Computational Physics on Graphics Processing Units

The use of graphics processing units for scientific computations is an emerging strategy that can significantly speed up various different algorithms. In this review, we discuss advances made in the field of computational physics, focusing on classical molecular dynamics, and on quantum simulations for electronic structure calculations using the density functional theory, wave function techniques, and quantum field theory.Comment: Proceedings of the 11th International Conference, PARA 2012, Helsinki, Finland, June 10-13, 201

Fast and Robust Parametric Estimation of Jointly Sparse Channels

We consider the joint estimation of multipath channels obtained with a set of receiving antennas and uniformly probed in the frequency domain. This scenario fits most of the modern outdoor communication protocols for mobile access or digital broadcasting among others. Such channels verify a Sparse Common Support property (SCS) which was used in a previous paper to propose a Finite Rate of Innovation (FRI) based sampling and estimation algorithm. In this contribution we improve the robustness and computational complexity aspects of this algorithm. The method is based on projection in Krylov subspaces to improve complexity and a new criterion called the Partial Effective Rank (PER) to estimate the level of sparsity to gain robustness. If P antennas measure a K-multipath channel with N uniformly sampled measurements per channel, the algorithm possesses an O(KPNlogN) complexity and an O(KPN) memory footprint instead of O(PN^3) and O(PN^2) for the direct implementation, making it suitable for K << N. The sparsity is estimated online based on the PER, and the algorithm therefore has a sense of introspection being able to relinquish sparsity if it is lacking. The estimation performances are tested on field measurements with synthetic AWGN, and the proposed algorithm outperforms non-sparse reconstruction in the medium to low SNR range (< 0dB), increasing the rate of successful symbol decodings by 1/10th in average, and 1/3rd in the best case. The experiments also show that the algorithm does not perform worse than a non-sparse estimation algorithm in non-sparse operating conditions, since it may fall-back to it if the PER criterion does not detect a sufficient level of sparsity. The algorithm is also tested against a method assuming a "discrete" sparsity model as in Compressed Sensing (CS). The conducted test indicates a trade-off between speed and accuracy.Comment: 11 pages, 9 figures, submitted to IEEE JETCAS special issue on Compressed Sensing, Sep. 201

Get Out of the Valley: Power-Efficient Address Mapping for GPUs

GPU memory systems adopt a multi-dimensional hardware structure to provide the bandwidth necessary to support 100s to 1000s of concurrent threads. On the software side, GPU-compute workloads also use multi-dimensional structures to organize the threads. We observe that these structures can combine unfavorably and create significant resource imbalance in the memory subsystem causing low performance and poor power-efficiency. The key issue is that it is highly application-dependent which memory address bits exhibit high variability. To solve this problem, we first provide an entropy analysis approach tailored for the highly concurrent memory request behavior in GPU-compute workloads. Our window-based entropy metric captures the information content of each address bit of the memory requests that are likely to co-exist in the memory system at runtime. Using this metric, we find that GPU-compute workloads exhibit entropy valleys distributed throughout the lower order address bits. This indicates that efficient GPU-address mapping schemes need to harvest entropy from broad address-bit ranges and concentrate the entropy into the bits used for channel and bank selection in the memory subsystem. This insight leads us to propose the Page Address Entropy (PAE) mapping scheme which concentrates the entropy of the row, channel and bank bits of the input address into the bank and channel bits of the output address. PAE maps straightforwardly to hardware and can be implemented with a tree of XOR-gates. PAE improves performance by 1.31 x and power-efficiency by 1.25 x compared to state-of-the-art permutation-based address mapping

A Study of High Performance Multiple Precision Arithmetic on Graphics Processing Units

Multiple precision (MP) arithmetic is a core building block of a wide variety of algorithms in computational mathematics and computer science. In mathematics MP is used in computational number theory, geometric computation, experimental mathematics, and in some random matrix problems. In computer science, MP arithmetic is primarily used in cryptographic algorithms: securing communications, digital signatures, and code breaking. In most of these application areas, the factor that limits performance is the MP arithmetic. The focus of our research is to build and analyze highly optimized libraries that allow the MP operations to be offloaded from the CPU to the GPU. Our goal is to achieve an order of magnitude improvement over the CPU in three key metrics: operations per second per socket, operations per watt, and operation per second per dollar. What we find is that the SIMD design and balance of compute, cache, and bandwidth resources on the GPU is quite different from the CPU, so libraries such as GMP cannot simply be ported to the GPU. New approaches and algorithms are required to achieve high performance and high utilization of GPU resources. Further, we find that low-level ISA differences between GPU generations means that an approach that works well on one generation might not run well on the next. Here we report on our progress towards MP arithmetic libraries on the GPU in four areas: (1) large integer addition, subtraction, and multiplication; (2) high performance modular multiplication and modular exponentiation (the key operations for cryptographic algorithms) across generations of GPUs; (3) high precision floating point addition, subtraction, multiplication, division, and square root; (4) parallel short division, which we prove is asymptotically optimal on EREW and CREW PRAMs

Fast algorithms for biophysically-constrained inverse problems in medical imaging

We present algorithms and software for parameter estimation for forward and inverse tumor growth problems and diffeomorphic image registration. Our methods target the following scenarios: automatic image registration of healthy images to tumor bearing medical images and parameter estimation/calibration of tumor models. This thesis focuses on robust and scalable algorithms for these problems. Although the proposed framework applies to many problems in oncology, we focus on primary brain tumors and in particular low and high-grade gliomas. For the tumor model, the main quantity of interest is the extent of tumor infiltration into the brain, beyond what is visible in imaging. The inverse tumor problem assumes that we have patient images at two (or more) well-separated times so that we can observe the tumor growth. Also, the inverse problem requires that the two images are segmented. But in a clinical setting such information is usually not available. In a typical case, we just have multimodal magnetic resonance images with no segmentation. We address this lack of information by solving a coupled inverse registration and tumor problem. The role of image registration is to find a plausible mapping between the patient's tumor-bearing image and a normal brain (atlas), with known segmentation. Solving this coupled inverse problem has a prohibitive computational cost, especially in 3D. To address this challenge we have developed novel schemes, scaled up to 200K cores. Our main contributions is the design and implementation of fast solvers for these problems. We also study the performance for the tumor parameter estimation and registration solvers and their algorithmic scalability. In particular, we introduce the following novel algorithms: An adjoint formulation for tumor-growth problems with/without mass-effect; The first parallel 3D Newton-Krylov method for large diffeomorphic image registration; A novel parallel semi-Lagrangian algorithm for solving advection equations in image registration and its parallel implementation on shared and distributed memory architectures; and Accelerated FFT (AccFFT), an open-source parallel FFT library for CPU and GPUs scaled up to 131,000 cores with optimized kernels for computing spectral operators. The scientific outcomes of this thesis, has appeared in the proceedings of three ACM/IEEE SCxy conferences (two best student paper finalist, and one ACM SRC gold medal), two journal papers, two papers in review, four papers in preparation (coupling, mass effect, segmentation, and multi-species tumor model), and seven conference presentations.Computational Science, Engineering, and Mathematic