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In this paper, we consider an approach to improve the performance of exponential Runge–Kutta integrators and Lawson schemes in cases where the solution of a related, but usually much simpler, problem can be computed efficiently. While for implicit methods such an approach is common (e.g., by using preconditioners), for exponential integrators this has proven more challenging. Here we propose to extract a constant coefficient differential operator from the semilinear advection-diffusion-reaction equation for which, in many situations, efficient methods are known to compute the required matrix functions. Both a linear stability analysis and extensive numerical experiments show that the resulting schemes can be unconditionally stable. In fact, we find that exponential integrators of Runge–Kutta type and Lawson schemes can have better stability properties than similarly constructed implicit-explicit schemes. We also derive two new Lawson-type integrators that further improve on these stability properties. The overall effectiveness of the approach is highlighted by a number of performance comparisons on examples in two and three space dimensions. [ABSTRACT FROM AUTHOR]

This paper studies the convergence of mean field games (MFGs) with finite state space to MFGs with a continuous state space. We examine a space discretization of a diffusive dynamics, which is reminiscent of the Markov chain approximation method in stochastic control, but also of finite difference numerical schemes; time remains continuous in the discretization, and the time horizon is arbitrarily long. We are mainly interested in the convergence of the solution of the associated master equations as the number of states tends to infinity. We present two approaches, to treat the case without or with common noise, both under monotonicity assumptions. The first one uses the system of characteristics of the master equation, which is the MFG system, to establish a convergence rate for the master equations without common noise and the associated optimal trajectories, both when there is a smooth solution to the limit master equation and when there is not. The second approach relies on the notion of monotone solutions introduced by [8, 9]. In the presence of common noise, we show convergence of the master equations, with a convergence rate if the limit master equation is smooth, and by compactness arguments otherwise. [ABSTRACT FROM AUTHOR]

In this paper, we provide modulated interaction energy estimates for the kernel K(x) = ∣≈∣-α with a E (0,d) and its applications to quantified asymptotic analyses for kinetic equations. The proof relies on a dimension extension argument for an elliptic operator and its commutator estimates. For the applications, we first discuss the quantified small inertia limit of kinetic equations with singular nonlocal interactions. The aggregation equations with singular interaction kernels are rigorously derived. We also study the rigorous quantified hydrodynamic limit of the kinetic equation to derive the isothermal Euler or pressureless Euler system with the nonlocal singular interaction forces. [ABSTRACT FROM AUTHOR]

The paper gives a survey of several directions of research connected with the works of A. N. Kolmogorov on parabolic and elliptic Fokker--Planck--Kolmogorov equations for transition and stationary probabilities of diffusion processes. We present the fundamental results on existence of solutions, their uniqueness, and the properties of solution densities. Open questions in this area are mentioned. [ABSTRACT FROM AUTHOR]

The bidomain model is the standard model for cardiac electrophysiology. In this paper, we study the bidomain Allen--Cahn equation, in which the Laplacian of the classical Allen-- Cahn equation is replaced by the bidomain operator, a Fourier multiplier operator whose symbol is given by a homogeneous rational function of degree two. The bidomain Allen--Cahn equation supports planar front solutions much like the classical case. In contrast to the classical case, however, these fronts are not necessarily stable due to a lack of maximum principle; they can indeed become unstable depending on the parameters of the system. In this paper, we prove nonlinear stability and instability results for bidomain Allen--Cahn fronts on an infinite two-dimensional strip. We show that previously established spectral stability/instability results in L² imply stability/instability in the space of bounded uniformly continuous functions by establishing suitable decay estimates of the resolvent kernel of the linearized operator. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our accompanying paper [M. Talbi, N. Touzi, and J. Zhang, Dynamic Programming Equation for the Mean Field Optimal Stopping Problem, https://arxiv.org/abs/2103.05736, 2021], where we characterized the mean field optimal stopping problem by an obstacle equation on the Wasserstein space of probability measures, provided that the value function is smooth. Our purpose here is to establish this characterization under weaker regularity requirements. We shall define a notion of viscosity solutions for such an equation and prove existence, stability, and the comparison principle. [ABSTRACT FROM AUTHOR]

