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This paper presents an interior point method for pure state and mixed-constraint optimal control problems for dynamics, mixed constraints, and cost function all affine in the control variable. This method relies on resolving a sequence of two-point boundary value problems of differential and algebraic equations. This paper establishes a convergence result for primal and dual variables of the optimal control problem. A primal and a primal-dual solving algorithm are presented, and a challenging numerical example is treated for illustration. [ABSTRACT FROM AUTHOR]

This paper is the second of two articles, in which we aim to prove the convergence of the Hiptmair--Xu (HX) preconditioner (originally proposed by [R. Hiptmair and J. Xu, SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509]) for H(curl)-elliptic boundary value problems with jump coefficients. In this paper, based on the auxiliary results obtained in our first article [Q. Hu, SIAM J. Numer. Anal., 59 (2021), pp. 2500--2535], we establish two new regular decompositions for lowdimensional edge finite element functions in three dimensions, which, under suitable assumptions on the distribution of the coefficients, are stable up to a polylogarithmic factor of the meshwidth with respect to weighted norms defined by the coefficients. Using these regular decompositions, we analyze the convergence of the HX preconditioner for the case of strongly discontinuous coefficients. We show that the HX preconditioner is asymptotically optimal up to a polylogarithmic factor in the meshwidth and is not severely affected by large jumps of the coefficients across the interface between two neighboring subdomains. [ABSTRACT FROM AUTHOR]

Probabilistic domain decomposition (PDD) is an alternative paradigm for solving boundary value problems (BVPs) in parallel with excellent scalability properties, thanks to its reliance on stochastic representations of the BVP. However, there are cases when the latter is less numerically convenient, or unknown. Semilinear elliptic BVPs and the Helmholtz equation are prominent examples of either class. In this paper, we overcome this issue by designing suitable iterative schemes for either problem. These schemes not only retain the desirable properties of PDD but also are optimally suited for pathwise variance reduction, resulting in a systematic, nearly cost-free reduction of the statistical error through the iterations. Numerical tests carried out on the supercomputer Marconi100 are presented. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms, and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such an implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators. [ABSTRACT FROM AUTHOR]

This paper considers an inverse problem on determining coefficients of piecewise homogeneous piezoelectric equations from a localized Dirichlet-to-Neumann map on partial boundaries. Assume the bounded domain can be divided into finite subdomains, in which the unknown coefficients including the anisotropic elastic tensor, the piezoelectric tensor, and the dielectric tensor are constants. In this paper, two different cases are considered: the subdomains are either known and Lipschitz or unknown and subanalytic. For both cases, the unknown coefficients can be uniquely determined from a given localized Dirichlet-to-Neumann map. [ABSTRACT FROM AUTHOR]

Boundary value problems for diffusion in singularly perturbed domains is a topic of considerable current interest. Applications include intracellular diffusive transport and the spread of pollutants or heat from localized sources. In a previous paper, we introduced a new method for characterizing the approach to steady state in the case of two-dimensional (2D) diffusion. This was based on a local measure of the relaxation rate known as the accumulation time T(x). The latter was calculated by solving the diffusion equation in Laplace space using a combination of matched asymptotics and Green's function methods. We thus obtained an asymptotic expansion of T(x) in powers of \nu = 1/ln \ε, where \ε specifies the relative size of the holes. In this paper, we develop the corresponding theory for three-dimensional (3D) diffusion. The analysis is a nontrivial extension of the 2D case due to differences in the singular nature of the Laplace transformed Green's function. In particular, the asymptotic expansion of the solution of the 3D diffusion equation in Laplace space involves terms of order O((\epsilon/s)n), where s is the Laplace variable. These s-singularities have to be removed by partial series resummations in order to obtain an asymptotic expansion of T(x) in powers ofε. [ABSTRACT FROM AUTHOR]

Inspired by applications, we study the effect of interface slip on the effective wave propagation in poroelastic materials, which are composites consisting of elastic frames whose pore space is filled with fluid. The current literature on the homogenization for the poroelastic wave equations are all based on the no-slip interface condition posed on the microscale. However, for certain pore fluids, the no-slip condition is known to be physically invalid. In the literature, slip boundary conditions have been considered for porous materials with rigid solid frames. For these rigid porous materials, the wave can only propagate in the pore fluid and hence the equations for the microscale are posed only in the pore space. Consequently, the slip on the interface involves only the fluid velocity and the fluid stress. In contrast, for poroelastic materials, the wave can propagate not only in the pore fluid but also in the solid frame; hence the slip conditions involve the velocities on both sides of the interface, rather than just the fluid side. With this slip condition, a variational boundary value problem governing the small vibrations of a periodic mixture of an elastic solid and a slightly viscous fluid is studied in this paper. The method of two-scale convergence is used to obtain the macroscopic behavior of the solution and to identify the role played by the slip interface condition. [ABSTRACT FROM AUTHOR]

This paper describes a locally corrected trapezoidal quadrature method for the discretization of singular and hypersingular boundary integral operators (BIOs) that arise in solving boundary value problems for elliptic partial differential equations. The quadrature is based on a uniform grid in parameter space coupled with the standard punctured trapezoidal rule. A key observation is that the error incurred by the singularity in the kernel can be expressed exactly using generalized Euler--Maclaurin formulas that involve the Riemann zeta function in 2 dimensions (2D) and the Epstein zeta functions in 3 dimensions (3D). These expansions are exploited to correct the errors via local stencils at the singular point using a novel systematic moment-fitting approach. This new method provides a unified treatment of all common BIOs (Laplace, Helmholtz, Stokes, etc.). We present numerical examples that show convergence of up to 32nd-order in 2D and 9th-order in 3D with respect to the mesh size. [ABSTRACT FROM AUTHOR]

