Numerical solution and bifurcation analysis of nonlinear partial differential equations with extreme learning machines

We address a new numerical method based on a class of machine learning methods, the so-called Extreme Learning Machines (ELM) with both sigmoidal and radial-basis functions, for the computation of steady-state solutions and the construction of (one-dimensional) bifurcation diagrams of nonlinear partial differential equations (PDEs). For our illustrations, we considered two benchmark problems, namely (a) the one-dimensional viscous Burgers with both homogeneous (Dirichlet) and non-homogeneous boundary conditions, and, (b) the one- and two-dimensional Liouville–Bratu–Gelfand PDEs with homogeneous Dirichlet boundary conditions. For the one-dimensional Burgers and Bratu PDEs, exact analytical solutions are available and used for comparison purposes against the numerical derived solutions. Furthermore, the numerical efficiency (in terms of numerical accuracy, size of the grid and execution times) of the proposed numerical machine-learning method is compared against central finite differences (FD) and Galerkin weighted-residuals finite-element (FEM) methods. We show that the proposed numerical machine learning method outperforms in terms of numerical accuracy both FD and FEM methods for medium to large sized grids, while provides equivalent results with the FEM for low to medium sized grids; both methods (ELM and FEM) outperform the FD scheme. Furthermore, the computational times required with the proposed machine learning scheme were comparable and in particular slightly smaller than the ones required with FE

Mini-Workshop: Efficient and Robust Approximation of the Helmholtz Equation

The accurate and efficient treatment of wave propogation phenomena is still a challenging problem. A prototypical equation is the Helmholtz equation at high wavenumbers. For this equation, Babuška & Sauter showed in 2000 in their seminal SIAM Review paper that standard discretizations must fail in the sense that the ratio of true error and best approximation error has to grow with the frequency. This has spurred the development of alternative, non-standard discretization techniques. This workshop focused on evaluating and comparing these different approaches also with a view to their applicability to more general wave propagation problems

Multilevel Numerical Algorithms for Systems of Nonlinear Parabolic Partial Differential Equations

This thesis is concerned with the development of efficient and reliable numerical algorithms for the solution of nonlinear systems of partial differential equations (PDEs) of elliptic and parabolic type. The main focus is on the implementation and performance of three different nonlinear multilevel algorithms, following discretisation of the PDEs: the Full Approximation Scheme (FAS), Newton-Multigrid (Newton-MG) and a Newton-Krylov solver with a novel pre- conditioner that we have developed based on the use of Algebraic Multigrid (AMG). In recent years these algorithms have been commonly used to solve nonlinear systems that arise from the discretisation of PDEs due to the fact that their execution time can scale linearly (or close to linearly) with the number of degrees of freedom used in the discretisation. We consider two mathematical models: a thin film flow and the Cahn-Hilliard-Hele-Shaw model. These mathematical models consist of nonlinear, time-dependent and coupled PDEs systems. Using a Finite Difference Method (FDM) in space and Backward Differentiation For- mulae (BDF) in time, we discrete the two models, to produce nonlinear algebraic systems. We are able to solve these nonlinear systems implicitly in computationally demanding 2D situa- tions. We present numerical results, for both steady-state and time-dependent problems, that demonstrate the optimality of the three numerical algorithms for the thin film flow model. We show optimality of the FAS and Newton-Krylov approaches for the time-dependent Cahn- Hilliard-Hele-Shaw (CHHS) problem. The main contribution is to address the question of which of these three nonlinear solvers is likely to be the best (i.e. computationally most effective) in practice. In order to asses this, we discuss the careful implementation and timing of these algorithms in order to permit a fair direct comparison of their computational cost. We then present extensive numerical results in order to make this comparison between these nonlinear multilevel methods. The conclusion emerging from this investigation is that it does not appear that there is a single superior approach, but rather that the best approach is problem dependent. Specifically, we find that our optimally preconditioned Newton-Krylov approach is best for the thin film flow model in the steady-state and time-dependent form, whilst the FAS solver appears best for the time-dependent CHHS model

