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This paper tackles efficient methods for Bayesian inverse problems with priors based on Whittle–Matérn Gaussian random fields. The Whittle–Matérn prior is characterized by a mean function and a covariance operator that is taken as a negative power of an elliptic differential operator. This approach is flexible in that it can incorporate a wide range of prior information including nonstationary effects, but it is currently computationally advantageous only for integer values of the exponent. In this paper, we derive an efficient method for handling all admissible noninteger values of the exponent. The method first discretizes the covariance operator using finite elements and quadrature, and uses preconditioned Krylov subspace solvers for shifted linear systems to efficiently apply the resulting covariance matrix to a vector. This approach can be used for generating samples from the distribution in two different ways: by solving a stochastic partial differential equation, and by using a truncated Karhunen–Loève expansion. We show how to incorporate this prior representation into the infinite-dimensional Bayesian formulation, and show how to efficiently compute the maximum a posteriori estimate, and approximate the posterior variance. Although the focus of this paper is on Bayesian inverse problems, the techniques developed here are applicable to solving systems with fractional Laplacians and Gaussian random fields. Numerical experiments demonstrate the performance and scalability of the solvers and their applicability to model and real-data inverse problems in tomography and a time-dependent heat equation. [ABSTRACT FROM AUTHOR]

We present a novel solver technique for the anisotropic heat flux equation, aimed at the high level of anisotropy seen in magnetic confinement fusion plasmas. Such problems pose two major challenges: (i) discretization accuracy and (ii) efficient implicit linear solvers. We simultaneously address each of these challenges by constructing a new finite element discretization with excellent accuracy properties, tailored to a novel solver approach based on algebraic multigrid (AMG) methods designed for advective operators. We pose the problem in a mixed formulation, introducing the directional temperature gradient as an auxiliary variable. The temperature and auxiliary fields are discretized in a scalar discontinuous Galerkin space with upwinding principles used for discretizations of advection. We demonstrate the proposed discretization's superior accuracy over other discretizations of anisotropic heat flux, achieving error 1000 x smaller for anisotropy ratio of 109, for closed field lines. The block matrix system is reordered and solved in an approach where the two advection operators are inverted using AMG solvers based on approximate ideal restriction, which is particularly efficient for upwind discontinuous Galerkin discretizations of advection. To ensure that the advection operators are nonsingular, in this paper we restrict ourselves to considering open (acyclic) magnetic field lines for the linear solvers. We demonstrate fast convergence of the proposed iterative solver in highly anisotropic regimes where other diffusion-based AMG methods fail. [ABSTRACT FROM AUTHOR]

In this paper, we tackle a topology optimization problem which consists in finding the optimal shape of a solid located inside a fluid that minimizes a given cost function. The motion of the fluid is modeled thanks to the Boussinesq system which involves the unsteady Navier-Stokes equation coupled with a heat equation. In order to cover several models presented in the literature, we choose a nonsmooth formulation for the outlet boundary conditions. This paper aims at proving the existence of solutions to the resulting equations, along with the study of a relaxation scheme of the nonsmooth conditions. A second part covers the topology optimization problem itself for which we proved the existence of optimal solutions and provides the definition of first order necessary optimality conditions. [ABSTRACT FROM AUTHOR]

This paper concerns numerical solution of the diffusion equation with strong anisotropy on meshes not aligned with the anisotropic vector field. In order to resolve the numerical pollution for simulations on a non-anisotropy-aligned mesh and reduce the associated high computational cost we propose an effective preconditioner, extending our previous work [D. Green et al., Comput. Phys. Commun., 9 (2022), 108333]. Similar to the anisotropy-aligned mesh case, we apply the auxiliary space preconditioning framework to design a preconditioner where a continuous finite element space is used as the auxiliary space for the discontinuous finite element space. The key component is an effective line smoother that can mitigate the high-frequency errors perpendicular to the magnetic field. We design a graph-based approach to find such a line smoother that is approximately perpendicular to the vector fields when the mesh does not align with the anisotropy. Numerical experiments for several benchmark problems are presented, demonstrating the effectiveness and robustness of the proposed preconditioner when applied to Krylov iterative methods. [ABSTRACT FROM AUTHOR]

This paper considers the immediate blowup of entropy-bounded classical solutions to the vacuum free boundary problem of non-isentropic compressible Navier-Stokes equations. The viscosities and the heat conductivity could be constants, or more physically, the degenerate, temperature-dependent functions which vanish on the vacuum boundary (i.e., ..., for constants ... and adiabatic exponent γ>1). With prescribed decaying rate of the initial density across the vacuum boundary, we prove that: (1) for three-dimensional spherically symmetric flows with non-vanishing bulk viscosity and zero heat conductivity, entropy-bounded classical solutions do not exist for any small time, provided the initial velocity is expanding near the boundary; (2) for three-dimensional spherically symmetric flows with non-vanishing heat conductivity, the normal derivative of the temperature of the classical solution across the free boundary does not degenerate, and therefore the entropy immediately blowups if the decaying rate of the initial density is not of 1/(γ-1) power of the distance function to the boundary; (3) for one-dimensional flow with zero heat conductivity, the non-existence result is similar but need more restrictions on the decaying rate. Together with our previous results on local or global entropy-bounded classical solutions (Liu and Yuan, SIAM J. Math. Anal. (2) 51, 2019; Liu and Yuan, Math. Models Methods Appl. Sci. (9) 12, 2019), this paper shows the necessity of proper degenerate conditions on the density and temperature across the boundary for the well-posedness of the entropy-bounded classical solutions to the vacuum boundary problem of the viscous gas. [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]

