Theoretical and numerical studies of some ill-posed problems in partial differential equations

Three nonlinear initial-boundary value problems are considered. A potential energy well theory applies for solutions of the hyperbolic problem u(,tt) = (DELTA)(,n) u in D x (0,T), u = 0 on (sigma) x (0,T), (PAR-DIFF)u/(PAR-DIFF)n = f(u) on (SIGMA) x (0,T), u(x,0) = U(x) and u(,t)(x,0) = V(x) in D, when the nonlinearity f is suitably restricted. Here D is a bounded, open, connected subset of R(\u27n); the boundary of D, (PAR-DIFF)D, consists of the disjoint (n-1)-dimen- sional submanifolds (sigma), (SIGMA), and their confluence; (DELTA)(,n) denotes the n-dimensional Laplacian; and (PAR-DIFF)/(PAR-DIFF)n denotes the outward normal derivative. The problem has a global weak solution in each dimen- sion n (GREATERTHEQ) 1 provided U lies in the potential well and the total initial energy is small. The global solution is obtained by expanding in normal modes in terms of the Helmholtz eigenfunctions and the eigenfunctions for a modified Steklov problem. Solutions of the hyperbolic problem which start in a region exterior to the potential well with sufficiently small total initial energy can only exist for a finite time;An analogous existence-nonexistence criterion obtains for glo- bal solutions of the parabolic problem u(,t) = (DELTA)(,n) u in D x (0,T), u = 0 on (sigma) x (0,T), (PAR-DIFF)u/(PAR-DIFF)n = f(u) on (SIGMA) x (0,T), and u(x,0) = U(x) in D;Let (phi) (ELEM) C(\u271)(- (INFIN),M) be nonnegative, increasing and satisfy;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI);The problem u(,tt) = (DELTA)(,n) u + (epsilon)(phi)(u) in D x (0,T), u = 0 on (PAR-DIFF)D x (0,T), u(x,0) = u(,0)(x) and u(,t)(x,0) = v(,0)(x) in D, has a unique local continuous solution for (epsilon) \u3e 0 sufficiently small in dimensions n = 1,2,3 under appropriate assumptions on (phi), u(,0), v(,0), and (PAR-DIFF)D. The solution u can be continued as long as u \u3c M. A potential well theory is unobtainable for this problem in the Sobolev space H(,0)(\u271)(D) for n (GREATERTHEQ) 2; however, an a priori inequality for solutions guarantees global existence via energy considerations. Numerical evidence indicates that such an a priori inequality is sometimes satisfied by solutions when n (GREATERTHEQ) 2

A semi-analytical approach utilising limit analysis for slope stability assessment and optimal design

An analytical upper bound method of limit analysis is adopted to derive generalized formulations for assessing the stability of slopes made of geomaterials obeying both the linear Mohr-Coulomb failure criterion and the non-linear Hoek-Brown failure criterion. The thesis is aimed at seeking slope profiles of optimal stability and facilitating the optimal design of pile reinforcement. The ef-fect of the presence of cracks, water pressure, seismic actions, non-homogeneous anisotropic ground and blast-induced damage is investigated. An extensive parametrical study was carried out. A large number of stability and design charts were provided for the benefit of practitioners. A software package to evaluate the safety of slope was created to overcome the limitations of chart-based design using analytical methods. The main findings of this study can be summarized as fol-lows. Firstly, to avoid potential local failure (sliding of the soil/rock mass behind cracks), the most critical failure mechanism should be determined under the constraint of maximum stable crack depth. Secondly, the application of the tangential technique to tackle the non linear Hoek-Brown failure criterion is an acceptable and convenient tool compared with the equivalent C – Ø method and the variational approach. The minimization of the least upper bound solution corresponding to the Hoek-Brown failure criterion has to be implemented under certain stress constraints to avoid any unrealistic selection of tangent lines. Thirdly, contrary to the previous literature assuming en-tirely concave shapes, the optimal profiles exhibit both a concave and a convex part. In comparison with the traditional planar profiles, the percentage of increase in the stability factor can reach (up to) 49%. In addition, for engineered slope excavation, given the same stability factor, the average slope inclination of an optimal slope is always higher than that of a planar slope. The amount of ground excavated for the optimal profile can be as little as 50% of that for a planar profile. Lastly, above-pile failure mechanisms must be taken into account when determining the optimal pile posi-tion otherwise the installation of piles may be completely ineffective