Numerical Relativity As A Tool For Computational Astrophysics

The astrophysics of compact objects, which requires Einstein's theory of general relativity for understanding phenomena such as black holes and neutron stars, is attracting increasing attention. In general relativity, gravity is governed by an extremely complex set of coupled, nonlinear, hyperbolic-elliptic partial differential equations. The largest parallel supercomputers are finally approaching the speed and memory required to solve the complete set of Einstein's equations for the first time since they were written over 80 years ago, allowing one to attempt full 3D simulations of such exciting events as colliding black holes and neutron stars. In this paper we review the computational effort in this direction, and discuss a new 3D multi-purpose parallel code called ``Cactus'' for general relativistic astrophysics. Directions for further work are indicated where appropriate.Comment: Review for JCA

Optimal fishery with coastal catch

In many spatial resource models, it is assumed that an agent is able to harvest the resource over the complete spatial domain. However, agents frequently only have access to a resource at particular locations at which a moving biomass, such as fish or game, may be caught or hunted. Here, we analyze an infinite time‐horizon optimal control problem with boundary harvesting and (systems of) parabolic partial differential equations as state dynamics. We formally derive the associated canonical system, consisting of a forward–backward diffusion system with boundary controls, and numerically compute the canonical steady states and the optimal time‐dependent paths, and their dependence on parameters. We start with some one‐species fishing models, and then extend the analysis to a predator–prey model of the Lotka–Volterra type. The models are rather generic, and our methods are quite general, and thus should be applicable to large classes of structurally similar bioeconomic problems with boundary controls. Recommedations for Resource Managers Just like ordinary differential equation‐constrained (optimal) control problems and distributed partial differential equation (PDE) constrained control problems, boundary control problems with PDE state dynamics may be formally treated by the Pontryagin's maximum principle or canonical system formalism (state and adjoint PDEs). These problems may have multiple (locally) optimal solutions; a first overview of suitable choices can be obtained by identifying canonical steady states. The computation of canonical paths toward some optimal steady state yields temporal information about the optimal harvesting, possibly including waiting time behavior for the stock to recover from a low‐stock initial state, and nonmonotonic (in time) harvesting efforts. Multispecies fishery models may lead to asymmetric effects; for instance, it may be optimal to capture a predator species to protect the prey, even for high costs and low market values of the predators

Three-dimensional inversion of transient-electromagnetic data: A comparative study

Inversion of transient-electromagnetic (TEM) data arising from galvanic types of sources is approached by two different methods. Both methods reconstruct the subsurface three-dimensional (3D) electrical conductivity properties directly in the time-domain. A principal difference is given by the scale of the inversion problems to be solved. The first approach represents a small-scale 3D inversion and is based upon well-known tools. It uses a stabilized unconstrained least-squares inversion algorithm in combination with an existing 3D forward modeling solver and is customized to invert for 3D earth models with a limited model complexity. The limitation to only as many model unknowns as typical for classical least-squares problems involves arbitrary and rather unconventional types of model parameters. The inversion scheme has mainly been developed for the purpose of refining a priori known 3D underground structures by means of an inversion. Therefore, a priori information is an important requirement to design a model such that its limited degrees of freedom describe the structures of interest. The inversion is successfully applied to data from a long-offset TEM survey at the active volcano Merapi in Central Java (Indonesia). Despite the restriction of a low model complexity, the scheme offers some versatility as it can be adapted easily to various kinds of model structures. The interpretation of the resistivity images obtained by the inversion have substantially advanced the structural knowledge about the volcano. The second part of this work presents a theoretically more elaborate scheme. It employs imaging techniques originally developed for seismic wavefields. Large-scale 3D problems arising from the inversion for finely parameterized and arbitrarily complicated earth models are addressed by the method. The algorithm uses a conjugate-gradient search for the minimum of an error functional, where the gradient information is obtained via migration or backpropagation of the differences between the data observations and predictions back into the model in reverse time. Treatment for electric field and time derivative of the magnetic field data is given for the specification of the cost functional gradients. The inversion algorithm is successfully applied to a synthetic TEM data set over a conductive anomaly embedded in a half-space. The example involves a total number of more than 376000 model unknowns. The realization of migration techniques for diffusive EM fields involves the backpropagation of a residual field. The residual field excitation originates from the actual receiver positions and is continued during the simulated time range of the measurements. An explicit finite-difference time-stepping scheme is developed in advance of the imaging scheme in order to accomplish both the forward simulation and backpropagation of 3D EM fields. The solution uses a staggered grid and a modified version of the DuFort-Frankel stabilization method and is capable of simulating non-causal fields due to galvanic types of sources. Its parallel implementation allows for reasonable computation times, which are inherently high for explicit time-stepping schemes