Deep learning (DL), in particular deep neural networks, by default is purely datadriven and in general does not require physics. This is the strength of DL but also one of its key limitations when applied to science and engineering problems in which underlying physical properties--such as stability, conservation, and positivity--and accuracy are required. DL methods in their original forms are often not capable of respecting the underlying mathematical models or achieving desired accuracy even in big-data regimes. On the other hand, many data-driven science and engineering problems, such as inverse problems, typically have limited experimental or observational data, and DL would overfit the data in this case. Leveraging information encoded in the underlying mathematical models, we argue, not only compensates for missing information in low data regimes but also provides opportunities to equip DL methods with the underlying physics, hence promoting better generalization. This paper develops a model-constrained DL approach and its variant TNet--a Tikhonov neural network--which are capable of learning not only information hidden in the training data but also in the underlying mathematical models to solve inverse problems governed by partial differential equations in low data regimes. We provide the constructions and some theoretical results for the proposed approaches for both linear and nonlinear inverse problems. Since TNet is designed to learn inverse solutions with Tikhonov regularization, it is interpretable: in fact it recovers Tikhonov solutions for linear cases while potentially approximating Tikhonov solutions for nonlinear inverse problems. We also prove that data randomization can enhance not only the smoothness of the networks but also their generalizations. Comprehensive numerical results confirm the theoretical findings and show that with even as little as 1 training data sample for one-dimensional (1D) deconvolution, 5 for an inverse 2D heat conductivity problem, 100 for inverse initial conditions for a time-dependent 2D Burgers's equation, and 50 for inverse initial conditions for 2D Navier--Stokes equations, TNet solutions can be as accurate as Tikhonov solutions while being several orders of magnitude faster. This is possible owing to the model-constrained term, replications, and randomization. [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]

The goal of this work is to introduce a new online data driven modeling solution for tracking the dynamics of systems, the behavior of which is governed by (nonlinear) partial differential equations (PDEs). In order to reach this goal, this paper introduces a recursive algorithm for black box time varying model identification along with an online algorithm for reducing the order of the identified model. The resulting system model is a linear time varying low order state space representation usable, e.g., for controller design or prediction. The model reduction step relies on the recursive determination of the orthonormal basis of a specific data based proper orthogonal decomposition. The system identification procedure is initialized with the dynamic mode decomposition with control and updated recursively via the introduction of a specific recursive least squares algorithm. In both steps, a forgetting factor is used to focus on the most recent variations of the system. As case studies, a linear heat diffusion equation with a time varying diffusion parameter is first considered. The Burgers equation, a nonlinear hyperbolic PDE, is then utilized to assess the introduced methods applicability with nonlinear systems. Identification results based on simulation show the high accuracy of this new approach. [ABSTRACT FROM AUTHOR]

In this paper we construct a quadrature scheme to numerically solve the nonlocal diffusion equation (Aα + bI)u = f with Aα the αth power of the regularly accretive operator A. Rigorous error analysis is carried out and sharp error bounds (up to some negligible constants) are obtained. The error estimates include a wide range of cases in which the regularity index and spectral angle of A, the smoothness of f, and the size of b and α are all involved. The quadrature scheme is exponentially convergent with respect to the step size and is root-exponentially convergent with respect to the number of solves. Some numerical tests are presented in the last section to verify the sharpness of our estimates. Furthermore, both the scheme and the error bounds can be utilized directly to solve and analyze time-dependent problems. [ABSTRACT FROM AUTHOR]

Diffusion in heterogeneous media partitioned by semipermeable interfaces has a wide range of applications in the physical and life sciences, ranging from thermal conduction in composite media, gas permeation in soils, diffusion magnetic resonance imaging, drug delivery, and intercellular gap junctions. Many of these systems involve three-dimensional diffusion in an array of parallel planes with homogeneity in the lateral directions, so that they can be reduced to effective onedimensional (1D) models. In this paper we develop a probabilistic model of single-particle diffusion in 1D multilayered media by constructing a multilayered version of so-called snapping out Brownian motion (BM). The latter sews together successive rounds of reflected BM, each of which is restricted to a single layer. Each round of reflected BM is killed when the local time at one end of the layer exceeds an independent, exponentially distributed random variable. (The local time specifies the amount of time a reflected Brownian particle spends in a neighborhood of a boundary.) The particle then immediately resumes reflected BM in the same layer or the layer on the other side of the boundary with equal probability, and the process is iterated We proceed by constructing a last renewal equation for multilayered snapping out BM that relates the full probability density to the probability densities of partially reflected BM in each layer. We then show how transfer matrices can be used to solve the Laplace transformed renewal equation, and prove that the renewal equation and corresponding multilayer diffusion equation are equivalent. We illustrate the theory by analyzing the first passage time (FPT) problem for escape at the exterior boundaries of the domain. Finally, we use the renewal approach to incorporate a generalization of snapping out BM based on the encounter-based method for surface absorption; each round of reflected BM is now killed according to a nonexponential distribution for each local time threshold. This is achieved by considering a corresponding first renewal equation that relates the full probability density to the FPT densities for killing each round of reflected BM. We show that for certain configurations, nonexponential killing leads to an effective time-dependent permeability that is normalizable but heavy-tailed. [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]