We prove that the approximate null-controllability with uniform cost of the hypoelliptic Ornstein-Uhlenbeck equations posed on Rn is characterized by an integral thickness geometric condition on the control supports. We also provide associated quantitative weak observability estimates. This result for the hypoelliptic Ornstein-Uhlenbeck equations is deduced from the same study for a large class of nonautonomous elliptic equations from moving control supports. We generalize in particular results known for parabolic equations posed on Rn, for which the approximate null-controllability with uniform cost is ensured by the notion of thickness, which is stronger than the integral thickness condition considered in the present paper. Examples of those parabolic equations are the fractional heat equations associated with the operator (-Δ)s in the regime s ≥ 1/2. Our strategy also allows us to characterize the approximate null-controllability with uniform cost from moving control supports for this class of fractional heat equations. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a novel efficient high-order finite volume method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the method simultaneously preserves two critical physical constraints: a discretely divergence-free (DDF) constraint on the magnetic field and the positivity-preserving (PP) property, which ensures the positivity of density, pressure, and internal energy. To enforce the DDF condition in each cell, we design a new discrete projection approach that projects the reconstructed point values at the cell interface into a DDF space, without using any approximation polynomials. This projection method is highly efficient, easy to implement, and particularly suitable for the high-order finite volume methods that return only the point values (no explicit approximation polynomials) in the reconstruction. Moreover, we also develop a new finite volume framework for constructing provably PP schemes for the ideal MHD system. The framework comprises the discrete projection technique, a suitable approximation to the Godunov--Powell source terms, and a simple PP limiter. We provide rigorous analysis of the PP property of the proposed finite volume method, demonstrating that the DDF condition and the proper approximation to the source terms eliminate the impact of magnetic divergence terms on the PP property. The analysis is challenging due to the internal energy function's nonlinearity and the intricate relationship between the DDF and PP properties. To address these challenges, we adopt the recently developed geometric quasilinearization approach [K. Wu and C.-W. Shu, SIAM Rev., 65 (2023), pp. 1031--1073], which transforms a nonlinear constraint into a family of linear constraints. Finally, we validate the effectiveness of the proposed method through several benchmark and demanding numerical examples. The results demonstrate that the proposed method is robust, accurate, and highly effective, confirming the significance of the proposed DDF projection and PP techniques. [ABSTRACT FROM AUTHOR]

In this paper, we present a second-order accurate and linear numerical scheme for the phase field crystal equation and prove its convergence in the discrete L\infty sense. The key ingredient of the error analysis is to justify the boundedness of the numerical solution, so that the nonlinear term, treated explicitly in the scheme, can be bounded appropriately. Benefiting from the existence of the sixth-order dissipation term in the model, we first estimate the discrete H2 norm of the numerical error. The error estimate in the supremum norm is then obtained by the Sobolev embedding, so that the uniform bound of the numerical solution is available. We also show the mass conservation and the energy stability in the discrete setting. The proposed scheme is linear with constant coefficients so that it can be solved efficiently via some fast algorithms. Numerical experiments are conducted to verify the theoretical results, and long-time simulations in two- and three-dimensional spaces demonstrate the satisfactory and high effectiveness of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

We present a priori estimates and unique solvability results in the mixed-norm Lebesgue spaces for the kinetic Kolmogorov-Fokker-Planck (KFP) equation in divergence form. The leading coefficients are bounded uniformly nondegenerate with respect to the velocity variable v and satisfy a vanishing mean oscillation (VMO) type condition. We consider the L2 case separately and treat more general equations, which include the relativistic KFP equation. This paper is a continuation of our previous work on Lp estimates for KFP equations in nondivergence form. [ABSTRACT FROM AUTHOR]

Structure-preserving doubling algorithms (SDAs) are efficient algorithms for solving Riccati-type matrix equations. However, breakdowns may occur in SDAs. To remedy this drawback, in this paper, we first introduce Ω -symplectic forms (Ω -SFs), consisting of symplectic matrix pairs with a Hermitian parametric matrix Ω. Based on Ω -SFs, we develop modified SDAs (MSDAs) for solving the associated Riccati-type equations. MSDAs generate sequences of symplectic matrix pairs in Ω -SFs and prevent breakdowns by employing a reasonably selected Hermitian matrix Ω. In practical implementations, we show that the Hermitian matrix Ω in MSDAs can be chosen as a real diagonal matrix that can reduce the computational complexity. The numerical results demonstrate a significant improvement in the accuracy of the solutions by MSDAs. [ABSTRACT FROM AUTHOR]

