Problem Of Initial Conditions Research Articles

Automatic adjoint-based inversion schemes for geodynamics: reconstructing the evolution of Earth's mantle in space and time

Abstract. Reconstructing the thermo-chemical evolution of Earth's mantle and its diverse surface manifestations is a widely recognised grand challenge for the geosciences. It requires the creation of a digital twin: a digital representation of Earth's mantle across space and time that is compatible with available observational constraints on the mantle's structure, dynamics and evolution. This has led geodynamicists to explore adjoint-based approaches that reformulate mantle convection modelling as an inverse problem, in which unknown model parameters can be optimised to fit available observational data. Whilst there has been a notable increase in the use of adjoint-based methods in geodynamics, the theoretical and practical challenges of deriving, implementing and validating adjoint systems for large-scale, non-linear, time-dependent problems, such as global mantle flow, has hindered their broader use. Here, we present the Geoscientific ADjoint Optimisation PlaTform (G-ADOPT), an advanced computational modelling framework that overcomes these challenges for coupled, non-linear, time-dependent systems by integrating three main components: (i) Firedrake, an automated system for the solution of partial differential equations using the finite-element method; (ii) Dolfin-Adjoint, which automatically generates discrete adjoint models in a form compatible with Firedrake; and (iii) the Rapid Optimisation Library, ROL, an efficient large-scale optimisation toolkit; G-ADOPT enables the application of adjoint methods across geophysical continua, showcased herein for geodynamics. Through two sets of synthetic experiments, we demonstrate the application of this framework to the initial condition problem of mantle convection, in both square and annular geometries, for both isoviscous and non-linear rheologies. We confirm the validity of the gradient computations underpinning the adjoint approach, for all cases, through second-order Taylor remainder convergence tests and subsequently demonstrate excellent recovery of the unknown initial conditions. Moreover, we show that the framework achieves theoretical computational efficiency. Taken together, this confirms the suitability of G-ADOPT for reconstructing the evolution of Earth's mantle in space and time. The framework overcomes the significant theoretical and practical challenges of generating adjoint models and will allow the community to move from idealised forward models to data-driven simulations that rigorously account for observational constraints and their uncertainties using an inverse approach.

Stochastic basins of attraction for uncertain initial conditions

Outcomes of nonlinear dynamical systems strongly depend on the initial conditions. However, it is common knowledge that it is not possible to fix the initial conditions in real experiments, with uncertainties being inherent to it. In this work, the problem of uncertain initial conditions in dynamical systems is investigated. These uncertainties are represented through four postulates, and a new definition of basin of attraction is deduced to quantify their effects on the global dynamics. It is shown that the new definition is a convolution between the uncertainty distribution and the deterministic basin of attraction, and a reduced-cost strategy to compute them is defined. A Helmholtz nonlinear oscillator is investigated by considering initial conditions with uniform and normal distributions, and a three-dimensional Hénon map is analyzed, too. Finally, the influence of the uncertainty level and the effectiveness of new basin definition are verified.

Effects of the wall temperature on the boiling process and the molecular dynamics behavior of the liquid hydrogen on a flat aluminum wall

Understanding the nucleation mechanism and boiling heat transfer characteristics of liquid hydrogen is a key issue in the development of liquid hydrogen storage systems. However, the current methods of experimental and mesoscopic or macroscopic scale continuous flow simulations suffer from the problems of large study scales and undetermined initial boundary conditions. The purpose of this study is to reveal the bubble nucleation mechanism and bubble development process of liquid hydrogen on the surface of heating plate at the molecular scale, and to investigate the effect of different heating plate temperature on bubble growth, during thin film boiling. In this research, molecular dynamics simulation is used to combine molecular position distribution with energy analysis and explain the molecular behavior. The effect of heating temperature on different phase transition stages of liquid hydrogen is also investigated by varying the heating surface temperature. In this study, the bubble nucleation, bubble development and merging of liquid hydrogen on the heating surface were successfully observed. It is shown that increasing the heating temperature has a small effect on the bubble growth rate and liquid layer thickness in the nucleation boiling stage, and mainly affects the gas film growth rate in the film boiling stage. The bubbles are extremely unstable at the initial stage of generation. Within 1 fs of the first appearance of bubbles, which are associated with random micro-pits caused by thermal motion of solid atoms, bubbles move on the surface of the heating plate. This study found a phenomenon that bubbles firstly generated and then disappeared. This paper provides a new idea for the nucleation mechanism and boiling heat transfer study of liquid hydrogen. Moreover, it provides a solution to the problems of liquid hydrogen phase change research.