This paper provides a probabilistic proof of the comparison result for viscosity solutions of path-dependent semilinear PDEs. We consider the notion of viscosity solutions introduced in [I. Ekren, et al., Ann. Probab., 42 (2014), pp. 204-236], which considers as test functions all those smooth processes which are tangent in mean. When restricted to the Markovian case, this definition induces a larger set of test functions and reduces to the notion of stochastic viscosity solutions analyzed in [E. Bayraktar and M. Sirbu, Proc. Amer. Math. Soc., 140 (2012), pp. 3645-3654; SIAM J. Control Optim., 51 (2013), pp. 4274-4294]. Our main result takes advantage of this enlargement of the test functions and provides an easier proof of comparison. This is most remarkable in the context of the linear path-dependent heat equation. As a key ingredient for our methodology, we introduce a notion of punctual differentiation, similar to the corresponding concept in the standard viscosity solutions [L. A. Caffarelli and X. Cabre, Amer. Math. Soc. Colloq. Publ., 43, AMS, Providence, RI, 1995], and we prove that semimartingales are almost everywhere punctually differentiable. This smoothness result can be viewed as the counterpart of the Aleksandroff smoothness result for convex functions. A similar comparison result was established earlier in [I. Ekren et al., Ann. Probab., 42 (2014), pp. 204-236]. The result of this paper is more general and, more importantly, the arguments that we develop do not rely on any representation of the solution. [ABSTRACT FROM AUTHOR]

The global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity have been obtained since Duan and Ma [Indiana Univ. Math. J., 5 (2008), pp. 2299-2319]. However, as far as we know, how to construct the global small solutions to the compressible Navier-Stokes equations without heat conductivity in R² is still an open problem. In the present paper, we give a positive answer in Rn (n=2,3) with the initial data restricted in the Besov spaces. [ABSTRACT FROM AUTHOR]

The nonequilibrium process in dislocation dynamics and its relaxation to the metastable transition profile are crucial for understanding the plastic deformation caused by line defects in materials. In this paper, we consider the full dynamics of a scalar dislocation model in two dimensions described by the bulk diffusion equation coupled with a dynamic boundary condition on the interface, where a nonconvex misfit potential, due to the presence of dislocation, yields an interfacial reaction term on the interface. We prove that the dynamic solution to this bulk-interface coupled system will uniformly converge to the metastable transition profile, which has a bistates with fat-tail decay rate at the far fields. This global stability for the metastable pattern is the first result for a bulk-interface coupled dynamics driven only by an interfacial reaction on the slip plane. [ABSTRACT FROM AUTHOR]

In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. First, we prove that there exists a unique L²-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the L²-bounded solution possesses the same recurrent properties (e.g., periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as the coefficients. Third, we prove that the recurrent solution of the original equation converges to the stationary solution of averaged equation under the compact-open topology as the time scale goes to zero-in other words, there exists a unique recurrent solution to the original equation in a neighborhood of the stationary solution of averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e., we show that the attractor of original system tends to that of the averaged equation in probability measure space as the time scale goes to zero. For illustration of our results, we give two applications, including stochastic reaction diffusion equations and stochastic generalized porous media equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the constrained approximate null controllability of the heat equation coupled with a real matrix P, where the controls are impulsive and periodically act into the system through a series of real matrices \{ Qk\} ~ k=1. The conclusions are given in two cases. In the case that the controls act globally into the system, we prove that the system is global constrained approximate null controllable under a spectral condition of P together with a rank condition of P and \{ Qk\} ~ k=1, while in the case that the controls act locally into the system, we prove the global constrained approximate null controllability under a stronger condition for P and the same rank condition as the above case. Moreover, we prove that the above-mentioned spectral condition of P is necessary for global constrained approximate null controllability of the control problem considered in this paper. [ABSTRACT FROM AUTHOR]