This paper is concerned with the state-constrained optimal control of the three-dimensional thermistor problem, a fully quasi-linear coupled system of a parabolic and elliptic PDEs with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Local existence, uniqueness, and continuity for the state system are derived by employing maximal parabolic regularity in the fundamental theorem of Prüss. Global solutions and controls admitting such are addressed and existence of optimal controls is shown if the temperature gradient is under control. This work is the rst of two papers on the three-dimensional thermistor problem. [ABSTRACT FROM AUTHOR]

Optimal balance is a nonasymptotic numerical method for computing a point on an elliptic slow manifold for two-scale dynamical systems with strong gyroscopic forces. It works by solving a modified differential equation as a boundary value problem in time, where the nonlinear terms are adiabatically ramped up from zero to the fully nonlinear dynamics. A dedicated boundary value solver, however, is often not directly available. The most natural alternative is a nudging solver, where the problem is repeatedly solved forward and backward in time and the respective boundary conditions are restored whenever one of the temporal end points is visited. In this paper, we show quasi-convergence of this scheme in the sense that the termination residual of the nudging iteration is as small as the asymptotic error of the method itself, i.e., under appropriate assumptions exponentially small. This confirms that optimal balance in its nudging formulation is an effective algorithm. Further, it shows that the boundary value problem formulation of optimal balance is well posed up to at most a residual error as small as the asymptotic error of the method itself. The key step in our proof is a careful two-component Gronwall inequality. [ABSTRACT FROM AUTHOR]

In this paper new innovative fourth order compact schemes for Robin and Neumann boundary conditions have been developed for boundary value problems of elliptic PDEs in two and three dimensions. Different from the traditional finite difference operator approach, which may not work for the flux type of boundary conditions, carefully designed undetermined coefficient methods are utilized in developing high order compact (HOC) schemes. The new methods not only can be utilized to design HOC schemes for flux type boundary conditions but also can be applied to general elliptic PDEs including Poisson, Helmholtz, diffusion-advection, and anisotropic equations with linear boundary conditions. In the newly developed HOC methods, the coefficient matrices are generally M-matrices, which guarantee the discrete maximum principle for well-posed problems, so the convergence of the HOC methods. The developed HOC methods are versatile and can cover most HOC schemes in the literature. The HOC methods for Robin boundary conditions and for anisotropic diffusion and advection equations with Robin or even Dirichlet boundary conditions are likely the first ones that have ever been developed. With the help of pseudoinverse or singular value decomposition, we have also observed that the developed HOC methods usually have smaller error constants compared with traditional HOC methods when applicable. Nontrivial examples with large wave numbers and oscillatory solutions are presented to confirm the performance of the new HOC methods. [ABSTRACT FROM AUTHOR]

Preconditioning for Krylov methods often relies on operator theory when mesh independent estimates are looked for. The goal of this paper is to contribute to the long development of the analysis of superlinear convergence of Krylov iterations when the preconditioned operator is a compact perturbation of the identity. Mesh independent superlinear convergence of GMRES and CGN iterations is derived for Galerkin solutions for complex non-Hermitian and noncoercive operators. The results are applied to noncoercive boundary value problems, including shifted Laplacian preconditioners for Helmholtz problems. [ABSTRACT FROM AUTHOR]

In this paper we prove uniqueness in the inverse boundary value problem for the three coefcient functions in the porous medium equation with an absorption term (um) + uq = 0 with m > 1, m 1 < q < m, with the space dimension 2 or higher. This is a degenerate parabolic type quasilinear PDE which has been used as a model for phenomena in felds such as gas fow (through a porous medium), plasma physics, and population dynamics. In the case when = 1 a priori, we prove unique identifability with data supported in an arbitrarily small part of the boundary. Even for the global problem we improve on previous work by obtaining uniqueness with a fnite (rather than infnite) time of observation and also by introducing the additional absorption term uq. [ABSTRACT FROM AUTHOR]

This paper studies the problem of exponential tracking and disturbance rejection for nonlinear reaction convection difusion equations via Dirichlet or Neumann boundary control. Disturbances are present both in the equation and on the boundary. The tracking problem is simply a nonlinear dynamic regulator problem. If an initial condition is not compatible with a reference, the nonlinear dynamic regulator problem does not have a solution. Thus, we split the tracking problem into two problems to relax the compatibility condition. We choose one problem to be exponentially stable and leave everything else in a dynamic regulator problem. Then a controller is just a solution component of the dynamic regulator problem, while the tracking error is given by the solution of the exponentially stable problem. The controller can be obtained by integrating the dynamic regulator equation and using the reference condition. The dynamic regulator problem with the obtained controller becomes an initial boundary value problem. We then transfer the initial boundary value problem into an equivalent integral equation and prove that the integral equation has a unique global solution if the nonlinearity is globally Lipschitz. In addition, a numerical example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with uniqueness of solution of the inverse diffraction by problems by a periodic curve with Dirichlet boundary condition in two dimensions. It is proved that the periodic curve can be uniquely determined by the near-field measurement data corresponding to infinitely many incident plane waves with distinct directions at a fixed frequency. Our proof is based on Schiffer's idea which consists of two ingredients: (i) the total fields for incident plane waves with distinct directions are linearly independent, and (ii) for a fixed wave number there exist only finitely many linearly independent Dirichlet eigenfunctions in a bounded domain or in a closed waveguide under additional assumptions on the waveguide boundary. Based on the Rayleigh expansion, we prove that the phased near-field data can be uniquely determined by the phaseless near-field data in a bounded domain, with the exception of a finite set of incident angles. Such a phase retrieval result leads to a new uniqueness result for the inverse grating diffraction problem with phaseless near-field data at a fixed frequency. Since the incident direction determines the quasi-periodicity of the boundary value problem, our inverse issues are different from the existing results of [F. Hettlich and A. Kirsch, Inverse Problems, 13 (1997), pp. 351--361], where fixed-direction plane waves at multiple frequencies were considered. [ABSTRACT FROM AUTHOR]