In this paper we establish a rigorous gradient flow structure for one-dimensional Kimura equations with respect to some Wasserstein--Shahshahani optimal transport geometry. This is achieved by first conditioning the underlying stochastic process to nonfixation in order to get rid of singularities on the boundaries, and then studying the conditioned Q-process from a more traditional and variational point of view. In doing so we complete the work initiated in [Chalub et al., Acta Appl. Math., 171 (2021), pp. 1--50], where the gradient flow was identified only formally. The approach is based on the energy dissipation inequality and evolution variational inequality notions of metric gradient flows. Building up on some convexity of the driving entropy functional, we obtain new contraction estimates and quantitative long-time convergence towards the stationary distribution. [ABSTRACT FROM AUTHOR]

In this paper we consider the application of implicit-explicit (IMEX) time discretizations for the incompressible Oseen equations. The pressure velocity coupling and the viscous terms are treated implicitly, while the convection term is treated explicitly. Both the second-order backward differentiation and the Crank-Nicolson methods are considered for time discretization, resulting in a scheme similar to Gear's method on the one hand and to the Adams-Bashforth of second order on the other. For the discretization in space, we consider finite element methods with stabilization on the gradient jumps. The stabilizing terms ensures inf-sup stability for equal-order interpolation and robustness at a high Reynolds number. Under suitable Courant conditions we prove stability of Gear's scheme in this regime. The stabilization allows us to prove error estimates of order O(hk+1/2 +τ²). Here h is the mesh parameter, k is the polynomial order, and τ the time step. Finally we show that for inviscid flow (or underresolved viscous flow) the IMEX scheme can be written as a fractional step method in which only a mass matrix is inverted for each velocity component and a Poisson-type equation is solved for the pressure. [ABSTRACT FROM AUTHOR]

This paper is devoted to the analysis of an energy-stable discontinuous Galerkin algorithm for solving the Cahn-Hilliard-Navier-Stokes equations within a decoupled splitting framework. We show that the proposed scheme is uniquely solvable and mass conservative. The energy dissipation and the L∞ stability of the order parameter are obtained under a CFL-like constraint. Optimal a priori error estimates in the broken gradient norm and in the L² norm are derived. The stability proofs and error analysis are based on induction arguments and do not require any regularization of the potential function. [ABSTRACT FROM AUTHOR]

This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg-Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Nedélec element and the linear element for spatial discretization, respectively; and a linearized backward Euler method and the first order exponential time differencing method for time discretization, respectively. The maximum bound principle (MBP) of the order parameter and the energy dissipation law in the discrete sense are proved. The discrete energy stability and MBP preservation can guarantee the stability and validity of the numerical simulations, and further facilitate the adoption of an adaptive time-stepping strategy, which often plays an important role in long-time simulations of vortex dynamics, especially when the applied magnetic field is strong. An optimal error estimate of the proposed scheme is also given. Numerical examples verify the theoretical results of the proposed scheme and demonstrate the vortex motions of superconductors in an external magnetic field. [ABSTRACT FROM AUTHOR]

This paper is concerned with new error analysis of a lowest-order backward Euler Galerkin-mixed finite element method for the time-dependent Ginzburg-Landau equations. The method is based on a commonly-used nonuniform approximation, in which a linear Lagrange element, the lowest-order Nédélec edge element, and the Raviart-Thomas face element are used for the order parameter ψ, the magnetic field curlA, and the magnetic potential A, respectively. This mixed method has been widely used in practical simulations due to its low cost and ease of implementation. In the Ginzburg-Landau model, the order parameter ψ is the most important variable, which indicates the state of the superconductor. An important feature of the method is the inconsistency of the approximation orders. A crucial question is how the first-order approximation of (curlA, A) influences the accuracy of ψh. The main purpose of this paper is to establish the second-order accuracy for the order parameter in a spatial direction, although the accuracy for (curlA, A) is first order only. Previous analysis only gave the first-order convergence for all three variables due to certain artificial pollution involved in the analysis. Our analysis is based on a nonstandard quasi-projection for ψ and the corresponding more precise estimates, including the H-1 -norm. With the quasi-projection, we prove that the lower-order approximation to (curlA, A) does not pollute the accuracy of ψh. Our numerical experiments confirm the optimal convergence of ψh. The approach can be extended to many other multiphysics models.. [ABSTRACT FROM AUTHOR]