A revertible kinetic equation for Brownian particles is introduced when the turning frequency and the collision kernel are spatially heterogeneous. We derive an anisotropic diffusion equation by taking the singular limit of the kinetic equation and then balancing out its curvature effect. We see that an extra information such as the turning frequency or waiting time is also needed together with the diffusivity to model a diffusion phenomenon correctly if the environment is spatially heterogeneous. A thought experiment is introduced to test the validity of diffusion laws. We observe that diffusion models with diffusivity alone fail the test and the anisotropic model driven in the paper passes the test. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the recovery of a spatially dependent potential in a (sub)diffusion equation from overposed final time data. We construct a monotone operator, one of whose fixed points is the unknown potential. The uniqueness of the identification is theoretically verified by using the monotonicity of the operator and a fixed point argument. Moreover, we show a conditional stability in Hilbert spaces under some suitable conditions on the problem data. Next, a completely discrete scheme is developed, by using a Galerkin finite element method in space and a finite difference method in time, and then a fixed point iteration is applied to reconstruct the potential. We prove the linear convergence of the iterative algorithm by the contraction mapping theorem and present a thorough error analysis for the reconstructed potential. Our derived a priori error estimate provides a guideline to choose discretization parameters according to the noise level. The analysis relies heavily on some suitable nonstandard error estimates for the direct problem as well as the aforementioned conditional stability. Numerical experiments are provided to illustrate and complement our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper deals with a strategy to solve numerically control problems of the Stackelberg-Nash kind for heat equations with Dirichlet boundary conditions. We assume that we can act on the system through several controls, respecting an order and a hierarchy: a first control (the leader) is assumed to choose the policy; then, a Nash equilibrium pair, determined by the choice of the leader and corresponding to a noncooperative multiple-objective optimization strategy, is found (these are the followers). Our method relies on a formulation inspired by the work of Fursikov and Imanuvilov. More precisely, we minimize over the class of admissible null controls a functional that involves weighted integrals of the state and the control, with weights that blow up at the final time. The use of the weights is crucial to ensure the existence of the controls and the associated state in a reasonable space. We present several mixed formulations of the problems and, then, associated mixed finite element approximations that are easy to handle. In a final step, we exhibit some numerical experiments making use of the Freefem++ package. [ABSTRACT FROM AUTHOR]

Fractional Gaussian noise models the time series with long-range dependence; when the Hurst index H(1/2, 1), it has positive correlation reflecting a persistent autocorrelation structure. This paper studies the numerical method for solving the stochastic fractional diffusion equation driven by fractional Gaussian noise. Using the operator theoretical approach, we present the regularity estimate of the mild solution and the fully discrete scheme with finite element approximation in space and backward Euler convolution quadrature in time. The O(H-pα) convergence rate in time in space are obtained, showing the relationship between the regularity of noise and convergence rates, where p is a parameter to measure the regularity of noise and α (0, 1). Finally, numerical experiments are performed to support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents a multilevel convergence framework for multigrid-reduction-intime (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time-stepping operator, analyzed directly and bounded in the norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time-stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge--Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge--Kutta schemes. [ABSTRACT FROM AUTHOR]

Parallel-in-time methods have become increasingly popular in the simulation of time-dependent numerical PDEs, allowing for the efficient use of additional message passing interface processes when spatial parallelism saturates. Most methods treat the solution and parallelism in space and time separately. In contrast, all-at-once methods solve the full space-time system directly, largely treating time as simply another spatial dimension. All-at-once methods offer a number of benefits over separate treatment of space and time, most notably significantly increased parallelism and faster time to solution (when applicable). However, the development of fast, scalable all-at-once methods has largely been limited to time-dependent (advection-)diffusion problems. This paper introduces the concept of space-time block preconditioning for the all-at-once solution of incompressible flow. By extending well-known concepts of spatial block preconditioning to the space-time setting, we develop a block preconditioner whose application requires the solution of a space-time (advection-)diffusion equation in the velocity block, coupled with a pressure Schur complement approximation consisting of independent spatial solves at each time-step, and a space-time matrix-vector multiplication. The new method is tested on four classical models in incompressible flow. Results indicate perfect scalability in refinement of spatial and temporal mesh spacing, perfect scalability in nonlinear Picard iteration count when applied to a nonlinear Navier--Stokes problem, and minimal overhead in terms of number of preconditioner applications compared with sequential time-stepping. [ABSTRACT FROM AUTHOR]

This paper addresses the complexity reduction of stochastic homogenization of a class of random materials for a stationary diffusion equation. A cost-efficient approximation of the correctors is obtained using a method designed to exploit quasi-periodicity. Accuracy and cost reduction are investigated for local perturbations or small transformations of periodic materials as well as for materials with no periodicity but a mesoscopic structure, for which the limitations of the method are shown. Finally, for materials outside the scope of this method, we propose to use the approximation of homogenized quantities as control variates for variance reduction of a more accurate and costly Monte Carlo estimator (using a multifidelity Monte Carlo method). The resulting cost reduction is illustrated in a numerical experiment and compared with a control variate method from weakly stochastic homogenization. The limits of this variance reduction technique are tested on materials without periodicity or mesoscopic structure. [ABSTRACT FROM AUTHOR]