In this paper, a parallel probabilistic method using the Feynman--Kac formula of killed Brownian motions is proposed to solve the mixed boundary value problems (BVPs) of 3D Laplace equations. To avoid using reflecting Brownian motions and the calculation of their local time L(t) in the Feynman--Kac representation of solutions for Neumann and Robin BVPs, the proposed method uses an iterative approach to approximate the solutions where each iteration will use the Feynman--Kac formula to solve a pure Dirichlet problem, thus only involving killed Brownian motions. First, the boundary of the domain is decomposed with overlapping local patches formed by the intersection of hemispheres superimposed on the domain boundary. The iteration starts with an arbitrary initial guess for the Dirichlet data on the Neumann and Robin boundaries; then, using the Feynman--Kac formula for a pure Dirichlet problem with the current available Dirichlet data on the whole boundary, the solution over the hemispheres can be obtained by the Feynman--Kac formula for the killed Brownian motions, sampled by a walk-on-spheres (WOS) algorithm. Second, by solving a local boundary integral equation (BIE) over each hemisphere and a local patch on the domain boundary, the Dirichlet data on the Neumann and Robin boundaries can be updated. By continuing this process, the proposed hybrid probabilistic and deterministic BIE-WOS method gives a highly parallel algorithm for the global solution of any mixed-type BVPs of the Laplace equations. Numerical results of various mixed interior and exterior BVPs demonstrate the parallel efficiency and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a numerical method to solve the classic L²-optimal transport problem. Our algorithm is based on the use of multiple shooting, in combination with a continuation procedure, to solve the boundary value problem associated to the transport problem. Based on the viewpoint of Wasserstein Hamiltonian flow with initial and target densities, our algorithm reflects the Hamiltonian structure of the underlying problem and exploits it in the numerical discretization. Several numerical examples are presented to illustrate the performance of the method. [ABSTRACT FROM AUTHOR]

This paper studies the perfectly matched layer (PML) method for wave scattering in a half space of a homogeneous medium bounded by a two-dimensional, perfectly conducting, and locally defected periodic surface, and develops a high-accuracy boundary integral equation (BIE) solver. Along the vertical direction, we place a PML to truncate the unbounded domain onto a strip and prove that the PML solution converges to the true solution in the physical subregion of the strip with an error bounded by the reciprocal PML thickness. Laterally, we divide the unbounded strip into three regions: one region containing the defect and two semi-waveguide regions, separated by two vertical line segments. In both semi-waveguides, we prove the well-posedness of an associated scattering problem so as to well define a Neumann-to-Dirichlet (NtD) operator on the associated vertical segment. The two NtD operators, serving as exact lateral boundary conditions, reformulate the unbounded strip problem as a boundary value problem over the defected region. Due to the periodicity of the semi-waveguides, the two NtD operators are related to two different Neumann-marching operators, and either Neumann-marching operator is governed by a nonlinear Riccati equation. It is proved that the Neumann-marching operators are contracting, so that the PML solution decays exponentially fast along both lateral directions. The consequences culminate in two opposite aspects. Negatively, the PML solution cannot converge exponentially to the true solution in the whole physical region of the strip. Positively, from a numerical perspective, the Riccati equations can now be efficiently solved by a recursive doubling procedure and a high-accuracy PML-based BIE method so that the boundary value problem on the defected region can be solved efficiently and accurately. Numerical experiments demonstrate that the PML solution converges exponentially fast to the true solution in any compact subdomain of the strip. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the stability of the steady normal shock structure in potential flows under an unsteady perturbation. The dynamic stability problem is formulated as the well-posedness problem of an initial boundary value problem of a nonlinear wave equation in a cornered space domain with a free boundary. The corner singularity is the essential difficulty and there is no result available even for the linear problem without the symmetry assumptions, i.e., \even" or \odd" traces vanish on the solid boundary, which allows extension from the cornered space domain to the half-space domain, as in the previous works. In this paper, we first obtain an existence result for the initial boundary value problem of linear hyperbolic equations of second order in a cornered space domain without such symmetry assumptions. The key idea is based on the construction of a new auxilliary problem, which allows us to reduce the linear problem to a new one that can be even extended to a half-space domain such that the existence of H² η-solutions can be established. However, due to the lack of the symmetry assumptions, the low regularity of the extended coefficients block us from obtaining the higher regularity of the solutions in the extended domain, which is necessary for the iteration to the nonlinear problem. In order to deal with it, new hyperbolic type and elliptic type estimates in the cornered space domain are established carefully. The results on the general linear problems can be applied to the linearized problem that we are concerned with in this paper. Due to the loss of regularity in the estimates of the linearized problem, a modified Nash-Moser iteration is developed. [ABSTRACT FROM AUTHOR]

