A Regularised Total Least Squares Approach for 1D Inverse Scattering

We study the inverse scattering problem for a Schrödinger operator related to a static wave operator with variable velocity, using the GLM (Gelfand–Levitan–Marchenko) integral equation. We assume to have noisy scattering data, and we derive a stability estimate for the error of the solution of the GLM integral equation by showing the invertibility of the GLM operator between suitable function spaces. To regularise the problem, we formulate a variational total least squares problem, and we show that, under certain regularity assumptions, the optimisation problem admits minimisers. Finally, we compute numerically the regularised solution of the GLM equation using the total least squares method in a discrete sense.

ECOLOGICAL PLASTICITY AND STABILITY OF PROSPECTIVE PEA LINES

The studies were carried out in order to assess the parameters of the adaptability of promising pea samples in terms of yield to identify the best genotypes for the conditions of middle Volga region. The work was carried out in 2018-2020 in the central zone of Ulyanovsk region. The object of the research was 10 pea samples, the standard was Ukaz variety. According to the methods of S.A. Eberhart, W.A. Russell, V.V. Khangildina and S.P. Martynova determined the adaptability of breeding samples using the following indicators: coefficient of variation (V%), homeostaticity (Hom), breeding value (Sc), stability index (Sj2), linear coefficient regression (bi), point stability estimate (Hi). On average, over three years of research, the greatest increase in yield, compared to the standard, was noted for Ulyanovskiy yubileiny variety - 0.43 t/ha. The genotypes of Ulyanovskiy yubileiny, Viridis and line 657/14 with the smallest values of the coefficients of variation - 14.6, 22.4, 23.4%, respectively, are attributed to the most stable in terms of the coefficient of variation V. The most valuable in terms of plasticity and stability were the Ukaz variety (bi=1.15 and Sj2=0.02) and line 559/11 (1.14 and 0.00 respectively). Line 621/14 (bi=1.42 and Sj2=0.15) was recognized as an intensive variety with very low phenotypic stability and line 752/14 (1.29 and 0.11 respectively), with a reduced variety. Lines 215/11, 533/14, 657/14 were distinguished by very high phenotypic stability (bi=0,91…1,07, Sj2=0,00…0,03). The highest level of homeostaticity in combination with breeding value was observed in the promising pea cultivar Ulyanovskiy Yubileiny (Hom=15.33 and Sc=1.67) and line 215/11 (Hom=7.74 and Sc=1.22). According to the point assessment of Hi stability, significant advantages were revealed in Ulyanovskiy yubileiny variety (Hi =4.22) and line 215/11 (1.33). According to the sum of the ranks of the six parameters of adaptability, the leading positions were occupied by lines 533/14 (27), 215/11 (32) and promising varieties Ulyanovskiy yubileiny (32), Viridis (32). According to the test results, two samples in 2020 were submitted for state variety testing

Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

Non-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

Recovery of a Lamé parameter from displacement fields in nonlinear elasticity models

We study some inverse problems involving elasticity models by assuming the knowledge of measurements of a function of the displaced field. In the first case, we have a linear model of elasticity with a semi-linear type forcing term in the solution. Under the hypothesis the fluid is incompressible, we recover the displaced field and the second Lamé parameter from power density measurements in two dimensions. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement, with the background pressure fixed. On the other hand, we prove in dimensions two and three a stability result for the second Lamé parameter when the displacement field follows the (nonlinear) Saint-Venant model when we add the knowledge of displaced field solution measurements. The Saint-Venant model is the most basic model of a hyperelastic material. The use of over-determined elliptic systems is new in the analysis of linearization of nonlinear inverse elasticity problems.

An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

Conditional stability for an inverse coefficient problem of a weakly coupled time-fractional diffusion system with half order by Carleman estimate

Under a priori boundedness conditions of solutions and coefficients, we prove a Hölder stability estimate for an inverse problem of determining two spatially varying zeroth order non-diagonal elements of a coefficient matrix in a one-dimensional fractional diffusion system of half order in time. The proof relies on the conversion of the fractional diffusion system to a system of order 4 in the space variable and the Carleman estimate.

Stability estimates for an inverse Steklov problem in a class of hollow spheres

In this paper, we study an inverse Steklov problem in a class of n-dimensional manifolds having the topology of a hollow sphere and equipped with a warped product metric. Precisely, we aim at studying the continuous dependence of the warping function defining the warped product with respect to the Steklov spectrum. We first show that the knowledge of the Steklov spectrum up to an exponential decreasing error is enough to determine uniquely the warping function in a neighbourhood of the boundary. Second, when the warping functions are symmetric with respect to 1/2, we prove a log-type stability estimate in the inverse Steklov problem. As a last result, we prove a log-type stability estimate for the corresponding Calderón problem.