Backward Heat Equation Research Articles

Observability inequality of backward stochastic heat equations with Lévy process for measurable sets and its applications

This paper endeavours to directly establish the observability inequality of backward stochastic heat equations involving a Lévy process for measurable sets. As an immediate application, we attain the approximate controllability of forward stochastic heat equations featuring a Lévy process. Subsequently, we embark upon the formulation of a nuanced optimization problem, encompassing both the determination of an optimal actuator location and its associated minimum norm control. This formulation is elegantly cast as a two-person zero-sum game problem. We then proceed to articulate a set of conditions–both sufficient and necessary–required for optimizing the solution through the prism of a Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.

Matrix Li–Yau–Hamilton estimates under Ricci flow and parabolic frequency

We prove matrix Li–Yau–Hamilton estimates for positive solutions to the heat equation and the backward conjugate heat equation, both coupled with the Ricci flow. We then apply these estimates to establish the monotonicity of parabolic frequencies up to correction factors. As applications, we obtain some unique continuation results under the nonnegativity of sectional or complex sectional curvature.

Erratum: Lipschitz Stability for Backward Heat Equation with Application to Fluorescence Microscopy

Erratum: Lipschitz Stability for Backward Heat Equation with Application to Fluorescence Microscopy

An RBF-PUM finite difference scheme for forward–backward heat equation

An RBF-PUM finite difference scheme for forward–backward heat equation

A generalized midpoint-based boundary value method for unstable partial differential equations

We present a class of Boundary Value Methods (BVMs) that can be applied to semi-stable and unstable Partial Differential Equation (PDE) models. Step-by-step (initial value) methods, such as Runge–Kutta and linear multistep methods have numerical stability regions which do not intersect with certain regions of the complex plane that are significant to the time-integration of unstable PDEs. BVMs, which need extra numerical conditions at the final time, are global methods and are, in some sense, free of such barriers. We discuss BVMs based on generalized midpoint methods, combined with appropriate numerical initial and final conditions. The stability regions of these methods intersect with a significant part of the complex plane. In several numerical experiments we will illustrate the usefulness of such methods when applied to PDE models, such as a dispersive wave equation, a space-fractional PDE and the backward heat equation.

A non‐overlapping domain decomposition dual reciprocity method for solving the forward–backward heat equation in two‐dimension

AbstractWe apply a boundary element dual reciprocity method (DRBEM) to the numerical solution of the forward–backward heat equation in a two‐dimensional case. The method is employed for the spatial variable via the fundamental solution of the Laplace equation and the Crank–Nicolson finite difference scheme is utilized to treat the time variable. The physical domain is divided into two non‐overlapping subdomains resulting in two standard forward and backward parabolic equations. The subproblems are then treated by the underlying method assuming a virtual boundary in the interface and starting with an initial approximate solution on this boundary followed by updating the solution by an iterative procedure. In addition, we show that the time discrete scheme is unconditionally stable and convergent using the energy method. Furthermore, some computational aspects will be suggested to efficiently deal with the formulation of the proposed method. Finally, two forward–backward problems, for which the exact solution is available, will be numerically solved for two different domains to demonstrate the efficiency of the proposed approach.

Liouville type theorems and gradient estimates for nonlinear heat equations along ancient K-super Ricci flow via reduced geometry

In this paper, we use the techniques of Kunikawa-Sakurai in [11] and the approach of Dung-Dung in [6] to study gradient estimates for the positive bounded solutions to the nonlinear parabolic equation concerning Perelman's reduced distance∂∂tu(x,t)=Δu(x,t)+au(x,t)ln⁡u(x,t) along ancient K-super Ricci flow, where a∈R. As applications, we prove Liouville type results of a nonlinear backward heat equation. Our results generalize and improve many previous works.

Time Analyticity for the Heat Equation on Gradient Shrinking Ricci Solitons

On a complete non-compact gradient shrinking Ricci soliton, we prove the analyticity in time for smooth solutions of the heat equation with quadratic exponential growth in the space variable. This growth condition is sharp. As an application, we give a necessary and sufficient condition on the solvability of the backward heat equation in a class of functions with quadratic exponential growth on shrinkers.

Stability results for backward heat equations with time-dependent coefficient in the Banach space [formula omitted

In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space Lp(R), (1<p<∞). For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme.