Repeatedly solving the parameterized optimal mass transport (pOMT) problem is a frequent task in applications such as image registration and adaptive grid generation. It is thus critical to develop a highly efficient reduced solver that is equally accurate as the full order model. In this paper, we propose such a machine learning--like method for pOMT by adapting a new reduced basis (RB) technique specifically designed for nonlinear equations, the reduced residual reduced over-collocation (R2-ROC) approach, to the parameterized Monge--Ampère equation. It builds on top of a narrow-stencil finite difference method (FDM), a so-called truth solver, which we propose in this paper for the Monge--Ampère equation with the transport boundary. Together with the R2-ROC approach, it allows us to handle the strong and unique nonlinearity pertaining to the Monge--Ampère equation achieving online efficiency without resorting to any direct approximation of the nonlinearity. Several challenging numerical tests demonstrate the accuracy and high efficiency of our reduced solver for solving the parameterized Monge--Ampère equation, effectively transporting the nontrivial boundaries. [ABSTRACT FROM AUTHOR]

In this paper we study two different ways of coupling a local operator with a nonlocal one so that the resulting equation is related to an energy functional. In the first strategy the coupling is given via source terms in the equation, and in the second one a flux condition in the local part appears. For both models we prove existence and uniqueness of a solution that is obtained via direct minimization of the related energy functional. In the second part of this paper we extend these ideas to local/nonlocal elasticity models in which we couple classical local elasticity with nonlocal peridynamics. [ABSTRACT FROM AUTHOR]

In this paper, we consider the tensor absolute value equations (TAVEs). When one tensor is row diagonal with odd order, we show that the TAVEs can be reduced to an algebraic equation; when it is row diagonal and nonsingular with even order, we prove that the TAVEs is equivalent to a polynomial complementary problem. When no tensor is row diagonal, we formulate the TAVEs equivalently as polynomial optimization problems in two different ways. Each of them can be solved by Lasserre's hierarchy of semideflnite relaxations. The finite convergence properties are also discussed. Numerical experiments show the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The global well-posedness for the 2D inviscid magneto-micropolar fluid (MMPF) equations remains an open problem. It is well known that the global boundedness of the L∞ -norm of the vorticity ω yields the resolution. This paper studies the growth of || ω || L∞ from a different perspective and shows that there exists a sequence of small initial vorticity approaching zero in the L∞ -norm such that the corresponding || ω || L∞ can grow to be uniformly bounded from below in a short time for MMPF equations near a special steady state. Specifically, the vorticity is mildly ill-posed in the L∞ -norm. [ABSTRACT FROM AUTHOR]

In this paper we study the large time asymptotic behavior of (energy) conservative solutions to the Hunter--Saxton equation in a generalized framework that consists of the evolutions of solution and its energy measure. We describe the large time asymptotic expansions of the conservative solutions and rigorously verify the validity of the leading order term in L∞ (R) and H¹(R) spaces, respectively. The leading order term is given by a kink-wave that is determined by the total energy of the system only. As a corollary, we also show that the singular part of the energy measure converges to zero, as the time goes to either positive or negative infinity. Under some natural decay rate assumptions on the tails of the initial energy measure, we rigorously provide the optimal error estimates in L∞ (R) and H¹(R). As the time goes to infinity, the pointwise convergence and pointwise growth rate for the solution are also obtained under the same assumptions on the initial data. The proofs of our results rely heavily on the elaborate analysis of the generalized characteristics designed for the measure-valued initial data and explicit formulae for conservative solutions. [ABSTRACT FROM AUTHOR]

In this paper, we prove a global Carleman estimate for the complex Ginzburg-Landau operator with a cubic nonlinear term in a bounded domain of Rn, n = 2, 3. As applications, we study state observation problems for the Ginzburg-Landau equation. [ABSTRACT FROM AUTHOR]

In this paper we consider a constrained parabolic optimal control problem. The cost functional is quadratic and it combines the distance of the trajectory of the system from the desired evolution profile together with the cost of a control. The constraint is given by a term measuring the distance between the final state and the desired state towards which the solution should be steered. The control enters the system through the initial condition. We present a geometric analysis of this problem and provide a closed-form expression for the solution. This approach allows us to present the sensitivity analysis of this problem based on the resolvent estimates for the generator of the system. The numerical implementation is performed by exploring efficient rational Krylov approximation techniques that allow us to approximate a complex function of an operator by a series of linear problems. Our method does not depend on the actual choice of discretization. The main approximation task is to construct an efficient rational approximation of a generalized exponential function. It is well known that this class of functions allows exponentially convergent rational approximations, which, combined with the sensitivity analysis of the closed form solution, allows us to present a robust numerical method. Several case studies are presented to illustrate our results. [ABSTRACT FROM AUTHOR]