Solving the initial conditions problem for modified gravity theories

Solving the initial conditions problem for modified gravity theories

Step forward on nonlinear differential equations with the Atangana–Baleanu derivative: Inequalities, existence, uniqueness and method

In this study, we have found new results related to the fractional derivatives, integrals, and corresponding nonlinear ordinary differential and integral equations of Atangana–Baleanu. A comment on the initial condition problem when working with ordinary differential equations with the Atangana–Baleanu fractional derivative introduced the paper. We have developed a variety of new inequalities that are similar to the Gronwall and generalized-Gronwall inequalities using some criteria for the evaluation. The linear growth condition was utilized to obtain several new inequalities after the interval and initial condition dependency were established. We demonstrated a Picard–Tonelli predictor–corrector integration for the Atangana–Baleanu fractional nonlinear differential equation’s local solution. We established three distinct proofs of uniqueness using the Kransorsel’skii–Krein, Kooi’s, and Lipschitz conditions. The Nystrom midpoint approach has also been expanded to include nonlinear ordinary differential equations with the Atangana–Baleanu derivative; hence, the case of Caputo is straightforward. To assess the effectiveness of this method, some theoretical analyses and some practical cases were provided.

Initial conditions problem in cosmological inflation revisited

We present first results from a novel numerical relativity code based on a tetrad formulation of the Einstein-scalar field equations combined with recently introduced gauge/frame invariant diagnostics. The results provide support for the argument that inflation does not solve the homogeneity and isotropy problem beginning from generic initial conditions following a big bang.

Low-complexity tracking for autonomous underwater helicopters with event-triggered mechanism

This paper aims to investigate a low-complexity tracking control method for autonomous underwater helicopters (AUH), which is devoid of nonlinear approximation strategies, ensuring good efficiency and keeping a high accuracy in environmental disturbances and modeling uncertainty. The Nussbaum function and shifting function are introduced to solve the problem of unknown control direction and initial condition of system error, respectively. Besides, to reduce the controller's update rate and the thruster's mechanical loss, an incremental event-triggered control mechanism with two thresholds is added to the algorithm. Finally, the simulations on AUHs show the effectiveness of the proposed scheme.

Sturm–Liouville Differential Inclusions with Set-Valued Reaction Term Depending on a Parameter

In this paper we study the controllability for a Cauchy problem governed by a nonlinear differential inclusion driven by a Sturm–Liouville type operator. In particular, the considered second order differential inclusion involves a set-valued reaction term depending on a parameter. The key tool in the proof of the controllability result we provide is a multivalued version of the theorem recently proved by Haddad–Yarou, here established for an initial conditions problem monitored by a nonlinear second order differential inclusion presenting the sum of two multimaps on the right-hand side. We thereby deduce the existence of a local admissible pair for the considered control problem, that is the existence of a couple of functions consisting of a control, which is a measurable function, and the correspondent trajectory, which is an absolutely continuous function with absolutely continuous derivative. Secondly, under appropriate assumptions on the involved multimaps, we obtain an increased regularity for the solutions produced by our existence result. This regularity is the same of that recently tested by Bonanno, Iannizzotto and Marras for a different type of problem, which however involves the Sturm–Liouville operator.

Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

As one of the main governing equations in kinetic theory, the Boltzmann equation is widely utilized in aerospace, microscopic flow, etc. Its high-resolution simulation is crucial in these related areas. However, due to the high dimensionality of the Boltzmann equation, high-resolution simulations are often difficult to achieve numerically. The moment method which was first proposed in Grad (Commun Pure Appl Math 2(4):331–407, 1949) is among the popular numerical methods to achieve efficient high-resolution simulations. We can derive the governing equations in the moment method by taking moments on both sides of the Boltzmann equation, which effectively reduces the dimensionality of the problem. However, one of the main challenges is that it leads to an unclosed moment system, and closure is needed to obtain a closed moment system. It is truly an art in designing closures for moment systems and has been a significant research field in kinetic theory. Other than the traditional human designs of closures, the machine learning-based approach has attracted much attention lately in Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) and Huang et al. (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose a machine learning-based method to derive a moment closure model for the Boltzmann–BGK equation. In particular, the closure relation is approximated by a carefully designed deep neural network that possesses desirable physical invariances, i.e., the Galilean invariance, reflecting invariance, and scaling invariance, inherited from the original Boltzmann–BGK equation and playing an important role in the correct simulation of the Boltzmann equation. Numerical simulations on the 1D–1D examples including the smooth and discontinuous initial condition problems, Sod shock tube problem, the shock structure problems, and the 1D–3D examples including the smooth and discontinuous problems demonstrate satisfactory numerical performances of the proposed invariance preserving neural closure method.

Simulation of mooring Lines in complex bathymetries using a finite element method

This paper presents a novel method for the modeling of the seabed interaction of mooring lines in complex bathymetries, known as “continuous projection method”. This method is able to calculate ground normal and friction forces for any seafloor surface. This provides an improvement in the mooring systems simulation, as it captures additional non-linearities on the mooring line performance due to seabed interaction.The method is based on constructing a triangulation for the seabed and projecting mooring line nodes by using the vertex normal vectors of each triangle, ensuring the continuity of the projection. For the sake of the computational cost reduction, the line nodes are first projected into the closest triangles. Also, whenever the floor is flat, inclined or horizontal, a point-to-plane projection expression is used instead. The projection method described has been implemented to a finite element model. The initial condition problem was solved with a static approach, based on finding the static equilibrium with Newton–Armijo algorithm. This improves other static approaches which use the catenary equation, and that are only valid for a flat seabed.The model was successfully verified against the analytical solution of an inextensible catenary line in a slope. Furthermore, the simulation results were validated against experimental scale tests on a single chain mooring line with three different seafloor structures: one flat floor and two different sloped steps. For each of them, static and dynamic regular tests were performed. Moreover, high and low frequency fairlead movements were imposed in the dynamic tests, aiming to validate the model both in cases with and without snapping loads. Overall, the obtained results were coherent and allow to validate the accuracy of the proposed model. Finally, a mooring line over an irregular seabed surface was studied, comparing the results obtained by directly applying the developed method for complex bathymetries, by interpolating the surface by an inclined plane and using the constructed projection algorithm for that case or, by last, approximating by a flat seafloor. The comparison of the results among the different approaches illustrates the importance of considering the seabed slope and irregularities for the fairlead tension prediction. Also, this flat seafloor was evaluated with two different projection methods: the one specific for horizontal seafloors and the one developed for general seabed surfaces. This allows to compare the computational time required in both of them.

Dynamically induced topological inflation

We propose an inflation model in which the inflationary era is driven by the strong dynamics of $Sp(2)$ gauge theory. The quark condensation in the confined phase of $Sp(2)$ gauge theory generates the inflaton potential comparable to the energy of the thermal bath at the time of phase transition. Afterwards, with a super-Planckian field value at the global minimum, the inflation commences at a false vacuum region lying between true vacuum regions and hence the name ``topological inflation.'' Featured by the huge separation between the scale of the false vacuum ($V(0{)}^{1/4}\ensuremath{\sim}{10}^{15}\text{ }\text{ }\mathrm{GeV}$) and the field value at the global minimum (${\ensuremath{\phi}}_{\mathrm{min}}\ensuremath{\sim}{M}_{P}$), the model can be consistent with cosmic microwave background (CMB) observables without suffering from the initial condition problems. Crucially, this is achieved only with some mild tuning of parameters in $V(\ensuremath{\phi})$. In addition to $Sp(2)$, this model is based on an anomaly free ${Z}_{6R}$ discrete $R$ symmetry. Remarkably, while all parameters are fixed by CMB observations, the model predicts a hierarchy of energy scales including the inflation scale, supersymmetry-breaking scale, R-symmetry breaking scale, Higgsino mass and the right-handed neutrino mass given in terms of the dynamical scale of $Sp(2)$.

