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We consider the model Y = X + ξ where Y is observable, ξ is a noise random variable with density f ξ , X has an unknown mixed density such that P ( X = X c ) = 1 − p , P ( X = a ) = p with X c being continuous and p ∈ ( 0 , 1 ) , a ∈ R . Typically, in the last decade, the model has been widely considered in a number of papers for the case of fully known quantities a , f ξ . In this paper, we relax the assumptions and consider the parametric error ξ ∼ σ N ( 0 , 1 ) with an unknown σ > 0 . From i.i.d. copies Y 1 , … , Y m of Y we will estimate ( σ , p , a , f X c ) where f X c is the density of X c . We also find the lower bound of convergence rate and verify the minimax property of established estimators.

This work is concerned with a mathematical study of backward problem for time-space fractional diffusion equations associated with the observed data measured in Hilbert scales. Transforming the original problem into an operator equation, we investigate the existence, the uniqueness and the instability for the problem. In order to overcome the ill-posedness of the problem, we apply a modified version of quasi-boundary value method to construct stable approximation problem. Using a Holder-type smoothness assumption of the exact solution it is shown that estimates achieve optimal rates of convergence in Hilbert scales both for an a-priori and for an a-posteriori parameter choice strategies.

This paper investigates the nonlinear backward fractional diffusion equations (BFDEs) having a variable order and variable diffusion coefficient. It is well-known that the BFDEs are ill-posed in the sense of Hadamard. Moreover, we investigate the problem taking into account the disturbance both variable diffusion coefficient and variable order. This may lead to some common regularization strategy to failure. Therefore, we propose a regularization method for this kind of problem. Under some appropriate regularity assumptions of the exact solution, a convergence estimate of Holder-type is proved.

The heat distribution on the surface of a layer inside a heat conducting body can be recovered from two interior temperature measurements. It is reasonable to take discrete measured data with random noise into account. In the paper, approximations of interior measurements from the discrete data are established. First, a one-dimensional Sinc series is applied to represent the approximation function on x standing for the length of the object and then the trigonometric estimators on time-variable t is obtained. We also construct an estimator for the heat distribution which converges in the sense of integrated mean squared error (MISE). Some numerical experiments are given as demonstrations of the ability of numerical implementation of our regularization.

We study the parameter estimation for ergodic diffusion process Xt from perturbed observations Y t j = X t j + e t j , j = 1 , … , n where t 1 , … , t n are observation times and the noise e t is a...

In this paper, we study a free boundary problem of two competing species in the left-shifting environment. This model may be used to describe the interaction of the spreading phenomena of two competing species over a one dimensional habitat influenced by an external effect such as the effect of global warming. Here, we assume that only the habitat of inferior competitor is eroded away by the left moving boundary at constant speed c and we consider the how its spreading influences to the spreading of the superior competitor. We prove, as $c_{2}^{\ast }

We investigate the linear but ill-posed inverse problem of determining a multi-dimensional space-dependent heat source in a time-fractional diffusion equation. We show that the problem is ill-posed in the Hilbert scale H r ( R n ) and establish global order optimal lower bound for the worst case error. Next, we use the Tikhonov regularization method to deal with this problem in the Hilbert scale H r ( R n ) . Locally optimal choices of parameters for the family of regularization operator in the Hilbert scales H r ( R n ) are analyzed by a-priori and a-posteriori methods. Numerical implementations are presented to illustrate our theoretical findings.

In this paper, we focus on the backward diffusion problem with the Caputo fractional derivative operator in time and a general spatial nonlocal operator. For T > 0 and s ∈ [ 0 , T ) , we consider the problem ( P s ) of recovering the distribution u ( x , s ) from a measure of the final data u ( x , T ) for the following non-homogeneous time–space fractional diffusion equation D t α u ( x , t ) + K β L γ u ( x , t ) = f ( x , t ) in R n × ( 0 , T ) subject to the final condition u ( x , T ) = u T ( x ) in R n . The derivative orders and the nonlocal operator are perturbed with noises. Firstly, for 0 s T , we prove the well-posedness of Problem ( P s ) by studying the unique existence and continuity with respect to the derivative orders, the source term as well as the final value of the solution. Secondly, for s = 0 , we verify the ill-posedness of Problem ( P 0 ) and use the method of modified iterated Lavrentiev to construct a regularization solution from inexact data and inexact derivative orders. We apply a modified form of the discrepancy principle to choose regularization parameter and establish new optimal convergence estimates between the exact solution and its regularized approximation.

This paper is devoted to the study of nonparametric estimation of random effects densities in a linear mixed-effects model. In the first case where noise distribution is fully known, we apply nonpa...

In this paper, we consider the backward diffusion problem for a space-fractional diffusion equation (SFDE) with a nonlinear source, that is, to determine the initial data from a noisy final data. Very recently, some papers propose new modified regularization solutions to solve this problem. To get a convergence estimate, they required some strongly smooth conditions on the exact solution. In this paper, we shall release the strongly smooth conditions and introduce a stepwise regularization method to solve the backward diffusion problem. A numerical example is presented to illustrate our theoretical result.

The aim of this paper is of studying the stability of solution of a backward problem of a timefractional diffusion equation with perturbed order. We investigate the well-posedness of the backward problem with perturbed order for t>0. The results on the unique existence and continuity with respect to the fractional order, the source term as well as the final value of the solution are given. At t=0 the backward problem is ill-posed and we introduce a truncated method to regularize the backward problem with respect to inexact fractional order. Some error estimates are provided in Holder type.

