PDE Problems Research Articles

An Approximation Theorist’s view on solving operator equations—With special attention to Trefftz, MFS, MPS, and DRM methods

When an Approximation Theorist looks at well-posed PDE problems or operator equations, and standard solution algorithms like Finite Elements, Rayleigh–Ritz or Trefftz techniques, methods of fundamental or particular solutions and their combinations, they boil down to approximation problems and stability issues. These two can be handled by Approximation Theory, and this paper shows how, with special applications to the aforementioned algorithms. The intention is that the Approximation Theorist’s viewpoint is helpful for readers who are somewhat away from that subject.

On Locally and Globally Optimal Solutions in Scalar Variational Control Problems

In this paper, optimality conditions are studied for a new class of PDE and PDI-constrained scalar variational control problems governed by path-independent curvilinear integral functionals. More precisely, we formulate and prove a minimal criterion for a local optimal solution of the considered PDE and PDI-constrained variational control problem to be its global optimal solution. The effectiveness of the main result is validated by a two-dimensional non-convex scalar variational control problem.

Second Refinement of Jacobi Iterative Method for Solving Linear System of Equations

In this paper, the new method called second refinement of Jacobi (SRJ) method for solving linear system of equations is proposed. The method can be used to solve ODE and PDE problems where the problems are reduced to linear system of equations with coefficient matrices which are strictly diagonally dominant (SDD) or symmetric positive definite matrices (SPD) or M-matrices. In this case, our new method minimizes the number of iterations as well as spectral radius and increases rate of convergence. Few numerical examples are considered to show the efficiency of SRJ over Jacobi (J) and refinement of Jacobi (RJ) methods.

Infinite dimensional compressed sensing from anisotropic measurements and applications to inverse problems in PDE

We consider a compressed sensing problem in which both the measurement and the sparsifying systems are assumed to be frames (not necessarily tight) of the underlying Hilbert space of signals, which may be finite or infinite dimensional. The main result gives explicit bounds on the number of measurements in order to achieve stable recovery, which depends on the mutual coherence of the two systems. As a simple corollary, we prove the efficiency of nonuniform sampling strategies in cases when the two systems are not incoherent, but only asymptotically incoherent, as with the recovery of wavelet coefficients from Fourier samples. This general framework finds applications to inverse problems in partial differential equations, where the standard assumptions of compressed sensing are often not satisfied. Several examples are discussed, with a special focus on electrical impedance tomography.

Solvability of the mixed problem of a high-order PDE with fractional time derivatives, Sturm-Liouville operators on spatial variables and non-local boundary conditions

In this paper, we study the solvability of a mixed problem of a partial differential equation of high order with fractional derivatives with respect to time and with Sturm-Liouville operators with spatial variables and non-local boundary conditions; the solution is found as a series of eigenfunctions of the Sturm-Liouville operator with non-local boundary conditions.

A fractional PDE for first passage time of time-changed Brownian motion and its numerical solution

We show that the First-Passage-Time probability distribution of a Lévy time-changed Brownian motion with drift is solution of a time fractional advection-diffusion equation subject to initial and boundary conditions; the Caputo fractional derivative with respect to time is considered. We propose a high order compact implicit discretization scheme for solving this fractional PDE problem and we show that it preserves the structural properties (non-negativity, boundedness, time monotonicity) of the theoretical solution, having to be a probability distribution. Numerical experiments confirming such findings are reported. Simulations of the sample paths of the considered process are also performed and used to both provide suitable boundary conditions and to validate the numerical results.

Two boundary integral equation methods for linear elastodynamics problems on unbounded domains

We consider (transient) 3D elastic wave propagation problems in unbounded isotropic homogeneous media, which can be reduced to corresponding 2D ones. For their solution, we propose and compare two boundary integral equation approaches, both based on the coupling of a discrete time convolution quadrature with a classical space collocation discretization. In the first approach, the PDE problem is preliminarily replaced by the equivalent well known (vector) space–time boundary integral equation formulation, while in the second, the same PDE is replaced by a system of two (coupled) wave equations, each one of which is then represented by the associated boundary integral equation. The construction of these two approaches is described and discussed. Some numerical testing are also presented.

The conditions of existence of a solution of the degenerate two-point in time problem for PDE

The problem with local nonhomogeneous two-point in time conditions for homogeneous PDE of the second order in time and, generally, infinite order in spatial variables is investigated. This problem is degenerated namely its characteristic determinant is identically zero. The condition of existence of a solution of the degenerate problem is established. Also, we proposed the differential-symbol method of constructing the solution of the problem in the classes of entire functions. Some examples of solving the degenerate two-point in time problems are presented.