On the generalized langevin equation for simulated annealing

In this paper, we consider the generalized (higher order) Langevin equation for the purpose of simulated annealing and optimization of nonconvex functions. Our approach modifies the underdamped Langevin equation by replacing the Brownian noise with an appropriate Ornstein–Uhlenbeck process to account for memory in the system. Under reasonable conditions on the loss function and the annealing schedule, we establish convergence of the continuous time dynamics to a global minimum. In addition, we investigate the performance numerically and show better performance and higher exploration of the state space compared to the underdamped Langevin dynamics with the same annealing schedule

Design and performance analysis of energy harvesting communications systems

The continuous growth of high data rates with huge increase in the number of mobile devices and communication infrastructure have led to greenhouse gas emission, higher pollution and higher energy costs. After the deployment of 4G and immense data rate and QoS requirements for 5G, there is an urgent need to design future wireless systems that aim to improve energy efficiency (EE) and spectral efficiency (SE). One of the possible solutions is to use energy harvesting (EH), which promises to reduce energy consumption issues in information and communication technology sector. In order to tackle these challenges, this thesis is focused on the design and performance analysis of EH systems. EH has emerged as a potential candidate for green wireless communication which not only provides solution to the energy limitation problem but also prolongs the lifetime of batteries. First, the performance evaluation of an EH-equipped dual-hop relaying system is proposed to improve the system throughput and the end-to-end signal-to-noise ratio (SNR). We derive novel closed-form expressions for cumulative distribution function of individual link's SNR and of the end-to-end SNR. In addition, the proposed model analyses the ergodic capacity which is an important performance metric for delay-sensitive services. Further, these closed-form expressions reduce the computational complexity of the receiver architecture for practical systems. An insight through system parameters provide significant improvement in end-to-end SNR especially when both transmitter and relay nodes are equipped with harvesting sources. Second, performance analysis and optimal transmission power allocation techniques for EH-equipped system are studied. Our proposed model investigates and provides the conditions under which the harvesting can improve the system performance. In this work, novel closed-form expressions are calculated for the maximum achievable EE, SE and EH beneficialness condition. We studied two cases such as power is adapted to variations in the channel and when transmit power is fixed. We proved that EE-optimum input power decreases with EH power level. Also, system parameters demonstrate the conditions under which EH improves overall system performance. Finally, a multi-objective optimization problem is formulated that jointly maximizes EE and SE for point-to-point EH-equipped system. We introduce new importance weight which set the priority levels of EE versus SE of the system. The formulated problem is solved by using convex optimization method to achieve optimal solution. The proposed system model provides freedom to choose any value for importance weight to satisfy quality of service (QoS) requirements and the flexibility of balancing between EE and SE performance metrics

Firing Rate Analysis for an Integrate-and-Fire Neuronal Model

We investigate a stochastic linear integrate-and-fire (IF) neuronal model and use the corresponding Fokker-Planck equation (FPE) to study the mean firing rate of a population of IF neurons. The firing rate(or emission rate) function, is given in terms of an eigenfunction expansion solution of the FPE. We consider two parameter regimes of current input and prove the existence of infinitely many branches of eigenvalues and derive their asymptotic properties. We use the eigenfunction expansion solution to prove asymptotic properties of the firing rate function. We also perform a numerical experiment of 10,000 IF neurons and show that our simulation is in agreement with our theoretical results. Finally, we state several open problems for future research

Tools to analyze cell signaling models

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2004.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Includes bibliographical references (v. 2, leaves 345-369).Diseases such as diabetes, some forms of cancer, hyper-tension, auto-immune diseases, and some viral diseases are characterized by complex interactions within the human body. Efforts to understand and treat these diseases have only been partially successful. There is currently a huge commercial and academic effort devoted to computational biology to address the shortfalls of qualitative biology. This research has become relevant due to the vast amounts of data now available from high-throughput techniques such as gene-chips, combinatorial chemistry, and fast gene sequencing. The goal of computational biology is to use quantitative models to test complex scientific hypotheses or predict desirable interventions. Consequently, it is important that the model is built to the minimum fidelity required to meet a specific goal, otherwise valuable effort is wasted. Unlike traditional chemical engineering, computational biology does not solely depend on deterministic models of chemical behavior. There is also widespread use of many types of statistical models, stochastic models, electro-static models, and mechanical models. All of these models are inferred from noisy data. It is therefore important to develop techniques to aide the model builder in their task of verifying and using these models to make quantitative predictions. The goal of this thesis is to develop tools for analysing the qualitative and quantitative characteristics of cell-signaling models. The qualitative behavior of deterministic models is studied in the first part of this thesis and the quantitative behavior of stochastic models is studied in the second part. A kinetic model of cell signaling is a common example of a deterministic model used in computational biology.(cont.) Usually such a model is derived from first-principles. The differential equations represent species conservation and the algebraic equations represent rate equations and equations to estimate rate constants. The researcher faces two key challenges once the model has been formulated: it is desirable to summarize a complex model by the phenomena it exhibits, and it is necessary to check whether the qualitative behavior of the model is verified by experimental observation. The key result of this research is a method to rearrange an implicit index one DAE into state-space form efficiently, amenable to standard control engineering analysis. Control engineering techniques can then be used to determine the time constants, poles, and zeros of the system, thus summarizing all the qualitative behavior of the system. The second part of the thesis focuses on the quantitative analysis of cell migration. It is hypothesized that mammalian cell migration is driven by responses to external chemical, electrical and mechanical stimulus. It is desirable to be able to quantify cell migration (speed, frequency of turning) to correlate output to experimental conditions (ligand concentration, cell type, cell medium, etc). However, the local concentration of signaling molecules and receptors is sufficiently low that a continuum model of cell migration is inadequate, i.e., it is only possible to describe cell motion in a probabilistic fashion ...by David Michael Collins.Ph.D

