相关论文: Two results on ill-posed problems
Consider the Dirichlet problem with respect to an elliptic operator \[ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 \] on a bounded Wiener regular open set…
Conditional stability estimates are a popular tool for the regularization of ill-posed problems. A drawback in particular under nonlinear operators is that additional regularization is needed for obtaining stable approximate solutions if…
Partial inverse problems are studied for Sturm-Liouville operators with a discontinuity. The main results of the paper are local solvability and stability of the considered inverse problems. Our approach is based on a constructive algorithm…
We consider the stationary semilinear Schr\"odinger equation $-\Delta u + a(x) u = f(x,u)$, $u\in H^1(\R^N)$, where $a$ and $f$ are continuous functions converging to some limits $a_\infty>0$ and $f_\infty=f_\infty(u)$ as $|x|\to\infty$. In…
We consider a general statistical linear inverse problem, where the solution is represented via a known (possibly overcomplete) dictionary that allows its sparse representation. We propose two different approaches. A model selection…
The article deals with gradient-like iterative methods for solving nonlinear operator equations on Hilbert and Banach spaces. The authors formulate a general principle of studying such methods. This principle allows to formulate simple…
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega$ in $R^N$ with Dirichlet boundary conditions. The operator $L$ is a uniformly elliptic operator of order $2m$. We assume that for…
Our focus is on the stable approximate solution of linear operator equations based on noisy data by using $\ell^1$-regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where…
A choice of first-order variables for the characteristic problem of the linearized Einstein equations is found which casts the system into manifestly well-posed form. The concept of well-posedness for characteristic problems invoked is that…
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…
We consider the $2m$-th order elliptic boundary value problem $Lu=f(x,u)$ on a bounded smooth domain $\Omega\subset\R^N$ with Dirichlet boundary conditions on $\partial\Omega$. The operator $L$ is a uniformly elliptic linear operator of…
We present a regularization method to approach a solution of the pessimistic formulation of ill -posed bilevel problems . This allows to overcome the difficulty arising from the non uniqueness of the lower level problems solutions and…
In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the…
In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…
In the framework of inverse linear problems on infinite-dimensional Hilbert space, we prove the convergence of the conjugate gradient iterates to an exact solution to the inverse problem in the most general case where the self-adjoint,…
We consider the composition of operators with non-closed range in Hilbert spaces and how the nature of ill-posedness is affected by their composition. Specifically, we study the \mbox{Hausdorff-,} Ces\`{a}ro-, integration operator, and…
We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a…
For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem.…
A convergent iterative process is constructed for solving any solvable linear equation in a Hilbert space.
In this paper we consider nonlinear problems with an operator depending only on the deformation tensor. We consider the class of operators derived from a potential and with $(p,\delta)$ structure, for $1<p\leq 2$ and for all $\delta\geq0$.…