A Markov chain-based multi-elimination preconditioner for elliptic PDE problems

Preconditioning is a simple but very powerful technique for accelerating the convergence of an iterative method for solving large, sparse linear systems. Traditionally, to design a good preconditioner implemented on a personal computer or a cluster of computer systems with CPUs, the preconditioner should meet the following three conditions. (a) the preconditioner is a good approximation of the inverse of the original linear coefficient matrix in some sense, (b) Setting up the components needed for the preconditioning operator is easy, and (c) the cost for the application of the preconditioner with a given vector is low. However, as new computer systems are rapidly invented, such designing strategies need to be reinvestigated. Motivated by the random walk technique, which is a computationally intensive task, we proposed a different preconditioner based on a multi-elimination algorithm, where the inversion operator is approximated explicitly by a partial product of the Neumann series of the iterative matrix for the original system. The computation of such an approximation takes advantages of the strengths of the GPU system. We report some numerical results implemented by CUDA PETSc. In our implementation, we adopt the hybrid GPU–CPU approach that preconditioning setup and application are performed on GPU, while Krylov subspace iteration, which requires a certain degree of communications is done on CPU. This method minimizes data communication between GPU–CPU, which is a bottleneck for many existing algorithms.

Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs

In this paper, we first present a sufficient condition(a variant) for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. The sufficient condition is particularly more suitable for stochastic differential/partial differential equations with reflection. We then apply the sufficient condition to establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will also play an important role.

Bouligand–Landweber iteration for a non-smooth ill-posed problem

This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not G\^ateaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is G\^ateaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand--Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated

Algorithm for overcoming the curse of dimensionality for state-dependent Hamilton-Jacobi equations

In this paper, we develop algorithms to overcome the curse of dimensionality in non-convex state-dependent Hamilton-Jacobi partial differential equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and they can be implemented in an embarrassingly parallel fashion. This is ideal for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality [1,2] when solving HJ PDE. The major contribution of the paper is to change either the solving of a PDE problem or an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond the work of [40]. We extend the method in [7,9,15], and conjecture a (Lax-type) minimization principle to solve state-dependent HJ PDE when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle to solve state-dependent HJ PDE when the Hamiltonian is non-convex, as a generalization of the well-known Hopf formula in [18,25,50]. We showed the validity of the formula under restricted assumption for the sake of completeness, and would like to bring our readers to [62] which validates our conjectures in a more general setting. We conjectured the weakest assumption of our formula to hold is a pseudoconvexity assumption similar to one stated in [50]. Our method is expected to have application in control theory, differential game problems and elsewhere.

Nonlocal problems in perforated domains

AbstractIn this paper, we analyse nonlocal equations in perforated domains. We consider nonlocal problems of the form $f(x) = \int \nolimits _{B} J(x-y) (u(y) - u(x)) {\rm d}y$ with x in a perforated domain $\Omega ^\epsilon \subset \Omega $. Here J is a nonsingular kernel. We think about $\Omega ^\epsilon $ as a fixed set Ω from where we have removed a subset that we call the holes. We deal both with the Neumann and Dirichlet conditions in the holes and assume a Dirichlet condition outside Ω. In the latter case we impose that u vanishes in the holes but integrate in the whole ℝN (B = ℝN) and in the former we just consider integrals in ℝN minus the holes ($B={\open R} ^N \setminus (\Omega \setminus \Omega ^\epsilon )$). Assuming weak convergence of the holes, specifically, under the assumption that the characteristic function of $\Omega ^\epsilon $ has a weak limit, $\chi _{\epsilon } \rightharpoonup {\cal X}$ weakly* in L∞(Ω), we analyse the limit as ε → 0 of the solutions to the nonlocal problems proving that there is a nonlocal limit problem. In the case in which the holes are periodically removed balls, we obtain that the critical radius is of the order of the size of the typical cell (that gives the period). In addition, in this periodic case, we also study the behaviour of these nonlocal problems when we rescale the kernel in order to approximate local PDE problems.

Construction of Finsler‐Lyapunov functions with meshless collocation

AbstractWe study the stability of invariant sets such as equilibria or periodic orbits of a Dynamical System given by a general autonomous nonlinear ordinary differential equation (ODE). A classical tool to analyse the stability are Lyapunov functions, i.e. scalar‐valued functions, which decrease along solutions of the ODE. An alternative to Lyapunov functions is contraction analysis. Here, stability (or incremental stability) is a consequence of the contraction property between two adjacent solutions, formulated as the local property of a Finsler‐Lyapunov function. This has the advantage that the invariant set plays no special role and does not need to be known a priori.In this paper, we propose a method to numerically construct a Finsler‐Lyapunov function by solving a first‐order partial differential equation using meshless collocation. Depending on the expected attractor, the contraction only takes place in certain directions, which can easily be implemented within the method. In the basin of attraction of an exponentially stable equilibrium or periodic orbit, we show that the PDE problem has a solution, which provides error estimates for the numerical method. Furthermore, we show how the method can also be applied outside the basin of attraction and can detect the stability as well as the stable/unstable directions of equilibria. The method is illustrated with several examples.