Iterative algorithms for optimal signal reconstruction and parameter identification given noisy and incomplete data

Originally published as thesis (Dept. of Electrical Engineering and Computer Science, Ph.D., 1982).Includes bibliographies.Supported in part by the Advanced Research Projects Agency monitored by ONR under Contract N00014-81-K-0742 NR-049-506 Supported in part by the National Science Foundation under Grant ECS80-07102Bruce Ronald Musicus

병렬화 용이한 통계계산 방법론과 현대 고성능 컴퓨팅 환경에의 적용

학위논문 (박사) -- 서울대학교 대학원 : 자연과학대학 통계학과, 2020. 8. 원중호.Technological advances in the past decade, hardware and software alike, have made access to high-performance computing (HPC) easier than ever. In this dissertation, easily-parallelizable, inversion-free, and variable-separated algorithms and their implementation in statistical computing are discussed. The first part considers statistical estimation problems under structured sparsity posed as minimization of a sum of two or three convex functions, one of which is a composition of non-smooth and linear functions. Examples include graph-guided sparse fused lasso and overlapping group lasso. Two classes of inversion-free primal-dual algorithms are considered and unified from a perspective of monotone operator theory. From this unification, a continuum of preconditioned forward-backward operator splitting algorithms amenable to parallel and distributed computing is proposed. The unification is further exploited to introduce a continuum of accelerated algorithms on which the theoretically optimal asymptotic rate of convergence is obtained. For the second part, easy-to-use distributed matrix data structures in PyTorch and Julia are presented. They enable users to write code once and run it anywhere from a laptop to a workstation with multiple graphics processing units (GPUs) or a supercomputer in a cloud. With these data structures, various parallelizable statistical applications, including nonnegative matrix factorization, positron emission tomography, multidimensional scaling, and ℓ1-regularized Cox regression, are demonstrated. The examples scale up to an 8-GPU workstation and a 720-CPU-core cluster in a cloud. As a case in point, the onset of type-2 diabetes from the UK Biobank with 400,000 subjects and about 500,000 single nucleotide polymorphisms is analyzed using the HPC ℓ1-regularized Cox regression. Fitting a half-million variate model took about 50 minutes, reconfirming known associations. To my knowledge, the feasibility of a joint genome-wide association analysis of survival outcomes at this scale is first demonstrated.지난 10년간의 하드웨어와 소프트웨어의 기술적인 발전은 고성능 컴퓨팅의 접근장벽을 그 어느 때보다 낮추었다. 이 학위논문에서는 병렬화 용이하고 역행렬 연산이 없는 변수 분리 알고리즘과 그 통계계산에서의 구현을 논의한다. 첫 부분은 볼록 함수 두 개 또는 세 개의 합으로 나타나는 구조화된 희소 통계 추정 문제에 대해 다룬다. 이 때 함수들 중 하나는 비평활 함수와 선형 함수의 합성으로 나타난다. 그 예시로는 그래프 구조를 통해 유도되는 희소 융합 Lasso 문제와 한 변수가 여러 그룹에 속할 수 있는 그룹 Lasso 문제가 있다. 이를 풀기 위해 역행렬 연산이 없는 두 종류의 원시-쌍대 (primal-dual) 알고리즘을 단조 연산자 이론 관점에서 통합하며 이를 통해 병렬화 용이한 precondition된 전방-후방 연산자 분할 알고리즘의 집합을 제안한다. 이 통합은 점근적으로 최적 수렴률을 갖는 가속 알고리즘의 집합을 구성하는 데 활용된다. 두 번째 부분에서는 PyTorch와 Julia를 통해 사용하기 쉬운 분산 행렬 자료 구조를 제시한다. 