Partial Differential Equation Framework Research Articles

Framework of fourth-order partial differential equation–based remote sensing image restoration

Framework of fourth-order partial differential equation–based remote sensing image restoration

Numerical PDE-Based Pricing of Convertible Bonds Under Two-Factor Models

Convertible bonds are popular financial instruments by which firms raise capital. Owing to the various features of such bonds, especially the early-exercise call, put, and conversion provisions, they can be valued by numerical techniques only. The price of a convertible bond is driven by both the underlying stock price and the interest rate, and these two factors are correlated. Under the partial differential equation framework, a two-dimensional convection-diffusion-reaction equation containing a mixed derivative must be solved. In this work, we employ an Alternating-Direction-Implicit method, namely the Craig-Sneyd scheme to solve the two-factor pricing equation. Comparison against the commonly employed Crank-Nicolson method shows the merit of the scheme. Besides, we analyze how the different contractual features of a convertible bond affect its price.

Well‐posedness of stochastic heat equation with distributional drift and skew stochastic heat equation

AbstractWe study stochastic reaction–diffusion equation where is a generalized function in the Besov space , and is a space‐time white noise on . We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever , and . This class includes equations with being measures, in particular, which corresponds to the skewed stochastic heat equation. For , we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020).

Bayesian learning of coupled biogeochemical–physical models

Predictive dynamical models for marine ecosystems are used for a variety of needs. Due to the sparse measurements and limited understanding of the myriad of ocean processes, there is however significant uncertainty. There is model uncertainty in the parameter values, functional forms with diverse parameterizations, and level of complexity needed, and thus in the state variable fields. We develop a Bayesian model learning methodology that allows interpolation in the space of candidate dynamical models and discovery of new models from noisy, sparse, and indirect observations, all while estimating state variable fields and parameter values, as well as the joint probability distributions of all learned quantities. We address the challenges of high-dimensional and multidisciplinary dynamics governed by partial differential equations (PDEs) by using state augmentation and the computationally efficient Gaussian Mixture Model — Dynamically Orthogonal filter. Our innovations include stochastic formulation parameters and stochastic complexity parameters to unify candidate models into a single general model as well as stochastic expansion parameters within piecewise function approximations to generate dense candidate model spaces. These innovations allow handling many compatible and embedded candidate models, possibly none of which are accurate, and learning elusive unknown functional forms that augment these models. Our new Bayesian methodology is generalizable and interpretable. It seamlessly and rigorously discriminates among existing models, but also extrapolates out of the space of models to discover new ones. We perform a series of twin experiments based on flows past a ridge coupled with three-to-five component ecosystem models, including flows with chaotic advection. We quantify the learning skill, and evaluate convergence and the sensitivity to hyper-parameters. Our PDE framework successfully discriminates among functional forms and model complexities, and learns in the absence of prior knowledge by searching in dense function spaces. The probabilities of known, uncertain, and unknown model formulations, and of biogeochemical–physical fields and parameters, are updated jointly using Bayes’ law. Non-Gaussian statistics, ambiguity, and biases are captured. The parameter values and the model formulations that best explain the noisy, sparse, and indirect data are identified. When observations are sufficiently informative, model complexity and model functions are discovered.

An EIM-based compression-extrapolation tool for efficient treatment of homogenized cross-section data