The Levenberg–Marquardt regularization for the backward heat equation with fractional derivative

The backward heat problem with time-fractional derivative in Caputo's sense is studied. The inverse problem is severely ill-posed in the case when the fractional order is close to unity. A Levenberg–Marquardt method with a new a posteriori stopping rule is investigated. We show that optimal order can be obtained for the proposed method under a Hölder-type source condition. Numerical examples for one and two dimensions are provided.

Strong error estimates for a space-time discretization of the linear-quadratic control problem with the stochastic heat equation with linear noise

Abstract We propose a time-implicit, finite-element-based space-time discretization of the necessary and sufficient optimality conditions for the stochastic linear-quadratic optimal control problem with the stochastic heat equation driven by linear noise of type $[X(t)+\sigma (t)]\,\,\textrm{d}W(t)$ and prove optimal convergence w.r.t. both space and time discretization parameters. In particular, we employ the stochastic Riccati equation as a proper analytical tool to handle the linear noise, and thus extend the applicability of the earlier work by Prohl &amp; Wang (2021, Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation. ESAIM Control Optim. Calc. Var., 27, 54), where the error analysis was restricted to additive noise.

Extended symmetry analysis of two-dimensional degenerate Burgers equation

We carry out the extended symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and all its generalized symmetries are proved equivalent to its Lie symmetries. We also prove that the space of conservation laws of this equation is infinite-dimensional and is naturally isomorphic to the solution space of the (1+1)-dimensional backward linear heat equation. Lie reductions of the two-dimensional degenerate Burgers equation are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We additionally consider solutions that also satisfy an analogous nondegenerate Burgers equation. In total, we construct four families of solutions of two-dimensional degenerate Burgers equation that are expressed in terms of arbitrary (nonzero) solutions of the (1+1)-dimensional linear heat equation. Various kinds of hidden symmetries and hidden conservation laws (local and potential ones) are discussed as well. As a by-product, we exhaustively describe generalized symmetries, cosymmetries and conservation laws of the transport equation, also called the inviscid Burgers equation, and construct new invariant solutions of the nonlinear diffusion and diffusion–convection equations with power nonlinearities of degree −1/2.

Least squares formulation for ill-posed inverse problems and applications

In this paper we propose a least squares formulation for ill-posed inverse problems. For example, ill-posed inverse problems in partial differential equations are those such that a solution of inverse problem exists for a smooth data but it does not depend continuously on data, or there is no solution for inverse problem, e.g. the Cauchy problem for elliptic equations and backward solution of parabolic equations. We develop the least squares formulation in which the sum of equations error over domain and data fitting criterion and Tikhonov regularization terms is minimized over entire solutions. In this way we can establish the existence and uniqueness of an inverse solution and establish the continuity of the inverse solution for noisy data in . The method can be applied to a general class of non-linear inverse problems and an operator theoretic stability analysis is developed. We describe how one can apply the method for various PDE inverse problems. Numerical tests using backward heat equation and Cauchy problem for elliptic equations are presented to demonstrate the applicability and performance of the proposed method.

Strong rates of convergence for a space-time discretization of the backward stochastic heat equation, and of a linear-quadratic control problem for the stochastic heat equation

We verify strong rates of convergence for a time-implicit, finite-element based space-time discretization of the backward stochastic heat equation, and the forward-backward stochastic heat equation from stochastic optimal control. The fully discrete version of the forward-backward stochastic heat equation is then used within a gradient descent algorithm to approximately solve the linear-quadratic control problem for the stochastic heat equation driven by additive noise. This work is thus giving a theoretical foundation for the computational findings in Dunst and Prohl, SIAM J. Sci. Comput. 38 (2016) A2725–A2755.