The a posteriori error estimate and greedy algorithms play central roles in the reduced basis method (RBM). In [M. Yano, Comput. Methods Appl. Mech. Engrg., 287 (2015), pp. 290--309; ESAIM Math. Model. Numer. Anal., 50 (2016), pp. 163--185; SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], several versions of RBMs based on exact error certifications and greedy algorithms with spatio-parameter adaptivities are developed. In this paper, with the parametric symmetric coercive elliptic boundary value problem as an example of the primal-dual variational problems satisfying the strong duality, we develop primal-dual RBMs (PD-RBM) with robust true error certifications and discuss three versions of greedy algorithms to balance the finite element error, the exact RB error, and the adaptive mesh refinements. For a class of convex minimization variational problems which has corresponding dual problems satisfying the strong duality, the primal-dual gap between the primal and dual functionals can be used as an a posteriori error estimator. This primal-dual gap error estimator is robust with respect to the parameters of the problem, and it can be used for both mesh refinements of finite element methods and the true RB error certification. With the help of an integration by parts formula, the primal-dual variational theory is developed for the symmetric coercive elliptic boundary value problems with nonhomogeneous boundary conditions by both the conjugate function and Lagrangian theories. A generalized Prager--Synge identity, which is the primal-dual gap error representation for this specific problem, is developed. RBMs for both the primal and dual problems with robust error estimates are developed. The dual variational problem often can be viewed as a constraint optimization problem. In the paper, different from the standard saddlepoint finite element approximation, the dual RBM is treated as a Galerkin projection by constructing RB spaces satisfying the homogeneous constraint. Inspired by the greedy algorithm with spatioparameter adaptivity of [M. Yano, SIAM J. Sci. Comput., 40 (2018), pp. A388--A420], adaptive balanced greedy algorithms with primal-dual finite element and RB error estimators are discussed. Numerical tests are presented to test the PD-RBM with adaptive balanced greedy algorithms. [ABSTRACT FROM AUTHOR]

Fourier extension is an approximation scheme in which a function on an arbitrary bounded domain is approximated using classical Fourier series on a bounding box. On the domain of interest the Fourier series exhibits redundancy, and it has the mathematical structure of a frame rather than a basis. It is not trivial to construct approximations in this frame using function evaluations in points that belong to the domain only, but one way to do so is through a discrete least squares approximation. The corresponding system is extremely ill-conditioned, due to the redundancy in the frame, yet its solution via a regularized singular value decomposition is known to be accurate to very high (and nearly spectral) precision. Still, this computation requires Q (N3) operations. In this paper we describe an algorithm to compute such Fourier extension frame approximations in only Q (N2 log2 N) operations for general 2D domains. The cost improves to Q (N log2 N) operations for simpler tensor-product domains. The algorithm exploits a phenomenon called the plunge region in the analysis of time-frequency localization operators, which manifests itself here as a sudden drop in the singular values of the least squares matrix. It is known that the size of the plunge region scales like Q (logN) in 1D problems. In this paper we show that for most 2D domains in the fully discrete case the plunge region scales like Q (N logN), proving a discrete equivalent of a result that was conjectured by Widom for a related continuous problem. The complexity estimate depends on the Minkowski or box-counting dimension of the domain boundary, and as such it is larger than Q (N logN) for domains with fractal shape. [ABSTRACT FROM AUTHOR]

The problem of output feedback boundary stabilization is considered for n coupled plants, distributed over the one-dimensional spatial domain [0; 1] where they are governed by linear reaction-diffusion partial differential equations (PDEs). All plants have constant parameters and each is equipped with its own scalar boundary control input, acting at one end of the domain. First, a state feedback law is designed to exponentially stabilize the closed-loop system with an arbitrarily fast convergence rate. Then, collocated and anticollocated observers are designed, using a single boundary measurement for each plant. The exponential convergence of the observed state towards the actual one is demonstrated for both observers, with a convergence rate that can be made as fast as desired. Finally, the state feedback controller and the preselected, either collocated or anticollocated, observer are coupled together to yield an output feedback stabilizing controller. The distinct treatments are proposed separately for the case in which all processes have the same diffusivity and for the more challenging scenario where each process has its own diffusivity. The backstepping method is used for both controller and observer designs. Two main classes of coupled PDEs are studied in the paper. In the first one, all processes possess a Dirichlet-type boundary condition (BC) at the uncontrolled side. With reference to this class, the state feedback and observer-based output feedback designs are successfully solved in both the equi-diffusivity and distinct-diffusivity scenarios, and, particularly, the kernel matrices of the underlying transformations are derived in analytical form by using the method of successive approximations to solve the corresponding kernel PDEs. Thus, the resulting control laws and observers become available in explicit form. The second and more general class of coupled PDEs considered in the paper entails a subset of the processes having Dirichlet-type BCs at the uncontrolled side, whereas all remaining processes possess Neumann-type BCs. With reference to this wider class of systems with heterogenous BCs, it turns out that the state feedback design can only be solved in the equi-diffusivity case, although the resulting kernel matrix is no longer available in explicit form, whereas the same approach yields an overdetermined kernel PDE admitting no solution in the distinct-diffusivity case. Anticollocated observer design and output feedback designs are additionally developed in the equi-diffusivity scenario. Interestingly, the observer gains are still available in explicit form also in this case. Capabilities of the proposed synthesis and its effectiveness are supported by a numerical study made for two coupled systems with heterogeneous BCs. [ABSTRACT FROM AUTHOR]

In anisotropic media, the standard perfectly matched layer (PML) technique suffers irrevocable instability in terminating the unbounded problem domains. It remains an open question whether a stable PML-like absorbing boundary condition exists. For wave scattering in a layered orthotropic medium, this question is affirmatively answered for the first time in this paper. In each orthotropic medium, the permittivity tensor uniquely determines a change of coordinates that transforms the governing anisotropic Helmholtz equation into an isotropic Helmholtz equation in the new coordinate system. This leads us to propose a novel Sommerfeld radiation condition (SRC) to rigorously characterize outgoing waves in the layered orthotropic medium. Naturally, the SRC motivates a regionalized PML (RPML) to truncate the scattering problem, in the sense that a standard PML is set up in the new coordinate system in each orthotropic region. It is revealed that the RPML is unconditionally stable compared with the unstable uniaxial PML. A high-accuracy boundary integral equation (BIE) method is developed to solve the resulting boundary value problem. Numerical experiments are carried out to validate the stability of the RPML and the accuracy of the BIE method, showing exponentially decaying truncation errors as the RPML parameters increase. [ABSTRACT FROM AUTHOR]