이 구조는 사용자들이 코드를 한 번 작성하면 이것을 노트북 한 대에서부터 여러 대의 그래픽 처리 장치 (GPU)를 가진 워크스테이션, 또는 클라우드 상에 있는 슈퍼컴퓨터까지 다양한 스케일에서 실행할 수 있게 해 준다. 아울러, 이 자료 구조를 비음 행렬 분해, 양전자 단층 촬영, 다차원 척 도법, ℓ1-벌점화 Cox 회귀 분석 등 다양한 병렬화 가능한 통계적 문제에 적용한다. 이 예시들은 8대의 GPU가 있는 워크스테이션과 720개의 코어가 있는 클라우드 상의 가상 클러스터에서 확장 가능했다. 한 사례로 400,000명의 대상과 500,000개의 단일 염기 다형성 정보가 있는 UK Biobank 자료에서의 제2형 당뇨병 (T2D) 발병 나이를 ℓ1-벌점화 Cox 회귀 모형을 통해 분석했다. 500,000개의 변수가 있는 모형을 적합시키는 데 50분 가량의 시간이 걸렸으며 알려진 T2D 관련 다형성들을 재확인할 수 있었다. 이러한 규모의 전유전체 결합 생존 분석은 최초로 시도된 것이다.Chapter1Prologue 1 1.1 Introduction 1 1.2 Accessible High-Performance Computing Systems 4 1.2.1 Preliminaries 4 1.2.2 Multiple CPU nodes: clusters, supercomputers, and clouds 7 1.2.3 Multi-GPU node 9 1.3 Highly Parallelizable Algorithms 12 1.3.1 MM algorithms 12 1.3.2 Proximal gradient descent 14 1.3.3 Proximal distance algorithm 16 1.3.4 Primal-dual methods 17 Chapter 2 Easily Parallelizable and Distributable Class of Algorithms for Structured Sparsity, with Optimal Acceleration 20 2.1 Introduction 20 2.2 Unification of Algorithms LV and CV (g ≡ 0) 30 2.2.1 Relation between Algorithms LV and CV 30 2.2.2 Unified algorithm class 34 2.2.3 Convergence analysis 35 2.3 Optimal acceleration 39 2.3.1 Algorithms 40 2.3.2 Convergence analysis 41 2.4 Stochastic optimal acceleration 45 2.4.1 Algorithm 45 2.4.2 Convergence analysis 47 2.5 Numerical experiments 50 2.5.1 Model problems 50 2.5.2 Convergence behavior 52 2.5.3 Scalability 62 2.6 Discussion 63 Chapter 3 Towards Unified Programming for High-Performance Statistical Computing Environments 66 3.1 Introduction 66 3.2 Related Software 69 3.2.1 Message-passing interface and distributed array interfaces 69 3.2.2 Unified array interfaces for CPU and GPU 69 3.3 Easy-to-use Software Libraries for HPC 70 3.3.1 Deep learning libraries and HPC 70 3.3.2 Case study: PyTorch versus TensorFlow 73 3.3.3 A brief introduction to PyTorch 76 3.3.4 A brief introduction to Julia 80 3.3.5 Methods and multiple dispatch 80 3.3.6 Multidimensional arrays 82 3.3.7 Matrix multiplication 83 3.3.8 Dot syntax for vectorization 86 3.4 Distributed matrix data structure 87 3.4.1 Distributed matrices in PyTorch: distmat 87 3.4.2 Distributed arrays in Julia: MPIArray 90 3.5 Examples 98 3.5.1 Nonnegative matrix factorization 100 3.5.2 Positron emission tomography 109 3.5.3 Multidimensional scaling 113 3.5.4 L1-regularized Cox regression 117 3.5.5 Genome-wide survival analysis of the UK Biobank dataset 121 3.6 Discussion 126 Chapter 4 Conclusion 131 Appendix A Monotone Operator Theory 134 Appendix B Proofs for Chapter II 139 B.1 Preconditioned forward-backward splitting 139 B.2 Optimal acceleration 147 B.3 Optimal stochastic acceleration 158 Appendix C AWS EC2 and ParallelCluster 168 C.1 Overview 168 C.2 Glossary 169 C.3 Prerequisites 172 C.4 Installation 173 C.5 Configuration 173 C.6 Creating, accessing, and destroying the cluster 178 C.7 Installation of libraries 178 C.8 Running a job 179 C.9 Miscellaneous 180 Appendix D Code for memory-efficient L1-regularized Cox proportional hazards model 182 Appendix E Details of SNPs selected in L1-regularized Cox regression 184 Bibliography 188 국문초록 212Docto