We correct assumption (4.6) in Theorem 4.1 in [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 2981-3012], and we consequently change the proof of Theorem 4.1. We correct the proof of Lemma 5.4 in [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 2981-3012]. Moreover, throughout the paper, starting from formulas (3.13) and (3.14), we notice that the current cost of the control problem cannot depend on the state for the results of the paper to work. This change in the current cost must be made in the paper [F. Gozzi and F. Masiero, SIAM J. Control Optim., 55 (2017), pp. 3013-3038], too. [ABSTRACT FROM AUTHOR]

In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semidiscrete system to produce a fully discrete numerical solution. Under certain regularity assumptions, error estimates measured in H¹-norm are successfully derived for the proposed schemes with one and two Runge-Kutta stages. More remarkably, the mass and coefficient matrices of the proposed method can be simultaneously diagonalized with an orthogonal matrix, which provides a fast solution process based on tensor product spectral decomposition and the fast Fourier transform. Various numerical experiments in two and three dimensions are also carried out to validate the theoretical results and demonstrate the excellent performance of the proposed method. [ABSTRACT FROM AUTHOR]

We study a new technique for reducing the number of iterations of the block Cimmino method by replicating rows in the partitioned system, so that we obtain a nondisjoint partitioning of the rows. Since rows in different partitions that are close to colinear produce a poorly conditioned iteration matrix for the block Cimmino method, row replication can get around this problem. With intelligent replication choices, we can reduce the number of iterations for convergence of the replicated block Cimmino method. The downside is a slight increase of the computational workload associated with each partition. In order to find a trade-off between a lower number of iterations and a higher cost per iteration, selecting the proper set of rows for replication is crucial. In this paper, we use graph-based techniques to find good candidates for replication. Since the block Cimmino method can be interpreted as a nonoverlapping additive Schwartz method applied to the normal equations, the replication techniques correspond to introducing an overlap between the subdomains defined by the partitions. We show analytically in the case of a two-block partitioning how the replication improves the condition number of the block Cimmino iteration matrix. We then use challenging two-dimensional PDE problems to show that our algebraic approach targets physically meaningful phenomena on the interface between partitions. We demonstrate the efficiency of the proposed method in improving the performance of the block Cimmino solver, even with a small amount of replication, on problems from the SuiteSparse Matrix Collection. Finally, we compare our approach to a BiCGStab preconditioned with an additive Schwartz method and show that our replication technique can be used to define the subdomains and overlaps in the context of domain decomposition methods. [ABSTRACT FROM AUTHOR]

In this paper, we consider the time integration of parabolic equations with block implicit methods (BIM). Depending on the size of the block, high-order BIM with A-stability are designed without the need of multiple initial guesses. Similar to Runge--Kutta methods, a BIM can be defined by a tableau including two matrices and two vectors. In addition to the general methodology of BIM, we show a special scheme defined by a positive definite matrix and a positive diagonal matrix; both matrix properties are desirable but not available in Runge--Kutta methods. Moreover, we show that the traditional finite element theory for parabolic problems discretized by the backward Euler or Crank--Nicolson schemes can also be extended to BIM. Finally, we introduce some domain decomposition preconditioners for the linear systems of algebraic equations arising from the block implicit discretization in time and the finite element in space. Some numerical results are also reported to show the effectiveness of BIM. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new exponential-type integrator for solving the gKdV equation under rough data. By introducing new frequency approximation techniques and a key gauge transform, the proposed scheme is explicit, stable, and efficient in practice. The optimal convergence result of the scheme is established, which says that it is first order accurate in the Hγ -norm for the solution in Hγ+2 with any γ > 0. The required regularity is lower than classical approaches. The theoretical results are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pointwise time analyticity of solutions to nonlocal parabolic equations in the settings of ℝd and a complete Riemannian manifold M. On the one hand, in ℝd, we prove that any solution u=u(t,x) to ..., where ... is a nonlocal operator of order α, is time analytic in (0,1] if u satisfies the growth condition ... for any ... and ... . We also obtain pointwise estimates for ..., where ... is the fractional heat kernel. Furthermore, under the same growth condition, we show that the mild solution is the unique solution. On the other hand, in a manifold M, we also prove the time analyticity of the mild solution under the same growth condition and the time analyticity of the fractional heat kernel when M satisfies the Poincaré inequality and the volume doubling condition. Moreover, we also study the time and space derivatives of the fractional heat kernel in Rd using the method of Fourier transform and contour integrals. We find that when ..., the fractional heat kernel is time analytic at t=0 when x≠0, which differs from the standard heat kernel. As corollaries, we obtain a sharp solvability condition for the backward nonlocal parabolic equations and time analyticity of some nonlinear nonlocal parabolic equations with power nonlinearity of order p . These results are related to those in [9] and [22], which deal with local equations. [ABSTRACT FROM AUTHOR]