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realized through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in two dimensions) or two cubes (in three dimensions), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analyzing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle. [ABSTRACT FROM AUTHOR]

We study the equation ut = [φ(u)]xx +ε[ψ(u)]txx with suitable boundary conditions and a nonnegative Radon measure as initial datum. Here φ(0) = φ(∞) = 0, φ is increasing in (0, α) and decreasing in (α,∞), and the regularizing term ψ is increasing but bounded. It is natural to study measure-valued solutions since singularities may appear spontaneously in finite time. Positive measure-valued solutions are known to exist and to not be unique. In this paper we study qualitative properties shared by all solutions of the problem. We prove, among other things, that the singular part of a solution is nondecreasing with respect to time, so its support is nonshrinking, and, due to the possible appearance of singularities, may even expand. This phenomenon sharply distinguishes the case of bounded ψ from those of power-type ψ, where the singular part remains constant in time, and logarithmic ψ, where the singular part may grow but its support does not expand. It also distinguishes the present case from the case of φ increasing in (0, α), decreasing in (α, β), increasing in (β,∞) for some 0 < α < β < ∞, and bounded (with ψ as in this paper), where the singular part of a solution is nonincreasing in time and singularities may disappear. [ABSTRACT FROM AUTHOR]

A delay Lyapunov matrix corresponding to an exponentially stable system of linear time-invariant delay differential equations can be characterized as the solution of a boundary value problem involving a matrix valued delay differential equation. This boundary value problem can be seen as a natural generalization of the classical Lyapunov matrix equation. Lyapunov matrices play an important role in constructing Lyapunov functionals and in H2 optimal control. In this paper we present a general approach for computing delay Lyapunov matrices and H2 norms for systems with multiple discrete delays, whose applicability extends toward problems where the matrices are large and sparse, and the associated positive semidefinite matrix has a low rank. The problems addressed are challenging, because the boundary value problem is matrix valued with a structure that is much harder to exploit than in the delay-free case, and its solution is in the generic situation nonsmooth. In contrast to existing methods that are based on solving the boundary value problem directly, our method is grounded in solving standard Lyapunov equations of increased dimensions. It combines several ingredients: (i) a spectral discretization of the system of delay equations, (ii) a targeted similarity transformation which induces a desired structure and sparsity pattern and, at the same time, favors accurate low-rank solutions of the corresponding Lyapunov equation, and (iii) a Krylov method for large-scale matrix Lyapunov equations. The structure of the problem is exploited in such a way that the final algorithm does not involve a preliminary discretization step and provides a fully dynamic construction of approximations of increasing rank. Interpretations are also given in terms of a projection method directly applied to a standard linear infinite-dimensional system, which is equivalent to the original time-delay system. Throughout the paper two didactic examples are used to illustrate the properties of the problem, the challenges and methodological choices, while numerical experiments are reported to illustrate the effectiveness of the algorithms. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate mathematically the fundamental mechanism of gas discharge. Townsend discovered that α- and γ-mechanisms are essential for a gas ionization process. Morrow, Degond, and Lucquin-Desreux derived a mathematical model taking these two mechanisms into account. This model consists of nonlinear hyperbolic, parabolic, and elliptic partial differential equations. In this paper, we establish a framework for analyzing this model rigorously, because no mathematical result has been announced. More precisely, we show the unique existence of a time-local solution to an initial-boundary value problem over an unbounded domain. The main difficulty of this problem lies in the fact that the direction of boundary characteristics of a hyperbolic equation degenerates at infinite distance. Generally speaking, the regularity of solutions to linearized hyperbolic equations with degenerate boundary characteristics may be lost near the boundary. This implies the possibility that loss on the derivatives arises at each step of the inductive scheme to solve the nonlinear problem. We first use the weighted Sobolev spaces in the construction of solutions to make clear the direction of characteristics. Furthermore, we reduce the initial-boundary value problem for the hyperbolic equation to an initial value problem by applying several extension operators. This reduction enables us to avoid the loss on the derivatives. [ABSTRACT FROM AUTHOR]

In this paper we provide a generalization of the Douglas--Rachford splitting (DRS) and the primal-dual algorithm [L. Condat, J. Optim. Theory Appl., 158 (2013), pp. 460-479; B. C. V\~u, Adv. Comput. Math., 38 (2013), pp. 667-681] for solving monotone inclusions in a real Hilbert space involving a general linear operator. The proposed method allows for primal and dual nonstandard metrics and activates the linear operator separately from the monotone operators appearing in the inclusion. In the simplest case when the linear operator has full range, it reduces to classical DRS. Moreover, the weak convergence of primal-dual sequences to a Kuhn--Tucker point is guaranteed, generalizing the main result in [B. F. Svaiter, SIAM J. Control Optim., 49 (2011), pp. 280-287]. Inspired by [D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems, M. Fortin and R. Glowinski, eds., Stud. Math. Appl. 15, North-Holland, Amsterdam, 1983, pp. 299-331], we also derive a new split alternating direction method of multipliers (SADMM) by applying our method to the dual of a convex optimization problem involving a linear operator which can be expressed as the composition of two linear operators. The proposed SADMM activates one linear operator implicitly and the other one explicitly, and we recover ADMM when the latter is set as the identity. Connections and comparisons of our theoretical results with respect to the literature are provided for the main algorithm and SADMM. The flexibility and efficiency of both methods is illustrated via numerical simulations in total variation image restoration and a sparse minimization problem. [ABSTRACT FROM AUTHOR]