Nuclear reactor simulators implementing the widespread two-steps deterministic calculation scheme tend to produce a large volume of intermediate data at the interface of their two subcodes – up to dozens or even hundred of gigabytes – which can be so cumbersome that it hinders the global performance of the code. The vast majority of this data consists of “few-groups homogenized cross-sections”, nuclear quantities stored in the form of tabulated multivariate functions which can be precomputed to a large extent.It has been noticed in Tomatis (2021) that few-groups homogenized cross-sections are highly redundant — that is, they exhibit strong correlations, which paves the way for the use of compression techniques. We here pursue this line of work by introducing a new coupled compression/surrogate modeling tool based on the Empirical Interpolation Method, an algorithm originally developed in the framework of partial differential equations (Barrault et al., 2004). This EIM-compression method is based on the infinite norm ∥⋅∥∞, and proceeds in a greedy manner by iteratively trying to approximate the data and incorporating the chunks of information which cause the largest error. In the process, it generates a vector basis and a set of interpolation points, which provide an elementary surrogate model that can be used to approximate future data from little information. The algorithm is also very suitable for parallelization and out-of-core computation (processing of data too large for the computer RAM) and very easy to apprehend and implement.This method enables us to both efficiently compress cross-sections and spare a large fraction of the required lattice calculations. We investigate its performance on large realistic nuclear data replicating the notorious VERA benchmark (Godfrey, 2014) (20 energy groups, pin-by-pin homogenization, 10 particularized isotopes). Compression loss, memory savings and speed are analyzed both from a data-centric point of view in the perspective of applications in neutronics, and by comparison with an existing and widely-used method – stochastic truncated SVD – to assess mathematical efficiency. We discuss the usage of our surrogate model and its sensitivity to the choice of the training set. The method is shown to be competitive in terms of accuracy and speed, provide important memory savings and spare a large amount of physics code computation; all this could facilitate the adoption of fine-grain modelization schemes (pin-by-pin and many-groups homogenization, particularization of many isotopes) in industrial setups. A Github repository is available,11Link: https://github.com/Otruffinet/EIM_surrogates.git. which contains all the methods used for the article.

Robust Output Regulation for Multi-Dimensional Heat Equation

In the past few years, we have developed internal model principle for some 1-d partial differential equations (PDEs) from PDE perspective, by which one can find the analytic form of the tracking error feedback control. However, the conditional robustness is only limited to some special cases because it is difficult to formulate the conditional robustness in the framework of PDEs. In this paper, we investigate output regulation for a boundary controlled multidimensional heat equation. We obtain not only an analytic tracking error feedback control but also prove the conditional robustness. This is a first time to combine the abstract result and PDE design, which takes advantages of both frameworks.

A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations

Within the framework of parameter dependent Partial Differential Equations (PDEs), we develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. The research is motivated by the limitations and drawbacks of state-of-the-art algorithms, such as the Reduced Basis method, when addressing problems that show a slow decay in the Kolmogorov n n -width. Our work is based on the use of deep autoencoders, which we employ for encoding and decoding a high fidelity approximation of the solution manifold. To provide guidelines for the design of deep autoencoders, we consider a nonlinear version of the Kolmogorov n n -width over which we base the concept of a minimal latent dimension. We show that the latter is intimately related to the topological properties of the solution manifold, and we provide theoretical results with particular emphasis on second order elliptic PDEs, characterizing the minimal dimension and the approximation errors of the proposed approach. The theory presented is further supported by numerical experiments, where we compare the proposed approach with classical Principal Orthogonal Decomposition (POD)-Galerkin reduced order models. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields, singular terms and stochastic coefficients.

Delay differential equations for the spatially resolved simulation of epidemics with specific application to COVID-19.

In the wake of the 2020 COVID‐19 epidemic, much work has been performed on the development of mathematical models for the simulation of the epidemic and of disease models generally. Most works follow the susceptible‐infected‐removed (SIR) compartmental framework, modeling the epidemic with a system of ordinary differential equations. Alternative formulations using a partial differential equation (PDE) to incorporate both spatial and temporal resolution have also been introduced, with their numerical results showing potentially powerful descriptive and predictive capacity. In the present work, we introduce a new variation to such models by using delay differential equations (DDEs). The dynamics of many infectious diseases, including COVID‐19, exhibit delays due to incubation periods and related phenomena. Accordingly, DDE models allow for a natural representation of the problem dynamics, in addition to offering advantages in terms of computational time and modeling, as they eliminate the need for additional, difficult‐to‐estimate, compartments (such as exposed individuals) to incorporate time delays. In the present work, we introduce a DDE epidemic model in both an ordinary and partial differential equation framework. We present a series of mathematical results assessing the stability of the formulation. We then perform several numerical experiments, validating both the mathematical results and establishing model's ability to reproduce measured data on realistic problems.

