Mourad Choulli
We study some hybrid inverse problems associated to BVP's for Schrödinger and Helmholtz type equations. The inverse problems we consider consist in the determination of coefficients from the knowledge of internal energies. We establish local Lipschitz stability inequalities as well as conditional Hölder stability inequalities.
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In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.
The primary objective of this research is to investigate an inverse problem of parameter identification in nonlinear mixed quasi-variational inequalities posed in a Banach space setting. By using a fixed point theorem, we explore properties of the solution set of the considered quasi-variational inequality. We develop a general regularization framework to give an existence result for the inverse problem. Finally, we apply the abstract framework to a concrete inverse problem of identifying the material parameter in an implicit obstacle problem given by an operator of $p$-Laplacian type.
We survey some of our recent results on inverse problems for evolution equations. The goal is to provide a unified approach to solve various types of evolution equations. The inverse problems we consider consist in determining unknown coefficients from boundary measurements by varying initial conditions. Based on observability inequalities and a special choice of initial conditions, we provide uniqueness and stability estimates for the recovery of volume and boundary lower order coefficients in wave and heat equations. Some of the results presented here are slightly improved from their origina…
We give necessary and sufficient conditions for the existence of weak solutions of a parabolic problem corresponding to the Kolmogorov operators perturbed by a multipolar inverse square potential with respect to the Gaussian probability measure which is the unique invariant measure for Ornstein-Uhlenbeck type operators. We state the optimality of the constant and, then, the nonexistence of positive exponentially bounded solutions to the parabolic problem.
Viscoelastic materials have the properties both of elasticity and viscosity. In a previous work we investigate glass relaxation in the framework of viscoelasticity. Here we consider the Burgers model, a first but meaningful step in the general analysis, showing a reachability theorem thanks the analysis of the gap between eigenvalues and the representation of the solution in Fourier series.
Observability and unique continuation inequalities for the Schrödinger equations with inverse-square potentials
该研究将自由 Schrödinger 方程的能观性与唯一延拓不等式推广到半轴上的反平方势情形，并利用解表示、Hankel 变换不确定性原理及紧支集插值不等式完成证明。
In the article we study mappings of Carnot groups satisfy moduli inequalities. We prove that homeomorphisms satisfy the moduli inequalities ($Q$-homeomor\-phisms) with a locally integrable function $Q$ are Sobolev mappings. On this base in the frameworks of the weak inverse mapping theorem we prove that mappings inverse to Sobolev homeomorphisms of finite distortion of the class $W^1_{ν,\text{loc}}(Ω;Ω')$ belong to the Sobolev class $W^1_{1,\text{loc}}(Ω';Ω)$.
In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measu…
In this paper, we consider the stability of quermassintegral inequalities along a inverse curvature flow. We choose a special rescaling of the flow such that the $k$-th quermassintegral is decreasing and the $k-1$-th quermassintegral is preserved. Along this rescaled flow, we prove that the decreasing rate of the $k$-th quermassintegral is faster than the Fraenkel asymmetry of the domain when approaching to the sphere. This leads to the stability inequality of quermassintegral inequalities for nearly spherical sets using the flow method.
Minkowski inequalities and constrained inverse curvature flows in warped spaces
本文研究黎曼扭曲空间中的局部受限逆曲率流，证明了其长时间存在性与平滑收敛性，并利用单调量及逆平均曲率流推导出多维空间中新的闵可夫斯基型不等式。
Let $\mathcal{L}_a$ be a Schrödinger operator with inverse square potential $a|x|^{-2}$ on $\mathbb{R}^d, d\geq 3$. The main aim of this paper is to prove weighted estimates for fractional powers of $\mathcal{L}_a$. The proof is based on weighted Hardy inequalities and weighted inequalities for square functions associated to $\mathcal{L}_a$. As an application, we obtain smoothing estimates regarding the propagator $e^{it\mathcal{L}_a}$.
We prove a rigidity result in the sphere which allows us to generalize a result about smooth convex hypersurfaces in the sphere by Do Carmo-Warner to convex $C^2$-hypersurfaces. We apply these results to prove $C^{1,β}$-convergence of inverse F-curvature flows in the sphere to an equator in \mathbb{S}^{n+1} for embedded, closed, strictly convex initial hypersurfaces. The result holds for large classes of curvature functions including the mean curvature and arbitrary powers of the Gauss curvature. We use this result to prove Alexandrov-Fenchel type inequalities in the sphere.