In this paper we prove well-posedness for a measure-valued continuity equation with solution-dependent velocity and ux boundary conditions, posed on a bounded one-dimensional domain. We generalize the results of an earlier paper [J. Differential Equations, 259 (2015), pp. 1068- 1097] to settings where the dynamics are driven by interactions. In a forward-Euler-like approach, we construct a time-discretized version of the original problem and employ those results as a building block within each subinterval. A limit solution is obtained as the mesh size of the time discretization goes to zero. Moreover, the limit is independent of the specific way of partitioning the time interval [0; T]. This paper is partially based on results presented in Chapter 5 of [Evolution Equations for Systems Governed by Social Interactions, Ph.D. thesis, Eindhoven University of Technology, 2015], while a number of issues that were still open there are now resolved. [ABSTRACT FROM AUTHOR]

In this paper, we address the temporal energy growth associated with numerical approximations of the perfectly matched layer (PML) for Maxwell's equations in first order form. In the literature, several studies have shown that a numerical method which is stable in the absence of the PML can become unstable when the PML is introduced. We demonstrate in this paper that this instability can be directly related to numerical treatment of boundary conditions in the PML. First, at the continuous level, we establish the stability of the constant coefficient initial boundary value problem for the PML. To enable the construction of stable numerical boundary procedures, we derive energy estimates for the PML in the Laplace space. Second, we develop a high order accurate and stable numerical approximation for the PML using summation-by-parts finite difference operators to approximate spatial derivatives and weak enforcement of boundary conditions using penalties. By constructing analogous discrete energy estimates we show discrete stability and convergence of the numerical method. Numerical experiments verify the theoretical results. [ABSTRACT FROM AUTHOR]

In a series of recent papers [B. Adcock, A. C. Hansen, and C. Poon, Appl. Comput. Harmon. Anal., 36 (2014), pp. 387-415; B. Adcock, M. Gataric, and A. C. Hansen, SIAM J. Imaging Sci., 7 (2014), pp. 1690-1723; Adcock et al., SIAM J. Math. Anal., 47 (2015), pp. 1196-1233], it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its Fourier transform via the method of generalized sampling. While these papers focused on the optimality of generalized sampling in terms of its stability and error bounds, the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm. In particular, we show that generalized sampling has a computational complexity of O (M(N) logN) when recovering the first N boundary-corrected wavelet coefficients of an unknown compactly supported function from M(N) Fourier samples. Therefore, due to the linear correspondences between the number of samples M and the number of coefficients N shown previously, generalized sampling offers a computationally optimal way of recovering wavelet coefficients from Fourier data. [ABSTRACT FROM AUTHOR]

We consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method. The collocation method can be applied in two versions: the finite element method and the spectral element method. We give convergence results for the collocation method deduced by the convergence theory developed in [S. Maset, Numer. Math., (2015), pp. 1-31] for a general discretization of an abstract reformulation of the problems. Such convergence results are then applied in Part II [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794-2821] of this paper to boundary values problems for a particular type of neutral functional differential equations, namely, differential equations with deviating arguments. [ABSTRACT FROM AUTHOR]

Shape gradients have been widely used in numerical shape gradient descent algorithms for shape optimization. The two types of shape gradients, i.e., the distributed one and the boundary type, are equivalent at the continuous level but exhibit different numerical behaviors after finite element discretization. To be more specific, the boundary type shape gradient is more popular in practice due to its concise formulation and convenience in combining with shape optimization algorithms but has lower numerical accuracy. In this paper we provide a simple yet useful boundary correction for the normal derivatives of the state and adjoint equations, motivated by their continuous variational forms, to increase the accuracy and possible effectiveness of the boundary shape gradient in PDE-constrained shape optimization. We consider particularly the state equation with Dirichlet boundary conditions and provide a preliminary error estimate for the correction. Numerical results show that the corrected boundary type shape gradient has comparable accuracy to that of the distributed one. Moreover, we give a theoretical explanation for the comparable numerical accuracy of the boundary type shape gradient with that of the distributed shape gradient for Neumann boundary value problems. [ABSTRACT FROM AUTHOR]

We consider a vector-Laplace problem posed on a two-dimensional surface embedded in a three-dimensional domain, which results from the modeling of surface fluids based on exterior Cartesian differential operators. The main topic of this paper is the development and analysis of a finite element method for the discretization of this surface partial differential equation. We apply the trace finite element technique, in which finite element spaces on a background shape-regular tetrahedral mesh that is surface independent are used for discretization. In order to satisfy the constraint that the solution vector field is tangential to the surface we introduce a Lagrange multiplier. We show well-posedness of the resulting saddle point formulation. A discrete variant of this formulation is introduced which contains suitable stabilization terms and is based on trace finite element spaces. For this method we derive optimal discretization error bounds. Furthermore algebraic properties of the resulting discrete saddle point problem are studied. In particular an optimal Schur complement preconditioner is proposed. Results of a numerical experiment are included. [ABSTRACT FROM AUTHOR]

The improved Boussinesq equation is studied in this paper. Control properties for this equation posed on a bounded interval are first considered. When the control acts through the Dirichlet boundary condition the linearized system is proved to be approximately but not spectrally controllable. In a second part, the equation is posed on the one-dimensional torus and distributed moving controls are considered. Under some condition on the velocity at which the control moves, exact controllability results for both linear and nonlinear improved Boussinesq equations are obtained applying the moment method and a fixed point argument. [ABSTRACT FROM AUTHOR]