The purpose of the current paper is to unveil the key mechanism which is responsible for the occurrence of Turing-type instability for a nonlocal Fisher-KPP model. In particular, we prove that the solution of the considered nonlocal Fisher-KPP equation in the neighborhood of a constant stationary solution is destabilized via a diffusion-driven blow-up. It is also shown that the observed diffusion-driven blow-up is complete, while its blow-up rate is completely classified. Finally, the detected diffusion-driven instability results in the formation of unstable blow-up patterns, which are also identified through the determination of the blow-up profile of the solution. [ABSTRACT FROM AUTHOR]

This paper focuses on an inverse problem associated with the plate equation which is derived from models in fluid mechanics and elasticity. We establish the unique identifying results in simultaneously determining both the unknown density and the internal sources from the passive boundary measurement. The proof mainly relies on the asymptotic analysis and harmonic analysis on integral transforms. [ABSTRACT FROM AUTHOR]

We consider the Kirchhoff equation ∂tt u -- △u (1 + ∫Td | ▽u|²) = 0 on the d-dimensional torus Td, and its Cauchy problem with initial data u(0, x), ∂t u(0, x) of size ε in the Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound ε-6 for the lifespan of the corresponding solutions (the standard local theory gives ε-2, and the normal form for the cubic terms gives ε-4). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the "superactions" of the effective equations on large time intervals is smaller (by a factor ε²) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper. [ABSTRACT FROM AUTHOR]

This paper describes the structure of solutions to Kolmogorov's equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller [Trans. Amer. Math. Soc., 48 (1940), pp. 488--515], who clarified in 1945 in the errata to that paper that some of its results covered only nonexplosive Markov processes. In this work, which is largely of a survey nature, the case of explosive processes is also considered. This paper is based on the invited talk presented by the authors at the conference "P. L. Chebyshev -- 200," and it describes the results of their joint studies with Manasa Mandava (1984--2019). [ABSTRACT FROM AUTHOR]

In this paper, we consider the least element problem of a nonnegative solution set of a system of inequalities defined by a homogeneous polynomial mapping and a vector. In the set under consideration, the homogeneous polynomial mapping is defined by a tensor. When the tensor involved is square, the set under consideration is just the feasible region of the tensor complementarity problem (TCP). We first introduce the concept of the generalized Z-tensor, which is a natural generalization of the Z-tensor. Then, under the assumption that the considered set is nonempty and the tensor involved is a generalized Z-tensor, we propose an iterative method for finding the least element of the considered set. Specifically, by recognizing the position indices of positive components in the least element continuously, we solve a series of lower-dimensional system of tensor equations corresponding to these indices and prove that the least element of the set can be obtained within finite step iterations. When the tensor involved is square, the least element obtained is also a solution of the TCP. Compared with the existing methods for finding the least-element solution of the TCP with a Z-tensor, our method does not require any additional assumptions and has lower computational cost. Preliminary numerical experiments show that the proposed method is effective. [ABSTRACT FROM AUTHOR]

In this paper we consider a generalized equation that is mainly characterized by a cone-valued mapping. It is well known that optimality conditions for different classes of optimization problems can be formulated as such a generalized equation. Moreover, we generalize Kojima's concept of strong stability and introduce appropriate constraint qualifications. We discuss corresponding properties between strong stability and these constraint qualifications. Finally, we apply these results to the particular class of mathematical programs with complementarity constraints and to that of mathematical programs with abstract constraints. [ABSTRACT FROM AUTHOR]