Imports-ICT Linkage in Colombian Manufacturing

The objective of this paper is to explore the ICT impact on firm’s import decision and on firm’s performance in import markets in Colombian manufacturing. This is the first document exploring this linkage in Latin America. We analyse firms’ imports of raw materials (not exports as the tradition) because of the decisive role imports play in firms’ exports. To deal with the problem of sample selection bias we replicate the Tobit II-Heckman procedure, by using a dynamic random effects probit model with panel data for decision process and a dynamic Generalized Linear Model (GLM) for performance process including Mills ratio. We include the influence of exports on imports, the initial conditions problem and the depreciation pattern of import experience. Our main results reveal a strong and direct impact of ICT on imports, especially for SMEs. Other results also suggest sunk costs/persistence, self-selection, cross-effects, and a decreasing import experience.

Cross-section asymptotic for random-effects panel data models with autoregressive errors

A useful strategy to examine serial correlation in panel regression models is to adopt serially correlated error terms. The random effect is also present to address the individual-specific heterogeneity. Familiar transformations can restructure the AR(1) process into a serially uncorrelated dynamic model. It may result in unfitting estimation results once the statistical inference is carried on the conditional likelihood. We discuss in the article the consequence of this action. In particular, in the case of large cross-section units and short time-sequences, we provide analytical expressions for the asymptotic bias of maximum likelihood estimates when the analyst naïvely neglects the initial conditions problem. Under several conditions, we compute the size of biases and measure the mean squared errors to illustrate this violation. We also construct a joint regression model of the initial and subsequent responses to handle the initial conditions. Then, we conduct simulation studies to determine the asymptotic behavior.

A Note on Some Weaker Notions of Cop-Win and Robber-Win Graphs

The game of pursuit and evasion, when played on graphs, is often referred to as the game of cops and robbers. This classical version of the game has been completely solved by Nowakowski and Winkler, who gave the exact class of graphs for which the pursuer can win the game (cop-win). When the graph does not satisfy the dismantlability property, Nowakowski and Winkler’s Theorem does not apply. In this paper, we give some weaker notions of cop-win and robber-win graphs. In particular, we fix the number of cops to be equal to one, and we ask whether there exist sets of initial conditions for which the graph can be cop-win or robber-win. We propose some open questions related to this initial condition problem with the goal of developing both structural characterizations and algorithms that are computable in polynomial time (P) and that can solve weakly cop-win and weakly- robber-win graphs.

More on emergent dark energy from unparticles

In a recent paper \cite{Artymowski:2020zwy} we suggested the possibility that the present acceleration of the Universe is due to thermodynamical behavior of unparticles. The model is free of scalar fields, modified gravity, a Cosmological Constant (CC), the coincidence problem, initial conditions problem and possesses interesting distinct predictions regarding the equation of state of Dark Energy, the growth rate and the number of relativistic degrees of freedom at BBN and CMB decoupling. In this work, we relate to a recent paper \cite{Abchouyeh:2021wey}, which discusses a similar setup of unparticles with and without a CC as an external source of late-time acceleration. The authors have shown how such a model is inconsistent with the data. We show that these claims are viable only in a particular part of the parameter space and that model \cite{Artymowski:2020zwy} stands tall. We further suggest a consistency condition in terms of observables. We then fit publicly available supernovae data to derive the expected Hubble parameter and constrain the parameters of the model.