The objectives of this paper are threefold. First, the paper defines a staged construction paradigm for the necessary conditions for locally optimal solutions to integro-differential boundary-value problems that mirrors the natural decomposition of the original constraints. Second, it details a library of realizations of the zero functions, monitor functions, and contributions to the adjoint equations associated with each stage of problem construction. Third, it illustrates the application of this framework to multisegment trajectory problems, for example, in the analysis of spacecraft optimal control problems. The proposed construction paradigm generalizes a formulation for the construction of integro-differential-algebraic continuation problems in the software package COCO to also include automatically generated adjoint terms. The implementation enables arbitrarily many levels of nested construction with larger problems assembled from smaller ones, and multiple instances of a problem class combined inside a composite problem with the help of suitably defined coupling conditions. The application of this paradigm to problems of constrained optimization relies on the successive continuation paradigm introduced by Kernévez and Doedel [Optimization in bifurcation problems using a continuation method, in Bifurcation: Analysis, Algorithms, Applications, Birkhäuser Verlag, Basel, 1987, pp. 153-160], in which solutions to the necessary conditions for locally optimal solutions are found at the end of a sequence of easily initialized separate stages of continuation. The numerical examples illustrate the paradigm of construction and the efficacy of the continuation approach in locating stationary solutions to high-dimensional problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the boundary integral equation method for solving the exterior Neumann boundary value problem of dynamic poroelasticity in two dimensions. The main contribution of this work consists of two aspescts: the proposal of a novel regularized boundary integral equation, and the presentation of new regularized formulations of the strongly singular and hyper-singular boundary integral operators. First, turning to the spectral properties of the doublelayer operator and the corresponding Calder\'on relation of the poroelasticity, we propose the novel low-GMRES-iteration integral equation whose eigenvalues are bounded away from zero and infinity. Second, with the help of the classical idea of using G\"unter derivatives and integration-by-parts, we reformulate the strongly singular and hyper-singular integral operators into combinations of the weakly singular operators and the tangential derivatives. The accuracy and efficiency of the proposed methodology are demonstrated through several numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space L²(0,T; H-1(Ω)). By duality, the related norm can be evaluated by means of the solution of an elliptic quasi-stationary boundary value problem. When eliminating the control, we end up with the reduced optimality system that is nothing but the variational formulation of the coupled forward-backward primal and adjoint equations. Using Babuska's theorem, we prove unique solvability in the continuous case. Furthermore, we establish the discrete inf-sup condition for any conforming space-time finite element discretization yielding quasi-optimal discretization error estimates. Various numerical examples confirm the theoretical findings. We emphasize that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an L²-regularization and with a sparse optimal control approach. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study of nonlinear Robin-type boundary value problems of second-order elliptic partial differential equations. Specifically, nonlinear boundary conditions are modeled by hemivariational inequalities, and the coefficients of the governing equations have a small periodic oscillating structure. We prove that the model problem can be homogenized; i.e., the sequence of solutions weakly converges to a homogenized solution as the periodicity approaches zero. We derive an O(∊1/2)-order estimate in H¹-norm for first-order asymptotic expansions. In particular, we provide an optimal O(∊)-order estimate in L²-norm for linear Robin boundary value problems. We analyze the numerical error of the Lagrange finite element method for the model problem and design a numerical experiment to validate the theoretical predictions. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to study a large class of differential variational-hemivariational inequalities involving history-dependent operators and constraints in a Banach space. First, we establish a well-posedness result, which includes existence, uniqueness, and continuous dependence on the initial data. Second, related penalized and regularized problems without constraints are introduced whose solutions approach the solution to the original inequality. Finally, these results are applied to an obstacle parabolic-elliptic system consisting of a nonlinear reaction-diffusion equation and a time-dependent mixed boundary value problem with generalized gradient and Volterra integral terms. [ABSTRACT FROM AUTHOR]

In many applications of practical interest, solutions of partial differential equation models arise as critical points of an underlying (energy) functional. If such solutions are saddle points, rather than being maxima or minima, then the theoretical framework is nonstandard, and the development of suitable numerical approximation procedures turns out to be highly challenging. In this paper, our aim is to present an iterative discretization methodology for the numerical solution of nonlinear variational problems with multiple (saddle point) solutions. In contrast to traditional numerical approximation schemes, which typically fail in such situations, the key idea of the current work is to employ a simultaneous interplay of a previously dev eloped local minimax approach and adaptive Galerkin discretizations. We thereby derive an adaptive local minimax Galerkin (LMMG) method, which combines the search for saddle point solutions and their approximation in finitedimensional spaces in a highly effective way. Under certain assumptions, we will prove that the generated sequence of approximate solutions converges to the solution set of the variational problem. This general framework will be applied to the specific context of finite element discretizations of (singularly perturbed) semilinear elliptic boundary value problems, and a series of numerical experiments will be presented. [ABSTRACT FROM AUTHOR]

The Patlak--Keller--Segel model can be used to model the nonlocal aggregation phenomena in the collective motion of cells or the evolution of the density of bacteria by chemotaxis. We consider the free boundary value problem for the Patlak--Keller--Segel model with homogeneous nonlinear degenerate diffusion in the present paper, which can be used to simulate the congested phenomena and the dynamical behaviors of cell motion with finite total mass and compactly supported density distribution. For the subcritical case, we prove that the cell density function exists globally in time and tends to the corresponding steady-state at an exponential time rate due to the balance between the nonlinear diffusion effect and nonlocal aggregation. For the supercritical case, we show that the global solution for the cell density exists and converges algebraically in time to the Barenblatt solution of the corresponding porous media equation due to the diffusion dominating mechanism. This provides a different viewpoint on the dynamical behaviors of the congested phenomena subject to the combined influences of nonlinear diffusion and nonlocal aggregation. [ABSTRACT FROM AUTHOR]