Inverse spectral problems for higher-order coefficients
本文利用边界谱数据与椭圆、双曲 DtN 映射的关系，证明由 BSD 恢复椭圆算子高阶系数的唯一性和稳定性，并推广至电导率与势的反演。
We consider $σ$-harmonic mappings, that is mappings $U$ whose components $u_i$ solve a divergence structure elliptic equation ${\rm div} (σ\nabla u_i)=0$, for $i=1,\ldots,n $. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.
This paper reviews recent results on hybrid inverse problems, which are also called coupled-physics inverse problems of multi-wave inverse problems. Inverse problems tend to be most useful in, e.g., medical and geophysical imaging, when they combine high contrast with high resolution. In some settings, a single modality displays either high contrast or high resolution but not both. In favorable situations, physical effects couple one modality with high contrast with another modality with high resolution. The mathematical analysis of such couplings forms the class of hybrid inverse problems. Hy…
For a general formulation of linearised hybrid inverse problems in impedance tomography, the qualitative properties of the solutions are analysed. Using an appropriate scalar pseudo-differential formulation, the problems are shown to permit propagating singularities under certain non-elliptic conditions, and the associated directions of propagation are precisely identified relative to the directions in which ellipticity is lost. The same result is found in the setting for the corresponding normal formulation of the scalar pseudo-differential equations. A numerical reconstruction procedure base…
Scale-uniform inverse inequalities for scaled kernel spaces
本文针对原生空间为Sobolev空间的缩放正定核空间，建立了尺度一致逆不等式与Bernstein不等式，为核近似方法的多尺度稳定性分析提供了关键工具。
This paper concerns the quantitative step of the medical imaging modality Thermo-acoustic Tomography (TAT). We model the radiation propagation by a system of Maxwell's equations. We show that the index of refraction of light and the absorption coefficient (conductivity) can be uniquely and stably reconstructed from a sufficiently large number of TAT measurements. Our method is based on verifying that the linearization of the inverse problem forms a redundant elliptic system of equations. We also observe that the reconstructions are qualitatively quite different from the setting where radiation…
Stability for an inverse flux and an inverse boundary coefficient problems
本文针对通量反问题建立Lipschitz和对数稳定性估计并应用于边界系数反问题，进一步利用其稳定性不等式解决腐蚀反问题，实现通过可访问边界测量确定不可访问边界的未知腐蚀系数。
There are several hybrid inverse problems for equations of the form $\nabla \cdot D \nabla u - σu = 0$ in which we want to obtain the coefficients $D$ and $σ$ on a domain $Ω$ when the solutions $u$ are known. One approach is to use two solutions $u_1$ and $u_2$ to obtain a transport equation for the coefficient $D$, and then solve this equation inward from the boundary along the integral curves of a vector field $X$ defined by $u_1$ and $u_2$. It follows from an argument of Guillaume Bal and Kui Ren that for any nontrivial choices of $u_1$ and $u_2$, this method suffices to recover the coeffic…
Nonlinear Diffusion, and Geometric and Functional Inequalities on Smooth Metric Measure spaces
本文研究了非线性扩散方程与光滑度量度量空间中的几何和函数不等式之间的联系，并提出了新的研究成果。
An inverse obstacle problem for the magnetic Schrödinger equation
本文研究磁薛定谔方程的反障碍物问题，利用边界上的两次测量重构障碍物边界上的未知函数，并证明了该反问题在时间局部具有Lipschitz稳定性、在时间全局具有对数稳定性。
Hybrid inverse problems are based on the interplay of two types of waves, in order to allow for imaging with both high resolution and high contrast. The inversion procedure often consists of two steps: first, internal measurements involving the unknown parameters and some related quantities are obtained, and, second, the unknown parameters have to be reconstructed from the internal data. The reconstruction in the second step requires the solutions of certain PDE to satisfy some non-zero constraints, such as the absence of nodal or critical points, or a non-vanishing Jacobian. In this work, we …
Stable recovery of coefficients in an inverse fault friction problem
本文研究Tresca摩擦模型中摩擦系数的反演问题，将其转化为微分不等式反问题，并证明利用破裂过程中的弹性波测量可唯一且稳定地恢复破裂面上的摩擦系数。
In this article we study the inverse problem of recovering a space-dependent coefficient of the Moore-Gibson-Thompson (MGT) equation, from knowledge of the trace of the solution on some open subset of the boundary. We obtain the Lipschitz stability for this inverse problem, and we design a convergent algorithm for the reconstruction of the unknown coefficient. The techniques used are based on Carleman inequalities for wave equations and properties of the MGT equation.