The controllability cost for the heat equation as the control time T goes to 0 is wellknown of the order eC/T for some positive constant C, depending on the controlled domain and for all initial datum. In this paper, we prove that the constant C can be chosen to be arbitrarily small if the support of the initial data is sufficiently close to the controlled domain, but not necessary inside the controlled domain. The proof is in the spirit on Lebeau and Robbiano's approach, in which a new spectral inequality is established. The main ingredient of the proof of the new spectral inequality is three-sphere inequalities with partial data. [ABSTRACT FROM AUTHOR]

In this paper, we study the Crank-Nicolson alternative direction implicit (ADI) method for two-dimensional Riesz space-fractional diffusion equations with nonseparable coefficients. Existing ADI methods are only shown to be unconditional stable when coefficients are some special separable functions. The main contribution of this paper is to show under mild assumptions the unconditional stability of the proposed Crank-Nicolson ADI method in discrete l² norm and the consistency of cross perturbation terms arising from the Crank-Nicolson ADI method. Also, we demonstrate that several consistent spatial discretization schemes satisfy the required assumptions. Numerical results are presented to examine the accuracy and the efficiency of the proposed ADI methods. [ABSTRACT FROM AUTHOR]

The stochastic Galerkin finite element method (SGFEM) provides an efficient alternative to traditional sampling methods for the numerical solution of linear elliptic partial differential equations with parametric or random inputs. However, computing stochastic Galerkin approximations for a given problem requires the solution of large coupled systems of linear equations. Therefore, an effective and bespoke iterative solver is a key ingredient of any SGFEM implementation. In this paper, we analyze a class of truncation preconditioners for SGFEM. Extending the idea of the mean-based preconditioner, these preconditioners capture additional significant components of the stochastic Galerkin matrix. Focusing on the parametric diffusion equation as a model problem and assuming affine-parametric representation of the diffusion coefficient, we perform spectral analysis of the preconditioned matrices and establish optimality of truncation preconditioners with respect to SGFEM discretization parameters. Furthermore, we report the results of numerical experiments for model diffusion problems with affine and nonaffine parametric representations of the coefficient. In particular, we look at the efficiency of the solver (in terms of iteration counts for solving the underlying linear systems) and compare truncation preconditioners with other existing preconditioners for stochastic Galerkin matrices, such as the mean-based and the Kronecker product ones. [ABSTRACT FROM AUTHOR]

In this paper, we resolve several long-standing issues dealing with optimal pointwise in time error bounds for proper orthogonal decomposition (POD) reduced order modeling of the heat equation. In particular, we study the role played by difference quotients (DQs) in obtaining reduced order model (ROM) error bounds that are optimal with respect to both the time discretization error and the ROM discretization error. When the DQs are not used, we prove that both the POD projection error and the ROM error are suboptimal. When the DQs are used, we prove that both the POD projection error and the ROM error are optimal. The numerical results for the heat equation support the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we use layer potentials and asymptotic analysis techniques to analyze the heat generation due to nanoparticles when illuminated at their plasmonic resonance. We consider arbitrary-shaped particles and the cases of both a single and multiple particles. We clarify the strong dependency of the heat generation on the geometry of the particles as it depends on the eigenvalues of the associated Neumann{Poincaré operator. For close-to-touching nanoparticles, we show that the temperature field deviates significantly from the one generated by two single nanoparticles. The results of this paper formally explain experimental results reported in the nanomedical literature. They open a door for solving the challenging problems of detecting plasmonic nanoparticles in biological media and monitoring temperature elevation in tissue generated by nanoparticle heating. [ABSTRACT FROM AUTHOR]

We consider a linear kinetic transport equation under a diffusive scaling that converges to a diffusion equation as the Knudsen number ε → 0. In [S. Boscarino, L. Pareschi, and G. Russo, SIAM J. Sci. Comput., 35 (2013), pp. A22--A51; Z. Peng et al., J. Comput. Phys., 415 (2020), 109485], to achieve the asymptotic preserving (AP) property and unconditional stability in the diffusive regime with ε ≪ 1, numerical schemes are developed based on an additional reformulation of the even-odd or micro-macro decomposed version of the equation. The key of the reformulation is to add a weighted diffusive term on both sides of one equation in the decomposed system. The choice of the weight function, however, is problem-dependent and ad-hoc, and it can affect the performance of numerical simulations. To avoid issues related to the choice of the weight function and still obtain the AP property and unconditional stability in the diffusive regime, we propose in this paper a new family of AP schemes, termed as IMEX-DG-S schemes, directly solving the micro-macro decomposed system without any further reformulation. The main ingredients of the IMEX-DG-S schemes include globally stiffly accurate implicit-explicit (IMEX) Runge--Kutta temporal discretizations with a new IMEX strategy, discontinuous Galerkin spatial discretizations, discrete ordinate methods for the velocity space, and the application of the Schur complement to the algebraic form of the schemes to control the overall computational cost. The AP property of the schemes is shown formally. With an energy type stability analysis applied to the first order scheme, and Fourier type stability analysis applied to the first to third order schemes, we confirm the uniform stability of the methods with respect to\varepsilon and the unconditional stability in the diffusive regime. A series of numerical examples are presented to demonstrate the performance of the new schemes. [ABSTRACT FROM AUTHOR]