Numerical analysis of some partial differential equations with fractal-fractional derivative

<abstract> <p>In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed.</p> </abstract>

3‐D Magnetic Unstructured Inversion

AbstractMagnetic inversion methods based on structured grids have been used extensively in geomagnetic research. Nevertheless, structured grids limit the modeling capability and accuracy of models with arbitrary geometries. To address these challenges, 3‐D magnetic numerical forward modeling and inversion methods using an unstructured tetrahedral grid based on a partial differential equation (PDE) framework are proposed. The methods are derived from Maxwell's partial equations and constructed using the finite element method. Arbitrary undulating topographies, complex geometries, and demagnetization effects can be represented exactly, leading to high‐accuracy forward modeling and inversion solutions. The proposed methods are suitable for both local‐ and global‐scale magnetic data interpretation, for example, in mineral exploration and tectonic research. A synthetic example and real airborne magnetic survey data collected from Mount Iliamna, Alaska, USA, are tested. The results demonstrate that the novel methods significantly improve the capability and accuracy of magnetic data interpretation.

Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

The compost bomb instability in the continuum limit

The ‘Compost Bomb’ instability refers to a proposed uncontrolled increase in soil temperature. This instability is caused when sufficiently rapid atmospheric warming increases soil heterotrophic respiration which, in turn, heats the soil further. This generates a runaway effect in which soil temperatures rise rapidly. We investigate this process, neglected in Earth system models, but which has thus far been analysed with a conceptual model using ordinary differential equations. That model is deliberately idealised without any representation of the spatial structure of soils. We confirm using a partial differential equation framework, this runaway effect still occurs when accounting for soil depth. Using this newer representation we investigate the forcing parameters that make soils vulnerable to this instability. In particular, we discover that the effect of dangerously large seasonal cycle variations in air temperature can create plausible conditions for a ‘compost bomb’ thermal instability.

Low-latitude reduction-to-the-pole and upward continuation between arbitrary surfaces based on the partial differential equation framework

SUMMARY Magnetic surveys conducted in complex conditions, such as low magnetic latitudes, uneven observation surfaces, or above high-susceptibility sources, pose significant challenges for obtaining stable solutions for reduction-to-the-pole (RTP) and upward-continuation processing on arbitrary surfaces. To tackle these challenges, in this study, we propose constructing an equivalent-susceptibility model based on the partial differential equation (PDE) framework in the space domain. A multilayer equivalent-susceptibility method was used for RTP and upward-continuation operations, thus allowing for application on undulating observation surfaces and strong self-demagnetization effect in a non-uniform mesh. A novel positivity constraint is introduced to improve the accuracy and efficiency of the inversion. We analysed the effect of the depth-weighting function in the inversion of equivalent susceptibility for RTP and upward-continuation reproduction. Iterative and direct solvers were utilized and compared in solving the large, sparse, non-symmetric and ill-conditioned system of linear equations produced by PDE-based equivalent-source construction. Two synthetic models were used to illustrate the efficiency and accuracy of the proposed method in processing both ground and airborne magnetic data. Aeromagnetic and ground data collected in Brazil at a low magnetic latitude region were used to validate the proposed method for processing RTP and upward-continuation operations on magnetic data sets with strong self-demagnetization.

Development and Evaluation of an ODE Representation of 3D Subsurface Tile Drainage Flow Using the HLM Flood Forecasting System