In this paper we analyze the optimal control of initial boundary value problems for entropy solutions of scalar hyperbolic balance laws with pointwise state constraints. Here, we suppose that the initial and the boundary data switch between different C¹-functions at certain switching points, where the C¹-functions and the switching points are considered as the control. For a class of cost functionals, we prove first order necessary optimality conditions for the corresponding optimal control problem with state constraints. Furthermore, we use a Moreau–Yosida type regularization to approximate the optimal control problem with state constraints. We derive optimality conditions for the regularized problems and finally prove convergence to the solution of the optimal control problem with state constraints. [ABSTRACT FROM AUTHOR]

In this paper we consider a viscoelastic plate (linear viscoelasticity of the Maxwell- Boltzmann type) and we compare its controllability properties with the (known) controllability of a purely elastic plate (the control acts on the boundary displacement or bending). By combining operator and moment methods, we prove that the viscoelastic plate inherits the controllability properties of the purely elastic plate. [ABSTRACT FROM AUTHOR]

This paper investigates the global dynamics of a mean field model of the electroencephalogram developed by Liley, Cadusch, and Dafilis [Network, 13 (2002), pp. 67-113]. The model is presented as a system of coupled ordinary and partial differential equations with periodic boundary conditions. Existence, uniqueness, and regularity of weak and strong solutions of the model are established in appropriate function spaces, and the associated initial-boundary value problems are proved to be well- posed. Sufficient conditions are developed for the phase spaces of the model to ensure nonnegativity of certain quantities in the model, as required by their biophysical interpretation. It is shown that the semigroups of weak and strong solution operators possess bounded absorbing sets for the entire range of biophysical values of the parameters of the model. Challenges involved in establishing a global attractor for the model are discussed and it is shown that there exist parameter values for which the constructed semidynamical systems do not possess a compact global attractor due to the lack of the compactness property. Finally, using the theoretical results of the paper, instructive insights are provided into the complexity of the behavior of the model and computational analysis of the model. [ABSTRACT FROM AUTHOR]

This paper introduces three sets of sufficient conditions for generating bijective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it either maintains the orientation of all elements or flips all the elements. However, this condition is known to be insufficient for injectivity of a simplicial map. In this paper we provide additional simple conditions that, together with the above-mentioned necessary condition, guarantee injectivity of the simplicial map. The first set of conditions generalizes classical global inversion theorems to the mesh (piecewise-linear) case. That is, it proves that in the case where the boundary simplicial map is bijective and the necessary condition holds, the map is injective and onto the target domain. The second set of conditions is concerned with mapping of a mesh to a polytope. It replaces the (often hard) requirement of a bijective boundary map with a collection of linear constraints and guarantees that the resulting map is injective over the interior of the mesh and onto. These linear conditions provide a practical tool for optimizing a map of the mesh onto a given polytope while allowing the boundary map to adjust freely and keeping the injectivity property in the interior of the mesh. Allowing more freedom in the boundary conditions is useful for two reasons: (a) it circumvents the hard task of providing a bijective boundary map, and (b) it allows one to optimize the boundary map together with the simplicial map to achieve lower energy levels. The third set of conditions adds to the second set the requirement that the boundary maps be orientation preserving as well (with a proper definition of boundary map orientation). This set of conditions guarantees that the map is injective on the boundary of the mesh as well as its interior. Several experiments using the sufficient conditions are shown for mapping triangular meshes. A secondary goal of this paper is to advocate and develop the tool of degree in the context of mesh processing. [ABSTRACT FROM AUTHOR]

This paper provides a class of optimal algorithms for the linear algebraic systems arising from direct finite element discretization of the fourth-order equation with different boundary conditions on any polygonal domains that are partitioned by unstructured grids. Several preconditioners are presented, and the preconditioned conjugate gradient methods applied with these preconditioners are proved to converge uniformly with respect to the size of the linear systems. As a result, for the first time in the literature, it is rigorously proved in this paper that a discretized finite element system for fourth-order PDEs on an unstructured grid of size N can be solved within O(N log N) operations. One main ingredient in constructing these optimal preconditioners is a mixed-form discretization of the fourth-order problem. Such a mixed-form discretization often leads to an approximation of the original solution that is either suboptimal or divergent, but it nevertheless provides optimal preconditioners for the direct finite element problem. It is further shown that the implementation of the preconditioners can be reduced to the solution of several discrete Poisson equations. Consequently, the existing optimal or nearly optimal solver, such as geometric or algebraic multigrid methods, for Poisson equations will lead to a nearly optimal solver for the discrete fourth-order system. A number of nonstandard Sobolev spaces and their discretizations defined on the boundary of polygonal domains are carefully studied and used for the analysis of these preconditioners. Numerical examples are also provided to confirm the theory developed in the paper. [ABSTRACT FROM AUTHOR]

In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order O(τq +hr+1) in the L² norm, where q = 2-β or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, β (0 < β < 1) is the order of the fractional derivative, τ and h are the step sizes in time and space, respectively, and r is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones. [ABSTRACT FROM AUTHOR]

In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method benefits from a direct discretization approach and is called the "direct RBF partition of unity (D-RBF-PU)" method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, the discontinuous PU weight functions can now be utilized to develop the method in a more efficient and less expensive way. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular two- and three-dimensional domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method. [ABSTRACT FROM AUTHOR]

The Anderson–Witting model is a relativistic generalization of the Bhatnagar–Gross– Krook model which is well-known as the relaxation-time approximation of the celebrated Boltzmann equation. In this paper, we address the stationary boundary value problems to the Anderson–Witting model in slab geometry. We prove the existence of unique stationary solution to the boundary value problem of the Anderson–Witting model in a bounded interval with the inflow boundary data when the gas is sufficiently rarefied. [ABSTRACT FROM AUTHOR]