This paper studies the time optimal control problem for systems of heat equations coupled by a pair of constant matrices. The control constraint is of the ball-type, while the target is the origin of the state space. We obtain an upper bound for the number of switching points of the optimal control over each interval with a fixed length. Also, we prove that at each switching point, the optimal control jumps from one direction to the reverse direction. [ABSTRACT FROM AUTHOR]

In this paper we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional. The cost functional is of a tracking type and contains a linear term in the control variables. We derive second order sufficient conditions relying on the Goh transform. [ABSTRACT FROM AUTHOR]

This paper is concerned with an optimal control problem governed by a semilinear, nonsmooth operator differential equation. The nonlinearity is locally Lipschitz-continuous and directionally differentiable but not Gateaux-differentiable. By employing the limited differentiability properties of the control-to-state map, first-order necessary optimality conditions in qualified form are established, which are equivalent to the purely primal condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. The paper ends with the application of the general results to a semilinear heat equation. [ABSTRACT FROM AUTHOR]

Bimolecular binding rate constants are often used to describe the association of large molecules, such as proteins. In this paper, we analyze a model for such binding rates that includes the fact that pairs of molecules can bind only in certain orientations. The model considers two spherical molecules, each with an arbitrary number of small binding sites on their surface, and the two molecules bind if and only if their binding sites come into contact (such molecules are often called "patchy particles" in the biochemistry literature). The molecules undergo translational and rotational diffusion, and the binding sites are allowed to diffuse on their surfaces. Mathematically, the model takes the form of a high-dimensional, anisotropic diffusion equation with mixed boundary conditions. We apply matched asymptotic analysis to derive the bimolecular binding rate in the limit of small, well-separated binding sites. The resulting binding rate formula involves a factor that depends on the electrostatic capacitance of a certain four-dimensional region embedded in five dimensions. We compute this factor numerically by modifying a recent kinetic Monte Carlo algorithm. We then apply a quasi chemical formalism to obtain a simple analytical approximation for this factor and find a binding rate formula that includes the effects of binding site competition/saturation. We verify our results by numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an inverse problem for the radiative transfer equation that is coupled with a heat equation in a nonscattering medium in ℝn, n ≥ 2. The two equations are coupled through a nonlinear blackbody emission term that is proportional to the fourth power of the temperature. By measuring the radiation intensity on the surface of the blackbody, we prove that the emission property of the system can be uniquely reconstructed. In particular, we design a reconstruction procedure that uses merely one set of experimental setup to fully recover the emission parameter. [ABSTRACT FROM AUTHOR]

Fourier analysis has been shown to provide valuable insight into the dispersion and dissipation characteristics of numerical schemes for PDEs. Applying Fourier analysis to discontinuous Galerkin (DG) methods results in an eigenvalue problem with multiple eigenmodes. It was often relied on one of these modes, the so-called physical-mode, for studying the dispersion and dissipation behavior. The effect of the other modes was considered spurious and typically neglected. Recently, a new approach, the combined-mode approach, was proposed for the linear wave equation, in which all modes are considered. In this paper, we apply the combined-mode approach to a number of DG methods for diffusion. We show that for the linear parabolic heat equation, the physical-mode behavior is completely different than the exact diffusion, over a wide range of wavenumbers, and sometimes nondissipative. In contrast, the combined-mode behavior is more consistent and sufficiently accurate in comparison with the exact diffusion for the whole wavenumber range. This approach also revealed that short time and long time diffusion behaviors are very different for high-order multi--degrees of freedom methods. Additionally, using this approach we conduct a comparative diffusion analysis between a number of popular DG methods for diffusion in one and two dimensions. We also provide a study on the influence of the penalty parameter on their behavior. The considered methods include the symmetric interior penalty, Bassi and Rebay, and the local and compact DG methods. The results are verified numerically through several test cases. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem with quasi-boundary value regularization for recovering a source term of the time fractional diffusion equations from the final observation data. In particular, a two-by-two block linear system arising from the problem is studied. We propose a fast preconditioning technique by approximating the Schur complement in the system using a product of some factors, motivated by an approximate diagonalization of one of the blocks. The eigenvalues of the preconditioned system are shown to be clustered around 1, and the fast convergence of the methods is guaranteed theoretically. We also present an approach for selecting the regularization parameter of the quasi-boundary value method. Numerical experiments are carried out to demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