An inverse Signorini obstacle problem
该研究针对各向同性弹性系统的逆Signorini障碍问题，证明通过单次边界测量可唯一确定障碍，并发展了适用于一般微分不等式逆问题的技术。
Forward and Inverse Problems for a Langevin-Type Fractional Equation Involving Non-Local Time Condition
本文针对带非局部时间条件的Langevin型分数阶方程，提出并分析正反问题，证明解的存在唯一性，导出稳定性不等式并给出唯一性准则。
Inverse problems with hybrid lenses
本文设计结合自由曲面与超表面的混合透镜，实现任意入射场到固定方向准直光束的折射，并研究近场下将物体成像至预定目标的存在性。
The Hardy Inequality (HI) for potentials with countably many singularities of the form $V=\sum_{k\in \mathbf{Z}}\frac{1}{|x-a_k|^2}$ is not a trivial issue. In principle, the more singular poles are, the less the Hardy constant is: it is well-known that in all the existing results about the HI with finite number of singularities the best constants converge to 0 with the number $n$ of singularities going to infinity. In this note we provide an example of nontrivial HI in right cylinders of fixed radius $R>0$ in $\mathbf{R}^d$, for a potential $V$ defined above having the singularities $\{a_k\}_…
On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space
研究双曲空间中逆平均曲率流的弱解，通过受启发的方法证明特定时间下解的性质，确定扩展球面为唯一弱IMCF，还将相关不等式扩展到外最小化域及平衡渐近双曲图。
A hybrid inverse problem in the fluorescence ultrasound modulated optical tomography in the diffusive regime
研究在扩散 regime 中的荧光超声调制光学断层成像的混合反问题，通过边界测量重构吸收系数和量子效率系数。
In this paper we introduce the randomised stability constant for abstract inverse problems, as a generalisation of the randomised observability constant, which was studied in the context of observability inequalities for the linear wave equation. We study the main properties of the randomised stability constant and discuss the implications for the practical inversion, which are not straightforward.
Fractional Calderón problems and Poincaré inequalities on unbounded domains
本文将分数阶 Calderón 问题的唯一性理论推广至具非空外部的无界区域，刻画部分数据反问题的唯一性与非唯一性，并证明 Poincaré 不等式、Runge 逼近及外部确定结果。
Feasibility of hybrid inverse problems in limited view
本文研究有限视角下混合反问题（如声电断层成像）的可行性，探讨如何选择边界函数以确保重建可行，并通过数值实验验证理论结果。
The purpose of this paper is to study spectral properties of non-self-adjoint Schrödinger operators $-Δ-\frac{(n-2)^2}{4|x|^{2}}+V$ on $\mathbb{R}^n$ with complex-valued potentials $V\in L^{p,\infty}$, $p>n/2$. We prove Keller type inequalities which measure the radius of a disc containing the discrete spectrum, in terms of the $L^{p,\infty}$ norm of $V$. Similar inequalities also hold if the inverse-square potential is replaced by a large class of subcritical potentials with critical singularities at the origin. The main new ingredient in the proof is the uniform Sobolev inequality of Kenig-R…
In this paper, we perform non-linear minimization using the Hybridizable Discontinuous Galerkin method (HDG) for the discretization of the forward problem, and implement the adjoint-state method for the computation of the functional derivatives. Compared to continuous and discontinuous Galerkin discretizations, HDG reduces the computational cost by working with the numerical traces, hence removing the degrees of freedom that are inside the cells. It is particularly attractive for large-scale time-harmonic quantitative inverse problems which make repeated use of the forward discretization as th…
In this paper, we study existence results for initial value problems for hybrid fractional integro-differential equations. Our investigation is based on the Dhage hybrid fixed point theorem. Some fundamental fractional differential inequalities are also established which are utilized to prove the existence of extremal solutions. Necessary tools are considered and the comparison principle is proved which will be useful for further study of qualitative behavior of solutions.
Reconstruction of cracks in Calderón's inverse conductivity problem using energy comparisons
针对Calderón反电导率问题中的Lipschitz超曲面型裂纹，提出两种重构方法：一种可通过算子不等式收缩上界得到精确位置，能同时处理两类裂纹；另一种针对单一类型裂纹生成保证为未知裂纹子集的近似结果，填补了相关严格重构方法的空白。
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