エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学

ADAPTIVE GRID BASED FINITE DIFFERENCE METHODS FOR SOLUTION OF HYPERBOLIC PDES: APPLICATION TO COMPUTATIONAL MECHANICS AND UNCERTAINTY QUANTIFICATION

Novel finite-difference based numerical methods for solution of linear and nonlinear hyperbolic partial differential equations (PDEs) using adaptive grids are proposed in this dissertation. The overall goal of this research is to improve the accuracy and/or computational efficiency of numerical solutions via the use of adaptive grids and suitable modifications of a given low-order order finite-difference scheme. These methods can be grouped in two broad categories. The first category of adaptive FD methods proposed in the dissertation attempt to reduce the truncation error and/or enhance the accuracy of the underlying numerical schemes via grid distribution alone. Some approaches for grid distribution considered include those based on (i) a moving uniform mesh/domain, (ii) adaptive gradient based refinement (AGBR) and (iii) unit local Courant-Freidrich-Lewy (CFL) number. The improvement in the accuracy which is obtained using these adaptive methods is limited by the underlying scheme formal order of accuracy. In the second category, the CFL based approach proposed in the first category was extended further using defect correction in order to improve the formal order of accuracy and computational efficiency significantly (i.e. by at least one order or higher). The proposed methods in this category are constructed based upon the analysis of the leading order error terms in the modified differential equation associated with the underlying partial differential equation and finite difference scheme. The error terms corresponding to regular and irregular perturbations are identified and the leading order error terms associated with regular perturbations are eliminated using a non-iterative defect correction approach while the error terms associated with irregular perturbations are eliminated using grid adaptation. In the second category of methods involving defect correction (or reduction of leading order terms of truncation error), we explored two different approaches for selection of adaptive grids. These are based on (i) optimal grid dis- tribution and (ii) remapping with monotonicity preserving interpolation. While the first category of methods may be preferred in view of ease of implementation and lower computational complexity, the second category of methods may be preferred in view of greater accuracy and computational efficiency. The two broad categories of methods, which have been applied to problems involving both bounded and unbounded domains, were also extended to multidimensional cases using a dimensional splitting approaches. The performance of these methods was demonstrated using several example problems in computational uncertainty quantification (CUQ) and computational mechanics. The results of the application of the proposed approaches all indicate improvement in both the accuracy and computational efficiency (by about three orders of magnitude in some selected cases) of underlying schemes. In the context of CUQ, all three proposed adaptive finite different solvers are combined with the Gauss-quadrature sampling technique in excitation space to obtain statistical quantities of interest for dynamical systems with parametric uncertainties from the solution of Liouville equation, which is a linear hyperbolic PDE. The numerical results for four canonical UQ problems show both enhanced computational efficiency and improved accuracy of the proposed adaptive FD solution of the Liouville equation compared to its standard/fixed domain FD solutions. Moreover, the results for canonical test problems in computational mechanics indicate that the proposed approach for increasing the formal order of the underlying FD scheme can be easily implemented in multidimensional spaces and gives an oscillation-free numerical solution with a desired order of accuracy in a reasonable computational time. This approach is shown to provide a better computational time compared to both the underlying scheme (by about three orders of magnitude) and standard FD methods of the same order of accuracy

Geometric Integrators for Schrödinger Equations

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control.Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716TESISPremios Extraordinarios de tesis doctorale

Gratings: Theory and Numeric Applications, Second Revisited Edition

International audienceThe second Edition of the Book contains 13 chapters, written by an international team of specialist in electromagnetic theory, numerical methods for modelling of light diffraction by periodic structures having one-, two-, or three-dimensional periodicity, and aiming numerous applications in many classical domains like optical engineering, spectroscopy, and optical telecommunications, together with newly born fields such as photonics, plasmonics, photovoltaics, metamaterials studies, cloaking, negative refraction, and super-lensing. Each chapter presents in detail a specific theoretical method aiming to a direct numerical application by university and industrial researchers and engineers.In comparison with the First Edition, we have added two more chapters (ch.12 and ch.13), and revised four other chapters (ch.6, ch.7, ch.10, and ch.11

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Synergies between Numerical Methods for Kinetic Equations and Neural Networks

The overarching theme of this work is the efficient computation of large-scale systems. Here we deal with two types of mathematical challenges, which are quite different at first glance but offer similar opportunities and challenges upon closer examination. Physical descriptions of phenomena and their mathematical modeling are performed on diverse scales, ranging from nano-scale interactions of single atoms to the macroscopic dynamics of the earth\u27s atmosphere. We consider such systems of interacting particles and explore methods to simulate them efficiently and accurately, with a focus on the kinetic and macroscopic description of interacting particle systems. Macroscopic governing equations describe the time evolution of a system in time and space, whereas the more fine-grained kinetic description additionally takes the particle velocity into account. The study of discretizing kinetic equations that depend on space, time, and velocity variables is a challenge due to the need to preserve physical solution bounds, e.g. positivity, avoiding spurious artifacts and computational efficiency. In the pursuit of overcoming the challenge of computability in both kinetic and multi-scale modeling, a wide variety of approximative methods have been established in the realm of reduced order and surrogate modeling, and model compression. For kinetic models, this may manifest in hybrid numerical solvers, that switch between macroscopic and mesoscopic simulation, asymptotic preserving schemes, that bridge the gap between both physical resolution levels, or surrogate models that operate on a kinetic level but replace computationally heavy operations of the simulation by fast approximations. Thus, for the simulation of kinetic and multi-scale systems with a high spatial resolution and long temporal horizon, the quote by Paul Dirac is as relevant as it was almost a century ago. The first goal of the dissertation is therefore the development of acceleration strategies for kinetic discretization methods, that preserve the structure of their governing equations. Particularly, we investigate the use of convex neural networks, to accelerate the minimal entropy closure method. Further, we develop a neural network-based hybrid solver for multi-scale systems, where kinetic and macroscopic methods are chosen based on local flow conditions. Furthermore, we deal with the compression and efficient computation of neural networks. In the meantime, neural networks are successfully used in different forms in countless scientific works and technical systems, with well-known applications in image recognition, and computer-aided language translation, but also as surrogate models for numerical mathematics. Although the first neural networks were already presented in the 1950s, the scientific discipline has enjoyed increasing popularity mainly during the last 15 years, since only now sufficient computing capacity is available. Remarkably, the increasing availability of computing resources is accompanied by a hunger for larger models, fueled by the common conception of machine learning practitioners and researchers that more trainable parameters equal higher performance and better generalization capabilities. The increase in model size exceeds the growth of available computing resources by orders of magnitude. Since $2012$, the computational resources used in the largest neural network models doubled every $3.4$ months\footnote{\url{https://openai.com/blog/ai-and-compute/}}, opposed to Moore\u27s Law that proposes a $2$-year doubling period in available computing power. To some extent, Dirac\u27s statement also applies to the recent computational challenges in the machine-learning community. The desire to evaluate and train on resource-limited devices sparked interest in model compression, where neural networks are sparsified or factorized, typically after training. The second goal of this dissertation is thus a low-rank method, originating from numerical methods for kinetic equations, to compress neural networks already during training by low-rank factorization. This dissertation thus considers synergies between kinetic models, neural networks, and numerical methods in both disciplines to develop time-, memory- and energy-efficient computational methods for both research areas