This paper derives and analyzes new diffusion synthetic acceleration (DSA) pre-conditioners for the SN transport equation when discretized with a high-order (HO) discontinuous Galerkin (DG) discretization. DSA preconditioners address the need to accelerate the SN transport equation when the mean free path ε of particles is small and the condition number of the SN transport equation scales like O(ε-2). By expanding the SN transport operator in " and employing a rigorous singular matrix perturbation analysis, we derive a DSA matrix that reduces to the symmetric interior penalty (SIP) DG discretization of the standard continuum diffusion equation when the mesh is first-order and the total opacity is constant. We prove that preconditioning the HO DG SN transport equation with the SIP DSA matrix results in an O(ε) perturbation of the identity, and fixed-point iteration therefore converges rapidly for optically thick problems. However, the SIP DSA matrix is conditioned like O(ε-1), making it difficult to invert for small ε. We further derive a new two-part, additive DSA preconditioner based on a continuous Galerkin discretization of diffusion-reaction, which has a condition number independent of ε, and prove that this DSA variant has the same theoretical efficiency as the SIP DSA preconditioner in the optically thick limit. The analysis is extended to the case of HO (curved) meshes, where so-called mesh cycles can result from elements both being upwind of each other (for a given discrete photon direction). In particular, we prove that performing two additional transport sweeps, with fixed scalar flux, in between DSA steps yields the same theoretical conditioning of fixed-point iterations as in the cycle-free case. Theoretical results are validated by numerical experiments on a HO, highly curved two- and three-dimensional meshes that are generated from an arbitrary Lagrangian-Eulerian hydrodynamics code, where the additional inner sweeps between DSA steps offer up to a 4x reduction in total number of sweeps required for convergence. [ABSTRACT FROM AUTHOR]

In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy localized in a subdomain, which is equidistributed over the whole space. The proof of this inequality is mainly adapted from the parabolic frequency function method, which plays an important role in proving the unique continuation property for solutions of parabolic equations. As an immediate application, we show that the null controllability holds for the heat equation with bounded potentials on the whole space. [ABSTRACT FROM AUTHOR]

The paper is concerned with the optimal harvesting of a marine park, which is described by a parabolic heat equation with Neumann boundary conditions and a nonlinear source term. We consider a cost functional, which is linear with respect to the control; hence the optimal solution can belong to the class of measure-valued control strategies. For each control function, we prove existence and stability estimates for solutions of the parabolic equation. Moreover, we prove the existence of an optimal solution. Finally, some numerical simulations conclude the paper. [ABSTRACT FROM AUTHOR]

In this paper, we derive a bang-bang property of a kind of time optimal control problem for some semilinear heat equation on bounded C2 domains (of the Euclidean space), with homogeneous Dirichlet boundary condition and controls distributed on an open and nonempty subset of the domain where the equation evolves. [ABSTRACT FROM AUTHOR]

In this paper, we consider a numerical method for the time-fractional diffusion equation. The method uses a high order finite difference method to approximate the fractional derivative in time, resulting in a time stepping scheme for the underlying equation. Then the resulting equation is discretized in space by using a spectral method based on the Legendre polynomials. The main body of this paper is devoted to carry out a rigorous analysis for the stability and convergence of the time stepping scheme. As a by-product and direct extension of our previous work, an error estimate for the spatial discretization is also provided. The key contribution of the paper is the proof of the (3-α)-order convergence of the time scheme, where α is the order of the time-fractional derivative. Then the theoretical result is validated by a number of numerical tests. To the best of our knowledge, this is the first proof for the stability of the (3 - α)-order scheme for the time-fractional diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper, we study the null controllability of linear heat and wave equations with spatial nonlocal integral terms. Under some analyticity assumpt ions on the corresponding kernel, we show that the equations are controllable. We employ compactness-uniqueness arguments in a suitable functional setting, an argument that is harder to apply for heat equations because of its very strong time irreversibility. Some possible extensions and open problem s concerning other nonlocal systems are presented. [ABSTRACT FROM AUTHOR]

In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits the Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bound of the Ricci curvature are presented related to the stochastic heat equation [ABSTRACT FROM AUTHOR]

In this paper, we study a macroscopic description on the ensemble of Kuramoto oscillators with finite inertia in a random media characterized by a white noise. In a mesoscopic regime, it is well known that the dynamics of a large Kuramoto ensemble in a random media is governed by the Kuramoto-Sakaguchi-Fokker-Planck (in short, parabolic Kuramoto{Sakaguchi) equation for one-oscillator distribution function. For this parabolic Kuramoto{Sakaguchi equation, we present a global existence of weak solutions in any finite-time interval. Furthermore, we rescale the kinetic equation using the diffusion scaling, and formally derive a drift-diffusion equation by using Hilbert-like expansion in a small parameter ". For the rigorous justification of this asymptotic limit, we introduce a new free energy functional ε consisting of total mass, kinetic energy, entropy functional, and interaction potential and show the uniform boundedness of this free energy with respect to the small parameter ε. This uniform boundedness of ε combined with L1-compactness argument enables us to derive the drift-diffusion equation. We also classified all C2-stationary solutions to the drift-diffusion equation in terms of synchronization parameters σ and ω. [ABSTRACT FROM AUTHOR]