Collocation Methods for Nonlinear Parabolic Partial Differential Equations

In this thesis, we present an implementation of a novel collocation method for solving nonlinear parabolic partial differential equations (PDEs) based on triangle meshes. The temporal partial derivative is discretized using the implicit Euler-backward finite difference scheme. The spatial domain of the PDEs discussed in this thesis is two-dimensional. The domain is first triangulated and then refined into appropriately sized triangular elements by the Rivara algorithm. The solution is approximated by piecewise polynomials in the elements. The polynomial in each element is requiredtosatisfythePDEatcollocationpointsoftheelementandkeepacertaindegreeofcontinuity with the polynomials in the neighboring elements via matching points. Nested dissection is used recursively, from the elements up to the entire domain, to merge all pairs of sibling sub-regions for eliminating the variables at the matching points on the common sides shared by the merged sub-regions. Then by applying global boundary conditions, we solve for the solution values at the boundary points of the entire domain. The solutions at the boundary points of the domain are backsubstituted to solve the variables at the matching points of the sub-regions. This back-substitution is repeated until every element is reached. The accuracy of the solution is affected by the time step, granularity of the subdivision, the number and location of matching points, and the number and location of collocation points. Increasing the number of matching points or collocation points does not always improve the accuracy. Instead, it may cause singularity. We have given several layouts of specific numbers of collocation and matching points which bring high accuracy. Our solution visualization algorithm directly renders mathematical surfaces instead of any approximation of them. Thus each pixel of the rendered surfaces exactly reflects the corresponding fragment on the mathematical surfaces

Microscopic Origin of the 0.7-Anomaly in Quantum Point Contacts

A Quantum point contact (QPC) is a one dimensional constriction, separating two extended electron systems allowing transport between them only though a short and narrow channel. The linear conductance of QPCs is quantized in units of the conductance quantum G_Q=2e^2/h, where e is the electron charge and h is Planck's constant. Thus the conductance shows a staircase when plotted as a function of gate-voltage which defines the width of the channel. In addition measured curves show a shoulder-like step around 0.7G_Q. In this regime QPCs show anomalous behaviour in quantities like electrical or thermal conductance, noise, and thermopower, as a function of external parameters such as temperature, magnetic field, or applied voltage. These phenomena, collectively known as the 0.7-anomaly in QPCs are subject of controversial discussion. This thesis offers a detailed description of QPCs in the parameter regime of the 0.7-anomaly. A model is presented which reproduces the phenomenology of the 0.7-anomaly. We give an intuitive picture and a detailed description of the microscopic mechanism leading to the anomalous behavior. Further, we offer detailed predictions for the behavior of the 0.7-anomaly in the presence of spin-orbit interactions. Our best theoretical results were achieved using an approximation scheme within the functional renormalization group (fRG) which we developed to treat inhomogeneous interacting fermi systems. This scheme, called the coupled ladder approximation (CLA), allows the flow of the two-particle vertex to be incorporated even if the number of interacting sites N, is large, by reducing the number of independent variables which represent the two-particle vertex from O(N^4) to O (N^2)