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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.

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 }

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.

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.

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.

In this paper, we investigate a backward problem for a time-fractional diffusion equation in a general abstract Hilbert space. We show that the problem is ill-posed and further apply a truncation regularization method to solve it. Based on a Holder-type smoothness assumption of the exact solution, asymptotically optimal estimates for the worst case error of the method in Hilbert scales are proved.

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.

We consider the inverse problem of finding the volatility σ∈L ρ (0, T) such that $U_{BS}(X,K,r,t,{{\int }_{0}^{t}}\sigma ^{2}(\tau )d\tau )=u(t)$ , 0≤t≤T, where U B S is the Black–Scholes formula and u(t) is the observable fair price of an European call option. The problem is ill-posed. Using the residual method, we shall regularize the problem. An explicit error estimate is given.

We consider the problem of reconstructing, from the interior data $u(x,1)$, a function $u$ satisfying a nonlinear elliptic equation $$ \Delta u = f(x,y,u(x,y)), x \in \RR, y > 0. $$, Comment: 24 pages

Let $Q$ be a heat conduction body and let $\varphi = \varphi(t)$ be given. We consider the problem of finding a two-dimensional heat source having the form $\varphi(t)f(x,y)$ in $Q$. The problem is ill-posed. Assuming $\partial Q$ is insulated and $\varphi \not\equiv 0$, we show that the heat source is defined uniquely by the temperature history on $\partial Q$ and the temperature distribution in $Q$ at the initial time $t = 0$ and at the final time $t = 1$. Using the method of truncated integration and the Fourier transform, we construct regularized solutions and derive explicitly error estimate.

The only source of ice formation in the Arctic during summer is a layer of ice called false-bottoms between an under-ice melt pond and the underlying ocean. Of interest is to give a mathematical model in order to determine the simultaneous growth and ablation of false-bottoms, which is governed by both of heat fluxes and salt fluxes. In one dimension, this problem may be considered mathematically as a two-phase Stefan problem with two free boundaries. Our main result is to prove the existence and uniqueness of the solution from the initial condition., 22 pages

Let $\Omega$ represent a two$-$dimensional isotropic elastic body. We consider the problem of determining the body force $F$ whose form $\phi(t)(f_1(x),f_2(x))$ with $\phi$ be given inexactly. The problem is nonlinear and ill-posed. Using the Fourier transform, the methods of Tikhonov's regularization and truncated integration, we construct a regularized solution from the data given inexactly and derive the explicitly error estimate. Numerical part is given, Comment: 23 pages

We consider the two-dimensional problem of recovering globally in time the heat distribution on the surface of a layer inside of a heat conducting body from two interior temperature measurements. The problem is ill-posed. The approximation function is represented by a two-dimensional Sinc series and the error estimate is given., 10 pages

We consider the problem of finding a function defined on $(0,\infty)$ from a countable set of values of its Laplace transform. The problem is severely ill-posed. We shall use the expansion of the function in a series of Laguerre polynomials to convert the problem in an analytic interpolation problem. Then, using the coefficients of Lagrange polynomials we shall construct a stable approximation solution., Comment: 14 pages

13 pages; International audience; We consider the two-dimensional problem of recovering globally in time the heat flux on the surface of a layer inside of a heat conducting body from two interior temperature measurements. The problem is ill-posed. The approximation function is represented by a two-dimensional Sinc series and the error estimate is given.