A New Look at the Initial Condition Problem

In this paper, some myths associated to the initial condition problem are studied and demystified. It is shown that the initial conditions provided by the one-sided Laplace transform are not those required for Riemann-Liouville and Caputo derivatives. The problem is studied and solved with generality as well as applied to continuous-time fractional autoregressive-moving average systems.

Cycle-tree guided attack of random K-core: Spin glass model and efficient message-passing algorithm

The K-core of a graph is the maximal subgraph within which each vertex is connected to at least K other vertices. It is a fundamental network concept for understanding threshold cascading processes with a discontinuous percolation transition. A minimum attack set contains the smallest number of vertices whose removal induces complete collapse of the K-core. Here we tackle this prototypical optimal initial-condition problem from the spin-glass perspective of cycle-tree maximum packing and propose a cycle-tree guided attack (CTGA) message-passing algorithm. The good performance and time efficiency of CTGA are verified on the regular random and Erd\"os-R\'enyi random graph ensembles. Our central idea of transforming a long-range correlated dynamical process to static structural patterns may also be instructive to other hard optimization and control problems.

The 21st Century Systems: An Updated Vision of Continuous-Time Fractional Models

This paper presents the continuous-time fractional linear systems and their main properties. Two particular classes of models are introduced: the fractional autoregressive-moving average type and the tempered linear system. For both classes, the computations of the impulse response, transfer function, and frequency response are discussed. It is shown that such systems can have integer and fractional components. From the integer component we deduce the stability. The fractional order component is always stable. The initial-condition problem is analyzed and it is verified that it depends on the structure of the system. For a correct definition and backward compatibility with classic systems, suitable fractional derivatives are also introduced. The Grünwald-Letnikov and Liouville derivatives, as well as the corresponding tempered versions, are formulated.

Computing the Load Margin of Power Systems Using Crow Search Algorithm

Power system operation centers require knowledge of the load level where the system may present instability characteristics. It is common practice to calculate the load margin where the system can collapse in voltage respecting a direction of load growth. However, the increase in load can also cause low damped oscillation modes to appear and cause small-signal stability problems of power systems. This research proposes a Crow Search Algorithm-based method to calculate the load margin of power systems considering small-signal stability and voltage stability requirements. To solve the problem of good initial conditions for the search problem, the Sine Cosine Algorithm was used. The proposed method was evaluated in the IEEE 39-bus system and the results achieved show the effectiveness of the method in determining the load margins quickly.

How the magnetic field behaves during the motion of a highly conducting fluid under its own gravity: A new theoretical, relativistic approach

Within the context of general relativity we study in a fully covariant way the so-called Euler-Maxwell system of equations. In particular, on decomposing the aforementioned system into its 1 temporal and 1 + 2 spatial components at the ideal magnetohydrodynamic limit, we bring it in a simplified form that favors physical insight to the problem of a self-gravitating, magnetized fluid. Of central interest is the decomposition of Faraday's equation which leads to a new general solution governing the evolution of the magnetic field during the motion of the highly conducting fluid. According to the latter relation, the magnetic field generally grows or decays in proportion to the inverse cube law of the scale factor--associated with the continuous contraction or expansion of the fluid's volume respectively. The magnetic field's law of variation, which has remarkable implications for the motion of the whole fluid, is subsequently applied to homogeneous (anisotropic-magnetized) cosmological models--especially to the Bianchi I case--as well as to the study of homogeneous and anisotropic gravitational collapse in a magnetized environment. Concerning the cosmological application, we derive the evolution equations of Bianchi I spacetime permeated by large-scale magnetic fields (these equations reduce to their FRW counterparts at the small/large--scale limit). Also, the compatibility of the new evolution formula for the magnetic field with the standard cosmic nucleosynthesis constraint is examined. As for the application in astrophysics, our results predict that homogeneous gravitational implosion is impeded when the electric Weyl tensor (associated with tidal forces) along the magnetic forcelines overwhelms the magnetic energy density. Lastly, our model denotes that the satisfaction of the aforementioned criterion is ultimately driven into a problem of initial conditions.