The Cahn--Hilliard equation was originally proposed to describe the phase separation phenomenon for a binary alloy in the quenching process and now has been widely applied in many other scientific fields. To investigate its solution behavior, one mainly depends on the numerical simulations because of the nonlinearity. However, it would require very small time steps to describe the phase separation procedure since it evolves very quickly in time, which implies that an efficient solver for the spatial problem at each time level is very important. In this paper, we investigate a coupled second order elliptic system of constant coefficients, which occurs in many different unconditionally energy stable time discretization schemes for the Cahn--Hilliard equation. For a min-max problem of convergence factor that is obtained via Fourier analysis in the case of two-subdomain domain decomposition, by using an asymptotic analysis we obtain the optimized transmission parameters in explicit form applied in the Robin and the two-sided Robin transmission conditions for both the overlapping and the nonoverlapping Schwarz domain decomposition algorithms, and we obtain as well the corresponding asymptotic convergence rates. Particularly, we also explore and analyze the many-subdomain cases directly and find that the algorithms are scalable until a certain number of subdomains is reached for small time steps, even if there is no coarse grid correction. We finally use numerical examples to illustrate our theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is the first in a series of two articles, aiming to prove the convergence of the HX preconditioner originally proposed by Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509] for Maxwell's equations with jump coefficients. In this paper, we establish several extensions of the discrete regular decomposition for edge finite element functions defined in threedimensional domains. The functions defined by the new discrete regular decompositions can inherit zero degrees of freedom of the considered edge finite element function on some faces and edges of polyhedral domains as well as of some non-Lipschitz domains and possess nearly optimal stability with only a logarithmic factor. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a time stepping method for the time fractional Allen-Cahn equation. The key property of the proposed method is its unconditional stability for general meshes, including the graded mesh commonly used for this type of equation. The unconditional stability is proved through establishing a discrete nonlocal free energy dissipation law, which is also true for the continuous problem. The main idea used in the analysis is to split the time fractional derivative into two parts: a local part and a history part, which are discretized by the well known L1, L1-CN, and L1+-CN schemes. Then an extended auxiliary variable approach is used to deal with the nonlinear and history term. The main contributions of the paper are first, it is found that the time fractional Allen-Cahn equation is a dissipative system related to a nonlocal free energy. Second, we construct efficient time stepping schemes satisfying the same dissipation law at the discrete level. In particular, we prove that the proposed schemes are unconditionally stable for quite general meshes. Finally, the efficiency of the proposed method is verified by a series of numerical experiments. [ABSTRACT FROM AUTHOR]

Spectral Barron spaces have received considerable interest recently, as it is the natural function space for approximation theory of two-layer neural networks with a dimension- free convergence rate. In this paper, we study the regularity of solutions to the whole-space static Schr\odinger equation in spectral Barron spaces. We prove that if the source of the equation lies in the spectral Barron space s(d) and the potential function admitting a nonnegative lower bound decomposes as a positive constant plus a function in s(d), then the solution lies in the spectral Barron space s+2(d). [ABSTRACT FROM AUTHOR]

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124-2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov's modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta. [ABSTRACT FROM AUTHOR]

This paper studies the convergence of a spatial semidiscretization for a backward semilinear stochastic parabolic equation. The fltration is general, and the spatial semidiscretization uses the standard continuous piecewise linear fnite element method. Firstly, higher regularity of the solution to the continuous equation is derived. Secondly, the frst-order spatial accuracy is derived for the spatial semidiscretization. Thirdly, an application of the theoretical results to a stochastic linear quadratic control problem is presented. [ABSTRACT FROM AUTHOR]

We consider a beam equation in the presence of a leading degenerate operator which is not in divergence form. We impose clamped conditions where the degeneracy occurs and dissipative conditions at the other endpoint. We provide some conditions for the uniform exponential decay of solutions for the associated problem. [ABSTRACT FROM AUTHOR]

Numerically solving the Liouville equation in classical mechanics with a discontinuous potential often leads to the challenges of how to preserve the Hamiltonian across the potential barrier and a severe time step constraint according to the Courant--Friedrichs--Lewy condition. Motivated by the Hamiltonian-preserving finite volume schemes by Jin and Wen [21], we introduce a Hamiltonian-preserving discontinuous Galerkin (DG) scheme for the Liouville equation with discontinuous potential in this paper. The DG method can be designed with arbitrary order of accuracy and offers many advantages including easy adaptivity, compact stencils, and the ability of handling complicated boundary conditions and interfaces. We propose to carefully design the numerical fluxes of the DG methods to build the behavior of a classical particle at the potential barrier into the numerical scheme, which ensures the continuity of the Hamiltonian across the potential barrier and the correct transmission and reflection condition. Our scheme is proved to be positive and stable in L¹ norm if the positivity-preserving limiter is applied. Numerical examples are provided to illustrate the accuracy and effectiveness of the proposed numerical scheme. [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]