In this paper, we study the initial and final value problems (FVPs) of a class of nonlinear diffusion equations including Riesz–Feller derivatives. A natural question on solution continuity with re...

Let X be a continuous random variable having an unknown cumulative distribution function F. We study the problem of estimating F based on i.i.d. observations of a continuous random variable Y from the model Y = X + Z. Here, Z is a random noise distributed with known density g and is independent of X. We focus on some cases of g in which its Fourier transform can vanish on a countable subset of ℝ. We propose an estimator $\hat F$ for F and then investigate upper bounds on convergence rate of $\hat F$ under the root mean squared error. Some numerical experiments are also provided.

In this paper, we reconstruct the solution u(x,t) of the backward space-fractional diffusion problem with a locally Lipschitzian nonlinear source This problem is severely ill-posed in the Hadamard sense, hence, a regularization is in order. In the paper, we introduce one spectral regularization method and establish stability error estimates with optimal order under an a priori choice of regularization parameter. Finally, numerical implementations are given to show the effectiveness of the proposed regularization methods.

This paper considers the generalized Langevin equation involving [Formula: see text]-Caputo fractional derivatives in a Banach space. The fractional derivative is generalized from the Caputo derivative ([Formula: see text]), the Caputo–Katugampola ([Formula: see text], the Hadamard derivative ([Formula: see text]). We investigate the existence of mild solutions [Formula: see text] of the problem, in which the source function is assumed to satisfy some weakly singular conditions. Before proceeding to the main results, we transform the problem into an integral equation. Based on the obtained integral equation, the main results are proved via the nonlinear Leray–Schauder alternatives and Banach fixed point theorems. To prove this end, a new generalized weakly Gronwall-type inequality is established. Further, we prove that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders and the friction constant. As a consequence, we deduce that the solution [Formula: see text] of the equation involving the Caputo–Katugampola derivative tends to the solution [Formula: see text] of the equation involving the Hadamard derivative as [Formula: see text].

We study the heteroscedastic deconvolution problem when random noises have compactly supported densities. In this context, the Fourier transforms of the densities can vanish on the real line. We propose a truncated type of estimator for target density and derive the convergence rate of the mean L1-error uniformly over a class of target densities. A lower bound for the mean L1-error is also established. Some simulations will be given to illustrate the performance of the proposed estimator.

In this paper, we consider a Riesz–Feller space-fractional backward diffusion problem with a time-dependent coefficient ut(x,t)=l(t)xDθγu(x,t)+f(x,t),(x,t)∈R×(0,T). We show that this problem is ill-posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.

An inverse problem to recover a space-dependent factor of a source term in the inexact order time-fractional diffusion equation from final data is considered. The problem arises in many applications, but it is in general ill-posed. The ill-posedness is since small errors in the input data cause large errors in the output solution. To overcome this instability we propose the stable approximation solution via a general modified quasi-boundary value regularization method. Order optimal convergence rates for the worst case error of the method are derived under the usual source condition by using an a-priori and an a-posteriori regularization parameter choice rules, respectively. Finally, several numerical examples are provided to illustrate the effectiveness of the proposed method.

We focus on the nonhomogeneous backward heat problem of finding the initial temperature θ = θ ⁢ ( x , y ) = u ⁢ ( x , y , 0 ) {\theta=\theta(x,y)=u(x,y,0)} such that { u t - a ⁢ ( t ) ⁢ ( u x ⁢ x + u y ⁢ y ) = f ⁢ ( x , y , t ) , ( x , y , t ) ∈ Ω × ( 0 , T ) , u ⁢ ( x , y , t ) = 0 , ( x , y ) ∈ ∂ ⁡ Ω × ( 0 , T ) , u ⁢ ( x , y , T ) = h ⁢ ( x , y ) , ( x , y ) ∈ Ω ¯ , \left\{\begin{aligned} \displaystyle u_{t}-a(t)(u_{xx}+u_{yy})&\displaystyle=f% (x,y,t),&\hskip 10.0pt(x,y,t)&\displaystyle\in\Omega\times(0,T),\\ \displaystyle u(x,y,t)&\displaystyle=0,&\hskip 10.0pt(x,y)&\displaystyle\in% \partial\Omega\times(0,T),\\ \displaystyle u(x,y,T)&\displaystyle=h(x,y),&\hskip 10.0pt(x,y)&\displaystyle% \in\overline{\Omega},\end{aligned}\right.\vspace*{-0.5mm} where Ω = ( 0 , π ) × ( 0 , π ) {\Omega=(0,\pi)\times(0,\pi)} . In the problem, the source f = f ⁢ ( x , y , t ) {f=f(x,y,t)} and the final data h = h ⁢ ( x , y ) {h=h(x,y)} are determined through random noise data g i ⁢ j ⁢ ( t ) {g_{ij}(t)} and d i ⁢ j {d_{ij}} satisfying the regression models g i ⁢ j ⁢ ( t ) = f ⁢ ( X i , Y j , t ) + ϑ ⁢ ξ i ⁢ j ⁢ ( t ) , \displaystyle g_{ij}(t)=f(X_{i},Y_{j},t)+\vartheta\xi_{ij}(t), d i ⁢ j = h ⁢ ( X i , Y j ) + σ i ⁢ j ⁢ ε i ⁢ j , \displaystyle d_{ij}=h(X_{i},Y_{j})+\sigma_{ij}\varepsilon_{ij}, where ( X i , Y j ) {(X_{i},Y_{j})} are grid points of Ω. The problem is severely ill-posed. To regularize the instable solution of the problem, we use the trigonometric least squares method in nonparametric regression associated with the projection method. In addition, convergence rate is also investigated numerically.