quantify endothelial function. Historically, FMD was determined at 60 seconds post-cuff deflation. We investigated whether FMD would be more accurate if determined at maximum dilatory peak (MDP) than at 60 seconds in healthy subjects and subjects with type 2 diabetes mellitus (T2DM). We studied 95 healthy and 72 subjects with T2DM and assessed FMD at MDP, 60 and 90 seconds. Twenty-four healthy and 12 subjects with T2DM underwent a repeat FMD after 28 days. In healthy subjects, FMD at MDP was higher than at 60 and 90 seconds, with mean difference MDP versus 60 seconds 1.14% (95% CI: 0.6-1.7); P < .0001 and MDP versus 90 seconds 1.9% (95% CI: 1.3-2.5) with similar results in T2DM, that is, 1.0% (95% CI: 0.1-1.9) and 2.3% (95% CI: 1.3-3.2), respectively. Intraindividual variability was lowest with MDP compared with 60 and 90 seconds, that is, 15.0 versus 23.2% and 40.0%, respectively, resulting in a more than 2-fold reduction in necessary sample size. In healthy subjects and subjects with T2DM, assessment of FMD using MDP results in a more accurate and precise assessment leading to a substantial reduction in sample size.

For nuclear reactor analysis such as the neutron eigenvalue calculations, the time consuming Monte Carlo (MC) simulations can be accelerated by using graphics processing units (GPUs). However, traditional MC methods are often history-based, and their performance on GPUs is affected significantly by the thread divergence problem. In this paper we describe the development of a newly designed event-based vectorized MC algorithm for solving the neutron eigenvalue problem. The code was implemented using NVIDIA’s Compute Unified Device Architecture (CUDA), and tested on a NVIDIA Tesla M2090 GPU card. We found that although the vectorized MC algorithm greatly reduces the occurrence of thread divergence thus enhancing the warp execution efficiency, the overall simulation speed is roughly ten times slower than the history-based MC code on GPUs. Profiling results suggest that the slow speed is probably due to the memory access latency caused by the large amount of global memory transactions. Possible solutions to improve the code efficiency are discussed.

Gravitational tree-code on graphics processing units: implementation in CUDA

Purpose: Monte Carlo (MC) simulation provides the most accurate results for CT dose calculations. Yet it is a time‐consuming task which often takes long time to run on CPUs. Using the recently developed hardware accelerators such as Graphics Processing Unit (GPU) and the Intel Xeon Phi coprocessor, one can run the MC simulations in massively parallel and reduce the simulation time significantly. In this work we present our effort of developing a new MC suite, Accelerated Radiation‐transport Computations in Heterogeneous EnviRonments (ARCHER), which utilizes NVIDIA GPUs and other accelerators. Methods: We first developed the radiation transport MC codes in C, and then adapted the code to ARCHERGPU and ARCHERMIC which are specific for NVIDIA GPU and Intel MIC architecture, respectively. Currently the code is capable of calculating the dose for CT X‐ray scan and electron radiotherapy. Our code also incorporated a whole CT scanner model and various computational human phantoms. Results: We tested our GPU‐based MC code on NVIDIA Tesla M2090 GPUs and compared its performance with other MC codes running on an Intel Xeon X5650 CPU. Two tests were carried out: 1) For a whole body helical CT scan with RPI‐Adult male obese phantom and 9e8 incoming photons, the ARCHER_GPU code and ARCHER_MIC code are 1200 and 561 times faster, respectively, than MCNPX running on the CPU. 2) For a simple water‐aluminum‐water phantom and 20MeV electron source, the ARCHER_GPU code takes 186 sec to simulate 6e6 electrons, and it is about 1000 times faster than MCNPX running on the CPU. Conclusion: Our preliminary results showed the efficiency of using different hardware accelerators to speed up MC radiation transport simulations for medical dosimetry. On‐going work includes further optimization of the code and implementation of more complex physics models.

Purpose: To present a Monte Carlo (MC) x‐ray transport simulation code that uses a massively parallel graphics processing units (GPU) to generate medical images of voxelized patient models and estimate organ doses. Methods: We updated MC‐GPU, a CUDA‐based open‐source code for the simulation of radiographic projections and CT, to include the capability to tally 3D dose distributions. Since electrons are not transported, deposited doses are assumed to be equivalent to photon KERMA. The validity of this assumption was investigated by comparing it with the general‐purpose MC code PENELOPE in the simulation of a posterior‐anterior chest examination with 1010, 60‐keV x rays and a realistic female phantom with 1‐mm voxels. The stability of the simulation in a 480‐core GPU was also benchmarked using a sequential execution in a single CPU core. A utility to post‐process the voxel doses and report average and peak organ doses and effective dose for the procedure (and their uncertainty) has been developed. Results: The accuracy of the dose values generated by MC‐GPU was successfully validated. Average relative differences in the doses from PENELOPE and MC‐GPU executed in CPU and GPU were below 2%, in the expected range for statistical fluctuations. The simulation speeds were 46554 x rays/s with PENELOPE (electron transport disabled), 158216 x rays/s with MC‐GPU in a CPU, and 5124867 x rays/s in a GPU (simulation time in the GPU: 32.5 minutes, 32‐fold speedup compared to the CPU). The average and peak doses in breast and skin were estimated as 1.1 and 3.4 eV/g, and 1.7 and 15.6 eV/g per history respectively. Conclusion: The presented code can estimate organ doses 110 times faster than a general‐purpose MC code with comparable accuracy. A multi‐GPU execution of MC‐GPU has the potential to provide accurate organ doses in a CT scan in near real‐time.

