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Ptychography promises diffraction limited resolution without the need for high resolution lenses. To achieve high resolution one has to solve the phase problem for many partially overlapping frames. Here we review some of the existing methods for solving ptychographic phase retrieval problem from a numerical analysis point of view, and propose alternative methods based on numerical optimization.
Plasma dynamics can be strongly influenced by fine-scale kinetic physics, which is not captured by standard fluid treatments, is challenging to simulate, and is poorly understood. The aim of this research was to investigate the fundamental fine-scale properties of collisionless plasmas in Z-pinch and pulsed power configurations by deploying novel noisefree high-accuracy computational algorithms. Significant advances were achieved through the development of robust numerical capabilities to exploit axisymmetry in kinetic simulations, the development of new methodologies to construct kinetic equilibria, and the theoretical and computational study of kinetic sheared-flow-driven transport in pulsed power systems. These new capabilities shed light on the anisotropies that can develop in Z-pinches and the transport processes that can lead to parasitic currents in pulsed power inertial confinement fusion experiments. The use of cutting-edge massively-parallel numerical algorithms to address outstanding scientific questions in plasma physics advances the mission of Lawrence Livermore National Laboratory in the areas of high energy density science and high performance computing. By bringing together kinetic theory, high-order numerical methods, high performance computing, and experimental applications, this research advances the state of the art in plasma physics modeling.
We consider numerical methods for initial value problems that employ a two stage approach consisting of solution on a relatively coarse discretization followed by solution on a relatively fine discretization. Examples include adaptive error control, parallel-in-time solution schemes, and efficient solution of adjoint problems for computing a posteriori error estimates. We describe a general formulation of two stage computations then perform a general a posteriori error analysis based on computable residuals and solution of an adjoint problem. The analysis accommodates various variations in the two stage computation and in formulation of the adjoint problems. We apply the analysis to compute “dual-weighted” a posteriori error estimates, to develop novel algorithms for efficient solution that take into account cancellation of error, and to the Parareal Algorithm. We test the various results using several numerical examples.
The efficient utilization of mixed-precision numerical linear algebra algorithms can offer attractive acceleration to scientific computing applications. Especially with the hardware integration of low-precision special-function units designed for machine learning applications, the traditional numerical algorithms community urgently needs to reconsider the floating point formats used in the distinct operations to efficiently leverage the available compute power. In this study, we provide a comprehensive survey of mixed-precision numerical linear algebra routines, including the underlying concepts, theoretical background, and experimental results for both dense and sparse linear algebra problems.
The Moment Preserving Constrained Resampling (MPCR) algorithm for particle resampling is introduced and applied to particle-in-cell (PIC) methods to increase simulation accuracy, reduce compute cost, and/or avoid numerical instabilities. The general algorithm partitions the system space into smaller subsets and resamples the distribution within each subset. Further, the algorithm is designed to conserve any number of particle and grid moments with a high degree of accuracy (i.e. machine accuracy). The effectiveness of MPCR is demonstrated with several numerical tests, including a use-case study in gyrokinetic fusion plasma simulations. Finally, the computational cost of MPCR is negligible compared to the cost of particle evolution in PIC methods, and the tests demonstrate that periodic particle resampling yields a significant improvement in the accuracy and stability of the results.
Gradient coding is a method for mitigating straggling servers in a centralized computing network that uses erasure-coding techniques to distributively carry out first-order optimization methods. Randomized numerical linear algebra uses randomization to develop improved algorithms for large-scale linear algebra computations. In this study, we propose a method for distributed optimization that combines gradient coding and randomized numerical linear algebra. The proposed method uses a randomized ℓ 2 -subspace embedding and a gradient coding technique to distribute blocks of data to the computational nodes of a centralized network, and at each iteration the central server only requires a small number of computations to obtain the steepest descent update. The novelty of our approach is that the data is replicated according to importance scores, called block leverage scores, in contrast to most gradient coding approaches that uniformly replicate the data blocks. Furthermore, we do not require a decoding step at each iteration, avoiding a bottleneck in previous gradient coding schemes. We show that our approach results in a valid ℓ 2 -subspace embedding, and that our resulting approximation converges to the optimal solution.
A tokamak-independent analysis suite has been developed to process data from Motional Stark Effect (mse) diagnostics. The software supports multi-spectral line-polarization mse diagnostics which simultaneously measure emission at the mse σ and π lines as well as at two "background" wavelengths that are displaced from the mse spectrum by a few nanometers. This analysis accurately estimates the amplitude of partially polarized background light at the σ and π wavelengths even in situations where the background light changes rapidly in time and space, a distinct improvement over traditional "time-interpolation" background estimation. The signal amplitude at many frequencies is computed using a numerical-beat algorithm which allows the retardance of the mse photo-elastic modulators (pem's) to be monitored during routine operation. It also allows the use of summed intensities at multiple frequencies in the calculation of polarization direction, which increases the effective signal strength and reduces sensitivity to pem retardance drift. The software allows the polarization angles to be corrected for calibration drift using a system that illuminates the mse diagnostic with polarized light at four known polarization angles within ten seconds of a plasma discharge. As a result, the software suite is modular, parallelized, and portable to other facilities.