AbstractWe developed a new ordinary differential equation (ODE) to represent subsurface flows from a hillslope with artificial subsurface drainage (tiling) into an adjacent channel. Our ODE is based on a derived storage‐discharge relationship from high‐resolutions HYDRUS3D simulations that solve 3D partial differential Richards' equations for a hillslope (plot scale) with internal drainage surfaces, variable soil depth, hydraulic conductivity, and varying slope in the direction of flow. The ODE does not capture the hysteresis in the storage‐drainage relation that can only be captured using partial differential equations (PDEs), however the simplification does not have major impacts on the modeled flows. The nondimensional representation of the equation is consistent with, and provides an explanation for, the empirical equation used in classical hydrological models such as the ARNO (Arno River) and Variable Infiltration Capacity (VIC) model to simulate subsurface flows, which removes the need for parameter calibration. Our equation captures the features not captured by other ODE simplifications such as DRAINMOD. The ODE can be coupled to the Hillslope Link Model (HLM) flood forecasting system allowing real‐time regional simulation of streamflow fluctuations and floods over large domains. We show that (i) the new equation improves the capability to model watershed scale hydrograph recessions and to better capture the timing above flood thresholds for multiple spatial scales from 0.1 to 18,000 km2; (ii)using realistic values of slope combined with typical values of tile separation and soil depth provides good predictive power at gauged basin outlets; and (iii) parameters can be derived for particular situations using the 3D PDE framework.

Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored.

Numerical Simulation of the Instability of a Non-Isothermal Liquid Film in Film Devices

This work presents results of numerical simulation of the instability of vertical liquid film flow at moderate Reynolds numbers in the framework of partial differential equation of the state of the free surface of the liquid film. Equation coefficients include different physicochemical factors such as surface tension parameter and Marangoni parameter. Python scripts are used for computation of wave characteristics, instability regions of liquid films and liquid films free surface state. Computational experiments were carried out for the vertical water and alcohol films. Results could be used in the design or modernization of existing equipment and in the design of technological processes in liquid films.

Geodesic Paths for Image Segmentation With Implicit Region-Based Homogeneity Enhancement.

Minimal paths are regarded as a powerful and efficient tool for boundary detection and image segmentation due to its global optimality and the well-established numerical solutions such as fast marching method. In this paper, we introduce a flexible interactive image segmentation model based on the Eikonal partial differential equation (PDE) framework in conjunction with region-based homogeneity enhancement. A key ingredient in the introduced model is the construction of local geodesic metrics, which are capable of integrating anisotropic and asymmetric edge features, implicit region-based homogeneity features and/or curvature regularization. The incorporation of the region-based homogeneity features into the metrics considered relies on an implicit representation of these features, which is one of the contributions of this work. Moreover, we also introduce a way to build simple closed contours as the concatenation of two disjoint open curves. Experimental results prove that the proposed model indeed outperforms state-of-the-art minimal paths-based image segmentation approaches.

Simulation of unstable liquid film flow in heat and mass transfer processes of film equipment

This work presents results of computational modeling of vertical water liquid film flow at moderate Reynolds numbers in the framework of partial differential equation of the state of the free surface of the liquid film. Equation coefficients include different physicochemical factors such as surface tension parameter and temperature gradients influence parameter. Utility developed in C++ is used for computation of wave characteristics and instability regions of liquid films. Computational experiments were carried out for the vertical water film both at free flow and at evaporation. Results could be used in the design or modernization of existing equipment and in the design of technological processes in liquid films.

Selection dynamics for deep neural networks

This paper presents a partial differential equation framework for deep residual neural networks and for the associated learning problem. This is done by carrying out the continuum limits of neural networks with respect to width and depth. We study the wellposedness, the large time solution behavior, and the characterization of the steady states of the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using standard variational calculus, the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between neural networks, PDE theory, variational analysis, optimal control, and deep learning.

Downward Continuation and Transformation of Total‐Field Magnetic Anomalies Into Magnetic Gradient Tensors Between Arbitrary Surfaces Using Multilayer Equivalent Sources

AbstractThe equivalent source method offers a potential means for continuing and manipulating magnetic data between arbitrary surfaces. However, it ultimately suffers from a limited continuation distance and significant errors, which restrict its applicability in practice. To address this problem, we reformulate the equivalent source method with a partial differential equation framework. A multilayer equivalent susceptibility method is innovatively derived based on this approach. The location of equivalent sources is designed for adaptive estimation through the inverse procedure. We introduce nonlinear optimization, regularization, and constraining technologies in inversion to improve the stability and accuracy of the solution. Synthetic, real airborne, and ground data are tested in experiments. The results show that the new method significantly improves the capability and accuracy of the downward continuation and transformation of irregularly located data between arbitrary undulating surfaces.