This article studies a Fokker--Planck type equation of fractional diffusion with conservative drift tf = Δα/2 f + div(Ef), where Δα/2 denotes the fractional Laplacian and E is a confining force field. The main interest of the present paper is that it applies to a wide variety of force fields with a polynomial growth at infinity. We first prove the existence and uniqueness of a solution in weighted Lebesgue spaces depending on E under the form of a strongly continuous semigroup. We also prove the existence and uniqueness of a stationary state, by using an appropriate splitting of the fractional Laplacian and by proving a weak and strong maximum principle. We then study the rate of convergence to equilibrium of the solution. The semigroup has a property of regularization in fractional Sobolev spaces, as well as a gain of integrability and positivity which we use to obtain polynomial or exponential convergence to equilibrium in weighted Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper, we first devise an ensemble hybridizable discontinuous Galerkin (HDG) method to efficiently simulate a group of parameterized convection diffusion PDEs. These PDEs have different coefficients, initial conditions, source terms, and boundary conditions. The ensemble HDG discrete system shares a common coefficient matrix with multiple right-hand-side vectors; it reduces both computational cost and storage. We have two contributions in this paper. First, we derive an optimal L² convergence rate for the ensemble solutions on a general polygonal domain, which is the first such result in the literature. Second, we obtain a superconvergent rate for the ensemble solutions after an element-by-element postprocessing under some assumptions on the domain and the coefficients of the PDEs. We present numerical experiments to confirm our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a fast (3 - α)-order numerical method for the Caputo fractional derivative based on the L2 scheme and the sum-of-exponentials (SOE) approximation to the convolution kernel involved in the fractional derivative. This work can be regarded as a continuation of previous works reported by one of the authors in [C.W. Lv and C.J. Xu, SIAM J. Sci. Comput., 38 (2016), pp. A2699-A2724], which constructed and analyzed a (3 - α)-order L2 time stepping scheme for the time-fractional diffusion equation. It is now extended to take into account the fast SOE evaluation method, which allows us to reduce the storage and overall computational cost from O(NT) and O(NT²) for the L2 scheme to O(Nε) and O(NTNε), respectively, with NT being the number of time steps and Nε being the number of fast evaluation terms. The proposed method is then used for the time-fractional diffusion equation in bounded domains. The stability as well as the accuracy of the resulting scheme are rigorously analyzed. Several numerical examples are provided to validate the theoretical results and to demonstrate the efficiency of the proposed method. Finally, we extend the discussion to a graded mesh to make the scheme more suitable for problems having weakly singular solutions at the initial time. [ABSTRACT FROM AUTHOR]

We consider the inverse problem of reconstructing the optical parameters of the radiative transfer equation (RTE) from boundary measurements in the diffusion limit. In the diffusive regime (the Knudsen number Kn ≪ 1), the forward problem for the stationary RTE is well approximated by an elliptic equation. However, the connection between the inverse problem for the RTE and the inverse problem for the elliptic equation has not been fully developed. This problem is particularly interesting because the former one is mildly ill-posed, with a Lipschitz type stability estimate, while the latter is well known to be severely ill-posed with a logarithmic type stability estimate. In this paper, we derive stability estimates for the inverse problem for RTE and examine its dependence on Kn. We show that the stability is Lipschitz in all regimes, but the coefficient deteriorates as e 1/eKn, making the inverse problem of RTE severely ill-posed when Kn is small. In this way we connect the two inverse problems. Numerical results agree with the analysis of worsening stability as the Knudsen number gets smaller. [ABSTRACT FROM AUTHOR]

We consider the initial value problem for the generalized Allen--Cahn equation, \partial t\Phi = \Delta \Phi \varepsilon 2\Phi (\Phi t\Phi I), x \in \Omega, t \geq 0, where \Phi is an n \times n real matrix-valued field, \Omega is a two-dimensional square with periodic boundary conditions, and \varepsilon > 0. This equation is the gradient flow for the energy, E(\Phi) := \int 1 2 \| \nabla \Phi \| 2 F + 1 4\varepsilon 2 \| \Phi t\Phi I\| 2 F, where \| \cdot \| F denotes the Frobenius norm. The primary contribution of this paper is to use asymptotic methods to describe the solution of this initial value problem. If the initial condition has a single-signed determinant, at each point of the domain, at a fast O(\varepsilon 2t) time scale, the solution evolves towards the closest orthogonal matrix. Then, at the O(t) time scale, the solution evolves according to the On diffusion equation. Stationary solutions to the On diffusion equation are analyzed for n = 2. If the initial condition has regions where the determinant is positive and negative, a free interface develops. Away from the interface, in each region, the matrix-valued field behaves as in the single-signed determinant case. At the O(t) time scale, the interface evolves in the normal direction by curvature. At a slow O(\varepsilon t) time scale, the interface is driven by curvature and the surface diffusion of the matrix-valued field. For n = 2, the interface is driven by curvature and the jump in the squared tangental derivative of the phase across the interface. In particular, we emphasize that the interface when n \geq 2 is driven by surface diffusion, while for n = 1, the original Allen--Cahn equation, the interface is only driven by mean curvature. A variety of numerical experiments are performed to verify, support, and illustrate our analytical results. [ABSTRACT FROM AUTHOR]