DIRECT METHOD OF LIE-ALGEBRAIC DISCRETE APPROXIMATIONS FOR SOLVING BACKWARD HEAT EQUATION

A direct method of Lie-algebraic discrete approximation for numerical solving the Cauchy problem for the backward heat equation is proposed in this paper. The key idea of direct method of Lie-algebraic discrete approximations is using analytical approaches, in particular the method of small parameter or Taylor series expansion, to construct analytical approximation of the solution for the problem in the form of power series with respect to the time variable. The conditions for convergence of analytical series are studied in particular. By means of small parameter method the recurrence relation for evaluation of each member of a sequence is provided. This approach enables fast computation and signicant reduction of computational cost in compare to Generalized method of Lie-algebraic discrete approximations which performs complete discretization by all variables. Thereafter, the discrete match of recurrence relation is built using quasi-representations of the Lie-algebra basis elements, which means, that each dierential operator is replaced by its analogue matrix which is quasi-representation of dierential operator in nite dimensional space. It is proved that computational scheme has a factorial rate of convergence. The proposed approach is applied to model case and obtained results are compared with nite dierence method, classical method of Lie-algebraic discrete approximations and Generalized method of Lie-algebraic discrete approximation. The convergence rates for all of these methods are compared in dierent functional spaces. In addition, we study the count of arithmetical operations for equal set of nodes. Demonstrated a possibility for reusability of the numerical scheme for heat equation.

Lipschitz Stability for Backward Heat Equation with Application to Fluorescence Microscopy

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 27 October 2020Accepted: 22 July 2021Published online: 21 October 2021Keywordsbackward heat equation, Lipschitz stability, inverse problem, fluorescence microscopyAMS Subject Headings35B35, 35K05, 35R30Publication DataISSN (print): 0036-1410ISSN (online): 1095-7154Publisher: Society for Industrial and Applied MathematicsCODEN: sjmaah

Time-dependent weak rate of convergence for functions of generalized bounded variation

Let W denote the Brownian motion. For any exponentially bounded Borel function g the function u defined by is the stochastic solution of the backward heat equation with terminal condition g. Let denote the corresponding approximation generated by a simple symmetric random walk with time steps and space steps where For a class of terminal functions g having bounded variation on compact intervals, the rate of convergence of to u(t, x) is considered, and also the behavior of the error as t tends to T.

Ill-posed problems associated with sectorial operators in Banach space

Ill-posed problems of the form u′(t)=Au+h, 0<t<T, u(0)=φ are studied in Banach space under two related scenarios, both of which consider a well-posed logarithmic approximation. In the first scenario, −A generates a bounded holomorphic semigroup of angle θ=π2, and so A is sectorial of angle 0. In the second, where θ may be relaxed to any value in (0,π2], we impose an additional assumption that e−A is sectorial. Regularization results for the original ill-posed problem are then established in the linear case where h=h(t), and subsequently extended to nonlinear problems where h is replaced by h(t,u(t)). We compare the two cases in terms of regularization convergence rates and apply our results to backward heat equations, both linear and nonlinear in Lp(Rn), 1<p<∞.

How symmetries yield non-invertible mappings of linear partial differential equations

Nonlocally related partial differential equation (PDE) systems are important in the analysis of a given PDE system. Useful nonlocally related systems can be constructed through conservation law and symmetry based methods. In this paper, we focus on an application of the symmetry-based method to linear PDE systems. In particular, we show how to obtain systematically non-invertible mappings of linear PDEs to linear PDEs. As examples, we obtain non-invertible mappings of the Kolmogorov equation with variable coefficients to the backward heat equation (a PDE with constant coefficients) as well as non-invertible mappings of linear hyperbolic PDEs with variable coefficients to linear hyperbolic PDEs with constant coefficients.

Time analyticity for the heat equation and Navier-Stokes equations

We prove the analyticity in time for solutions of two parabolic equations in the whole space, without any decaying or vanishing conditions. One of them involves solutions to the heat equation of exponential growth of order 2 on M. Here M is Rd or a complete noncompact manifold with Ricci curvature bounded from below by a constant. An implication is a sharp solvability condition for the Cauchy problem of the backward heat equation, which is a well known ill-posed problem. Another implication is a sharp criteria for time analyticity of solutions down to the initial time. The other pertains bounded mild solutions of the incompressible Navier-Stokes equations in the whole space.There are many long established analyticity results for the Navier-Stokes equations. See for example [17] and [10] for spatial and time analyticity in certain integral sense, [4] for pointwise space-time analyticity of 3 dimensional solutions to the Cauchy problem, and also the pointwise time analyticity results of [22] and [12] under zero boundary condition. Our result seems to be the first general pointwise time analyticity result for the Cauchy problem for all dimensions, whose proof involves only real variable method. The proof involves a method of algebraically manipulating the integral kernel, which appears applicable to other evolution equations.