A reduced basis approach for PDEs on parametrized geometries based on the shifted boundary finite element method and application to a Stokes flow

We propose a model order reduction technique integrating the Shifted Boundary Method (SBM) with a POD-Galerkin strategy. This approach allows to deal with complex parametrized domains in an efficient and straightforward way. The impact of the proposed approach is threefold.First, problems involving parametrizations of complex geometrical shapes and/or large domain deformations can be efficiently solved at full-order by means of the SBM. This unfitted boundary method permits to avoid remeshing and the tedious handling of cut cells by introducing an approximate surrogate boundary.Second, the computational effort is reduced by the development of a Reduced Order Model (ROM) technique based on a POD-Galerkin approach.Third, the SBM provides a smooth mapping from the true to the surrogate domain, and for this reason, the stability and performance of the reduced order basis are enhanced. This feature is the net result of the combination of the proposed ROM approach and the SBM. Similarly, the combination of the SBM with a projection-based ROM gives the great advantage of an easy and fast to implement algorithm considering geometrical parametrization with large deformations. The transformation of each geometry to a reference geometry (morphing) is in fact not required.These combined advantages will allow the solution of PDE problems more efficiently. We illustrate the performance of this approach on a number of two-dimensional Stokes flow problems.

Dynamic data-driven Bayesian GMsFEM

In this paper, we propose a Bayesian approach for multiscale problems with the availability of dynamic observational data. Our method selects important degrees of freedom probabilistically in a Generalized multiscale finite element method framework. Due to scale disparity in many multiscale applications, computational models cannot resolve all scales. Dominant modes in the Generalized Multiscale Finite Element Method are used as “permanent” basis functions, which we use to compute an inexpensive multiscale solution and the associated uncertainties. Through our Bayesian framework, we can model approximate solutions by selecting the unresolved scales probabilistically. We consider parabolic equations in heterogeneous media. The temporal domain is partitioned into subintervals. Using residual information and given dynamic data, we design appropriate prior distribution for modeling missing subgrid information. The likelihood is designed to minimize the residual in the underlying PDE problem and the mismatch of observational data. Using the resultant posterior distribution, the sampling process identifies important degrees of freedom beyond permanent basis functions. The method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations.

Optimal Investment with Vintage Capital: Equilibrium Distributions

The paper concerns the study of equilibrium points, or steady states, of economic systems arising in modelling optimal investment with vintage capital, namely, systems where all key variables (capitals, investments, prices) are indexed not only by time τ but also by age s. Capital accumulation is hence described as a partial differential equation (briefly, PDE), and equilibrium points are in fact equilibrium distributions in the variable s of ages. Investments in frontier as well as non-frontier vintages are possible. Firstly a general method is developed to compute and study equilibrium points of a wide range of infinite dimensional, infinite horizon boundary control problems for linear PDEs with convex criterion, possibly applying to a wide variety of economic problems. Sufficient and necessary conditions for existence of equilibrium points are derived in this general context. In particular, for optimal investment with vintage capital, existence and uniqueness of a long run equilibrium distribution is proved for general concave revenues and convex investment costs, and analytic formulas are obtained for optimal controls and trajectories in the long run, definitely showing how effective the theoretical machinery of optimal control in infinite dimension is in computing explicitly equilibrium distributions, and suggesting that the same method can be applied in examples yielding the same abstract structure. To this extent, the results of this work constitutes a first crucial step towards a thorough understanding of the behaviour of optimal controls and trajectories in the long run.

Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem

In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not G\^ateaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fr\'echet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration.

The work of Norman Dancer

In this article, a sample of Norman Dancer's published works are briefly described, to give the reader a taste of his deep and important research on nonlinear functional analysis, nonlinear ODE and PDE problems, and dynamical systems. The sample covers a variety of topics where Norman Dancer has made remarkable contributions. The author takes this opportunity to express his deep admiration of the work of Professor Norman Dancer, and to thank him for the kind help to the development of the author's career, which has been greatly influenced by him and his work.

Decoupling simulation accuracy from mesh quality

For a given PDE problem, three main factors affect the accuracy of FEM solutions: basis order, mesh resolution, and mesh element quality. The first two factors are easy to control, while controlling element shape quality is a challenge, with fundamental limitations on what can be achieved. We propose to use p -refinement (increasing element degree) to decouple the approximation error of the finite element method from the domain mesh quality for elliptic PDEs. Our technique produces an accurate solution even on meshes with badly shaped elements, with a slightly higher running time due to the higher cost of high-order elements. We demonstrate that it is able to automatically adapt the basis to badly shaped elements, ensuring an error consistent with high-quality meshing, without any per-mesh parameter tuning. Our construction reduces to traditional fixed-degree FEM methods on high-quality meshes with identical performance. Our construction decreases the burden on meshing algorithms, reducing the need for often expensive mesh optimization and automatically compensates for badly shaped elements, which are present due to boundary constraints or limitations of current meshing methods. By tackling mesh generation and finite element simulation jointly, we obtain a pipeline that is both more efficient and more robust than combinations of existing state of the art meshing and FEM algorithms.