Traveling oscillating fronts (TOFs) are specific one-dimensional waves of the form U (x, t) = e-iωV (x ct) with a complex-valued profile V that decays at -∞ but approaches a nonzero limit at +∞. In particular, we consider TOFs which connect two stable homogeneous rest states and which occur generically in complex Ginzburg Landau equations of the type Ut = \alpha Uxx + G(| U| ²)U. In this paper we prove a theorem on the asymptotic stability of TOFs, where we allow the initial perturbation to be the sum of an exponentially localized part and a front-like part which approaches a small but nonzero limit at +∞. The underlying assumptions guarantee that the operator, obtained from linearizing about the TOF in a co-moving and co-rotating frame, has essential spectrum touching the imaginary axis in a quadratic fashion and that further isolated eigenvalues are bounded away from the imaginary axis. The basic idea of the proof is to consider the problem in an extended phase space which couples the wave dynamics on the real line to the ODE dynamics at +∞. Using slowly decaying exponential weights, the framework allows one to derive appropriate resolvent estimates, semigroup techniques, and Gronwall estimates. [ABSTRACT FROM AUTHOR]

In this paper, we are first concerned with the exact number and expressions of the positive normalized solutions for the nonlinear Kirchhoff equation in which the frequency is unknown and to be determined by the solution. With these results, we further answer an open problem concerning the exact number of positive solutions to the Kirchhoff equation with fixed frequency. It is interesting that we observe some new phenomena, which are completely different from those for the corresponding nonlinear Schr\"odinger equation and which reveal the special influence of the nonlocal term. [ABSTRACT FROM AUTHOR]

This paper is devoted to establishing global Carleman estimates for refined stochastic beam equations. First, by establishing a fundamental weighted identity, two Carleman estimates are derived with different weight functions for the refined stochastic beam equation, which is a coupled system consisting of a stochastic ordinary differential equation and a stochastic partial differential equation. As applications of these Carleman estimates, the exact controllability of the refined system is proved by the least controls in some sense. Different from classical stochastic beam equations, the refined one is exactly controllable at any time. Meanwhile, the uniqueness of an inverse source problem for refined stochastic beam equations is obtained without any requirement on the initial and terminal data. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of solutions of stochastic McKean-Vlasov equations via feedback control based on discrete-time state observation. By using a specific Lyapunov function, the H∞ stability, asymptotic stability and exponential stability in mean square for the solution of the controlled systems are obtained. Since the distribution of the solution is difficult to observe, we study the corresponding particle system which can be observed for the feedback control. We prove that the exponential stability of control system is equivalent to the the exponential stability of the corresponding particle system. Finally, an example is provided to show the effectiveness of the theory. [ABSTRACT FROM AUTHOR]

In this paper we consider the Runge-Kutta discontinuous Galerkin (RKDG) method to solve linear constant-coefficient hyperbolic equations, where the fourth-order explicit Runge- Kutta time-marching is used. By the aid of the equivalent evolution representation with temporal differences of stage solutions, we make a detailed investigation on the matrix transferring process about the energy equations and then present a sufficient condition to ensure the L²-norm stability under the standard Courant-Friedrichs-Lewy condition. If the source term is equal to zero, we achieve the strong (boundedness) stability without the matrix transferring process to multiple-steps time-marching of the RKDG method. By carefully introducing the reference functions and their projections, we obtain the optimal (or suboptimal) error estimate under a mild smoothness assumption on the exact solution, which is independent of the stage number of the RKDG method. Some numerical experiments are also given to verify our conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we propose a novel method for solving high-dimensional spectral fractional Laplacian equations. Using the Caffarelli--Silvestre extension, the d-dimensional spectral fractional equation is reformulated as a regular partial differential equation of dimension d+1. We transform the extended equation as a minimal Ritz energy functional problem and search for its minimizer in a special class of deep neural networks. Moreover, based on the approximation property of networks, we establish estimates on the error made by the deep Ritz method. Numerical results are reported to demonstrate the effectiveness of the proposed method for solving fractional Laplacian equations up to 10 dimensions. Technically, in this method, we design a special network-based structure to adapt to the singularity and exponential decaying of the true solution. Also, a hybrid integration technique combining the Monte Carlo method and sinc quadrature is developed to compute the loss function with higher accuracy. [ABSTRACT FROM AUTHOR]