A numerical method for solving systems of nonlinear one-dimensional balance laws, based on multivariate sigmoidal functions approximation, is developed. Constructive approximation theorems are first established in both, uniform and L{sup p} norms. A priori as well as a posteriori error estimates are derived for the numerical solutions, when various kinds of sigmoidal functions, such as unit step, logistic and Gompertz functions, are chosen. The residual of the numerical method is also estimated. Numerical examples are given to test the performance of the algorithm, a comparison with the Godunov method is made concerning accuracy and computational cost. Finally, the numerical stability of the method is analyzed.
The physical designs of the new concept reactors which have complex structure, various materials and neutronic energy spectrum, have greatly improved the requirements to the calculation methods and the corresponding computing hardware. Along with the widely used parallel algorithm, heterogeneous platforms architecture has been introduced into numerical computations in reactor physics. Because of the natural parallel characteristics, the CPU-FPGA architecture is often used to accelerate numerical computation. This paper studies the application and features of this kind of heterogeneous platforms used in numerical calculation of reactor physics through practical examples. After the designed neutron diffusion module based on CPU-FPGA architecture achieves a 11.2 speed up factor, it is proved to be feasible to apply this kind of heterogeneous platform into reactor physics. (authors)
In this work, we propose a new scalable method to optimize the architecture of an artificial neural network. The proposed algorithm, called Greedy Search for Neural Network Architecture, aims to determine a neural network with minimal number of layers that is at least as performant as neural networks of the same structure identified by other hyperparameter search algorithms in terms of accuracy and computational cost. Numerical results performed on benchmark datasets show that, for these datasets, our method outperforms state-of-the-art hyperparameter optimization algorithms in terms of attainable predictive performance by the selected neural network architecture, and time-to-solution for the hyperparameter optimization to complete.
Abstract Variational quantum eigensolvers (VQEs) represent a powerful class of hybrid quantum-classical algorithms for computing molecular energies. Various numerical issues exist for these methods, however, including barren plateaus and large numbers of local minima. In this work, we consider the Adaptive, Problem-Tailored Variational Quantum Eiegensolver (ADAPT-VQE) ansätze, and examine how they are impacted by these local minima. We find that while ADAPT-VQE does not remove local minima, the gradient-informed, one-operator-at-a-time circuit construction accomplishes two things: First, it provides an initialization strategy that can yield solutions with over an order of magnitude smaller error compared to random initialization, and which is applicable in situations where chemical intuition cannot help with initialization, i.e., when Hartree-Fock is a poor approximation to the ground state. Second, even if an ADAPT-VQE iteration converges to a local trap at one step, it can still “burrow” toward the exact solution by adding more operators, which preferentially deepens the occupied trap. This same mechanism helps highlight a surprising feature of ADAPT-VQE: It should not suffer optimization problems due to barren plateaus and random initialization. Even if such barren plateaus appear in the parameter landscape, our analysis suggests that ADAPT-VQE avoids such regions by design.
This work extends our previous works [J. Liu and W. Z. Liang, J. Chem. Phys. 135, 014113 (2011); J. Liu and W. Z. Liang, J. Chem. Phys. 135, 184111 (2011)] on analytical excited-state energy Hessian within the framework of time-dependent density functional theory (TDDFT) to couple with molecular mechanics (MM). The formalism, implementation, and applications of analytical first and second energy derivatives of TDDFT/MM excited state with respect to the nuclear and electric perturbations are presented. Their performances are demonstrated by the calculations of adiabatic excitation energies, and excited-state geometries, harmonic vibrational frequencies, and infrared intensities for a number of benchmark systems. The consistent results with the full quantum mechanical method and other hybrid theoretical methods indicate the reliability of the current numerical implementation of developed algorithms. The computational accuracy and efficiency of the current analytical approach are also checked and the computational efficient strategies are suggested to speed up the calculations of complex systems with many MM degrees of freedom. Finally, we apply the current analytical approach in TDDFT/MM to a realistic system, a red fluorescent protein chromophore together with part of its nearby protein matrix. The calculated results indicate that the rearrangement of the hydrogen bond interactions between the chromophore and the protein matrix is responsible for the large Stokes shift.