Purpose: To develop a Graphics Processing Unit (GPU) based Monte Carlo (MC) code that uses a dual‐GPU system to accelerate radiographic simulation and CT imaging dose calculations. Methods: We considered two clinical cases, a chest x‐ray radiography and an abdominal CT scan. In the first case, a voxelized VIP‐Man phantom with detailed 3D anatomical information was used and an x‐ray beam of 120kVp was simulated. In the second case, a voxelized abdomen phantom derived from 120 CT slices was used, and a GE LightSpeed 16‐MDCT scanner was modeled. The CPU version of the MC code was written in C++ and run on Intel Xeon X5660 2.8GHz CPU, then translated into GPU code written in CUDA C and tested on a dual Tesla m2090 GPU system. The code was featured with automatic assignment of simulation task to multiple GPUs, as well as accurate calculation of energy‐ and material‐dependent cross‐sections. Results: Double‐precision floating point format was used for accuracy. In the first case, radiograph formation was simulated and doses to the organs listed in ICRP‐60 were calculated. When running on a single GPU, the MC GPU code was found to be x13 times faster than the CPU code and x29 times faster than MCNPX. In the second case, doses to the rectum, prostate, bladder and femoral heads were calculated. A speedup of x19 was observed compared to CPU code. These speedup factors were doubled on the dual‐GPU system. The imaging dose was benchmarked against MCNPX and a maximum difference of 1% was observed when the relative error is kept below 0.1%. Conclusions: A GPU‐based MC code was developed to simulate radiography and calculate imaging dose using detailed patient and CT scanner models. Efficiency and accuracy were both guaranteed in this code. Scalability of the code was confirmed on the dual‐GPU system.

Fast and accurate collision probability computations are essential for protecting space assets. Monte Carlo (MC) simulation is the most accurate but computationally intensive method. A Graphics Processing Unit (GPU) is used to parallelize the computation and reduce the overall runtime. Using MC techniques to compute the collision probability is common in literature as the benchmark. An optimized implementation on the GPU, however, is a challenging problem and is the main focus of the current work. The MC simulation takes samples from the uncertainty distributions of the Resident Space Objects (RSOs) at any time during a time window of interest and outputs the separations at closest approach. Therefore, any uncertainty propagation method may be used and the collision probability is automatically computed as a function of RSO collision radii. Integration using a fixed time step and a quartic interpolation after every Runge Kutta step ensures that no close approaches are missed. Two orders of magnitude speedups over a serial CPU implementation are shown, and speedups improve moderately with higher fidelity dynamics. The tool makes the MC approach tractable on a single workstation, and can be used as a final product, or for verifying surrogate and analytical collision probability methods.

Simulation of X-ray projections on GPU: Benchmarking gVirtualXray with clinically realistic phantoms

Monte Carlo (MC) simulation is commonly considered to be the most accurate dose calculation method in radiotherapy. However, its efficiency still requires improvement for many routine clinical applications. In this paper, we present our recent progress toward the development of a graphics processing unit (GPU)-based MC dose calculation package, gDPM v2.0. It utilizes the parallel computation ability of a GPU to achieve high efficiency, while maintaining the same particle transport physics as in the original dose planning method (DPM) code and hence the same level of simulation accuracy. In GPU computing, divergence of execution paths between threads can considerably reduce the efficiency. Since photons and electrons undergo different physics and hence attain different execution paths, we use a simulation scheme where photon transport and electron transport are separated to partially relieve the thread divergence issue. A high-performance random number generator and a hardware linear interpolation are also utilized. We have also developed various components to handle the fluence map and linac geometry, so that gDPM can be used to compute dose distributions for realistic IMRT or VMAT treatment plans. Our gDPM package is tested for its accuracy and efficiency in both phantoms and realistic patient cases. In all cases, the average relative uncertainties are less than 1%. A statistical t-test is performed and the dose difference between the CPU and the GPU results is not found to be statistically significant in over 96% of the high dose region and over 97% of the entire region. Speed-up factors of 69.1 ∼ 87.2 have been observed using an NVIDIA Tesla C2050 GPU card against a 2.27 GHz Intel Xeon CPU processor. For realistic IMRT and VMAT plans, MC dose calculation can be completed with less than 1% standard deviation in 36.1 ∼ 39.6 s using gDPM.

The stroke patient population is heterogeneous, leading to wide variation in outcome caused by differences in age, initial severity, and presence of concomitant disease. Setting an identical recovery target for all patients in intervention trials may conceal individually important therapeutic treatment effects. Instead, a variable end point that takes severity or likely prognosis into account may be more informative. We used data from the Glycine Antagonist in Neuroprotection (GAIN) International trial to assess statistical power of various primary end points for intervention trials. We selected prognosis-adjusted cut points based on Barthel Index (BI) or Rankin Scale (RS) using a prognostic model, or assigned a fixed end point within subgroups of patients defined by their Oxford category or National Institutes of Health Stroke Scale (NIHSS) score. We simulated a treatment effect and estimated statistical power with standard formulae. Assignment of end points using a prognostic model for individual patients increased statistical power, when compared with assigning end points using only the Oxford classification. For the BI, power was increased from 60% to 88% (equivalent to a 49% reduction in sample size if power remains unchanged). With the RS end points, power was increased from 84% to 92% (or a 24% reduction in sample size). Versus a fixed end point for all patients, model-based methods increased power by 22 percentage points for BI> or =95 and 14 percentage points for RS< or =1 (effective sample size reductions 43% and 34%). Prognosis-adjusted end points can increase statistical power compared with fixed end points. Assessment is based on realistic goals for individual patients and yet trial results remain generalizable.