Operation in a regime of zero off-normal events is an important goal for ITER, and after that DEMO. Edge Localized Modes (ELMs) are off-normal events of particular concern due to their ubiquity in the highest performing discharges and their localized energy deposition on vessel walls. To avoid damage from these events, ITER has invested in costly coils in addition to exploring other mitigation techniques. A more attractive approach is to find an operational regime that has the requisite high performance while avoiding ELMs entirely. DIII-D experimentalists have found such a regime, wherein the ELMs are prevented from occurring by the existence of a saturated instability. Although this regime is attractive, how this saturated instability occurs and how to extrapolate to ITER is not well understood. NIMROD is well suited for studying this mode, and significant progress has already been made in modeling the equilibria that give rise to this instability. Another off-normal event of concern to ITER is the sawtooth mode in the presence of energetic particles. These modes are ubiquitous in near-steady-state tokamaks, and their behavior in the presence of the energetic particles that will be found in burning plasma experiments is poorly understood. The internal kink is notorious for requiring all extended MHD effects (two-fluid, closures, vacuum regime, energetic particles) in a production level simulation. NIMROD has matured to the point where it can now address this problem, though significant expertise in the MHD physics, numerical algorithms, and large-scale computation will be required. The expertise of Tech-X in attacking the most difficult NIMROD problems has already resulted in significant improvements to the NIMROD code that benefit the larger fusion community, and we expect further improvements as we attack these challenging problems. To maximize the community benefits, Tech-X provided community outreach and code management efforts.
A tokamak-independent analysis suite has been developed to process data from Motional Stark Effect (MSE) diagnostics. The software supports multi-spectral line-polarization MSE diagnostics which simultaneously measure emission at the MSE sigma and pi lines as well as at two `background' wavelengths that are displaced from the MSE spectrum by a few nanometers. This analysis accurately estimates the amplitude of partially-polarized background light at the sigma and pi wavelengths even in situations where the background light changes rapidly in time and space. The signal amplitude at many frequencies is computed using a numerical-beat algorithm which allows the retardance of the MSE photo-elastic modulators (PEMs) to be monitored during routine operation. It also allows the use of summed intensities at multiple frequencies in the calculation of polarization direction, which increases the effective signal strength and reduces sensitivity to PEM retardance drift. The software allows the polarization angles to be corrected for calibration drift using a system that illuminates the MSE diagnostic with polarized light at four known polarization angles within ten seconds of a plasma discharge The software suite is modular, parallelized, and portable to other facilities. (2016-06-06)
A tokamak-independent analysis suite has been developed to process data from Motional Stark Effect (MSE) diagnostics. The software supports multi-spectral line-polarization MSE diagnostics which simultaneously measure emission at the MSE sigma and pi lines as well as at two `background' wavelengths that are displaced from the MSE spectrum by a few nanometers. This analysis accurately estimates the amplitude of partially-polarized background light at the sigma and pi wavelengths even in situations where the background light changes rapidly in time and space. The signal amplitude at many frequencies is computed using a numerical-beat algorithm which allows the retardance of the MSE photo-elastic modulators (PEMs) to be monitored during routine operation. It also allows the use of summed intensities at multiple frequencies in the calculation of polarization direction, which increases the effective signal strength and reduces sensitivity to PEM retardance drift. The software allows the polarization angles to be corrected for calibration drift using a system that illuminates the MSE diagnostic with polarized light at four known polarization angles within ten seconds of a plasma discharge The software suite is modular, parallelized, and portable to other facilities.
Highlights: • The PAAVLS algorithm can be employed for alignment of very long sequences. • The PAAVLS algorithm largely reduces memory demand as the Smith–Waterman algorithm. • The running times of the PAAVLS and Smith–Waterman algorithms are comparable. Sequence alignment is one of the fundamental problems in computational biology and has numerous applications. The Smith–Waterman algorithm generates optimal local alignment for pairwise alignment task and has become a standard algorithm in its field. However, the current version of the Smith–Waterman algorithm demands a significant amount of memory and is not suitable for alignment of very long sequences. On the hand, the recent DNA sequencing technologies have produced vast amounts of biological sequences. Some nucleic acid sequences are very long and cannot employ the Smith–Waterman algorithm. To this end, this study proposes the PAAVLS algorithm that follows the dynamic programming technique employed by the Smith–Waterman algorithm and largely reduces the demand of memory. The proposed PAAVLS algorithm can be employed for alignment of very long sequences, i.e., sequences contain more than 100,000,000 nucleotides, on a personal computer. Additionally, the running time of the proposed PAAVLS algorithm is comparable with the running time of the standard Smith–Waterman algorithm.
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We present an efficient numerical algorithm to approximate the statistical moments of stochastic problems, in the presence of models with different fidelities. The method extends the multi-fidelity approximation method developed in . By combining the efficiency of low-fidelity models and the accuracy of high-fidelity models, our method exhibits fast convergence with a limited number of high-fidelity simulations. We establish an error bound of the method and present several numerical examples to demonstrate the efficiency and applicability of the multi-fidelity algorithm.