Narrow Results

The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. In this study, we delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches:

• ⁽¹⁾Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials.
• ⁽²⁾Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces.
• ⁽³⁾Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional’s Euler–Lagrange equation.

As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. These networks undergo training using diverse methodologies. This approach allows us to obtain approximating functions that can be rapidly evaluated after training. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude. This allows evaluations over large datasets several times faster, including training, than traditional optimal transport algorithms. Moreover, our analytically designed deep learning architecture slightly outperforms the test error of state-of-the-art CNN architectures on datasets of images.

Rotating beams are often encountered in the wind turbines and the rotors, and detection of the damages in rotating beams as earlier as possible is central to ensuring the safety and serviceability of practical structures. To this end, a modal sensitivity approach in conjunction with the sparse regularization is proposed in this paper. First, the eigen equations for the flap-wise and chord-wise vibrations of a rotating beam are established upon Hamilton’s principle. Then, damage detection is formulated as a nonlinear least-squares problem that finds the damage coefficients to minimize the error between the measured and calculated data. To solve the nonlinear least-squares problem, the sensitivity method that requires the modal sensitivity analysis is developed. In real applications, damage detection is usually an ill-posed problem and to circumvent the ill-posedness, the sparse regularization is introduced due to the fact that the numbers of actual damage locations are often scarce. Numerical examples are studied and results show that the proposed approach is more accurate than the enhanced sensitivity approach and the flap-wise modal data outperforms the chord-wise modal data in damage detection of rotating beams.

Accurate identification of moving vehicle loads on bridges is one of the challenging tasks in bridge structural health monitoring, but lacks of intensive investigations to merge the heterogeneous data of vision-based vehicle spatiotemporal information (VVSI) and vehicle-induced bridge responses for moving force identification (MFI) in the existing time domain methods (TDM). In this study, a novel MFI method is proposed by integrating instantaneous VVSI and an improved TDM (iTDM). At first, a novel VVSI method combining background subtraction with template matching is presented to accurately track moving vehicles on bridges. With the calibration technique and camera perspective transformation model, the distribution of vehicles (DOV) on bridges is obtained and used as a priori information in the subsequent MFI. Then, the iTDM is developed based on the MFI equation re-formed in the form of instantaneous VVSI instead of the constant speed vehicle crossing bridges assumed in the traditional TDM. Finally, based on the redundant dictionary matrix composed of Haar functions for a moving load, the MFI problem is converted to explore a solution to the atom vectors and then solved by the Tikhonov regularization method. Experimental verifications in laboratory and a comparative study with the existing three methods are conducted to assess the feasibility of the proposed method. The results show that the proposed MFI method outperforms the existing methods and can effectively identify the moving vehicle loads with a higher and acceptable accuracy. It is successful for the proposed method to replace the assumption of constant speed vehicle crossing bridge in the traditional TDM with the instantaneous VVSI in the MFI problem.

We study the discretization of inverse problems defined by a Carleman operator. In particular, we develop a discretization strategy for this class of inverse problems and we give a convergence analysis. Learning from examples, as well as the discretization of integral equations, can be analyzed in our setting.

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating , a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖² + ‖Lx‖² ≥ γ‖x‖² for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation where y^δ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = B^s with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {X_t}_t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.

Despite a variety of available techniques, such as discrepancy principle, generalized cross validation, and balancing principle, the issue of the proper regularization parameter choice for inverse problems still remains one of the relevant challenges in the field. The main difficulty lies in constructing an efficient rule, allowing to compute the parameter from given noisy data without relying either on any a priori knowledge of the solution, noise level or on the manual input. In this paper, we propose a novel method based on a statistical learning theory framework to approximate the high-dimensional function, which maps noisy data to the optimal Tikhonov regularization parameter. After an offline phase where we observe samples of the noisy data-to-optimal parameter mapping, an estimate of the optimal regularization parameter is computed directly from noisy data. Our assumptions are that ground truth solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. We show that for our method to be efficient, the number of previously observed samples of the noisy data-to-optimal parameter mapping needs to scale at most linearly with the dimension of the solution subspace. We provide explicit error bounds on the approximation accuracy from noisy data of unobserved optimal regularization parameters and ground truth solutions. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach. We conclude with presenting numerical experiments verifying our theoretical results and illustrating the superiority of our method with respect to several state-of-the-art approaches in terms of accuracy or computational time for solving inverse problems of various types.

We consider a Cauchy problem for a modified Helmholtz equation, especially when we give the optimal error bound for this problem. Some spectral regularization methods and a revised Tikhonov regularization method are used to stabilize the problem from the viewpoint of general regularization theory. Hölder-type stability error estimates are provided for these regularization methods. According to the optimality theory of regularization, the error estimates are order optimal.

In this paper, an efficient numerical method is proposed to handle two-dimensional fractional diffusion equations on a finite domain. The proposed method combines the product of Legendre wavelet bases for two spatial dimensions and a time direction. The operational matrix of the proposed method is obtained. Tikhonov regularization is employed to stabilize the system in cases where the final linear system of equations is large. The convergence analysis of the method is studied and some numerical examples are presented to investigate the efficiency and accuracy of the method.

In this paper, we report on a strategy for computing the numerical approximate solution for a class of ill-posed operator equations in Hilbert spaces: $K:E\to F,Kf=g$. This approach is a combination of Tikhonov regularization method and the finite rank approximation of $K^{\ast }K$. Finally, numerical results are given to show the effectiveness of this method.

In this paper, we study the problem of reconstructing an unknown initial condition for nonlinear diffusion from integral observations which have many practical meanings. We reformulate this problem as a variational one to minimizing objective functional. We prove the existence of this minimized problem in our main result.

We consider an inverse problem for the Poisson equation $-\mathrm{\Delta }u=f$ in the square $\mathrm{\Omega }=(0,1)\times (0,1)$ which consists of determining the source $f$ from boundary measurements. Such problem is ill-posed. We restrict ourselves to a class of functions $f(x_1,x_2)={\phi }_1(x_2)g_1(x_1)+{\phi }_2(x_2)g_2(x_1)$. To illustrate our method, we first assume that $g_1$ and $g_2$ are known functions with partial data at the boundary. For the reconstruction, we consider approximations by the Fourier series, therefore we obtain an ill-posed linear system which requires a regularization strategy. In the general case, we propose an iterative algorithm based on the full data at the boundary. Finally, some numerical results are presented to show the effectiveness of the proposed reconstruction algorithms.

In this paper, the inverse problem of reconstructing the right-hand side from integral observation is studied by using variational method to minimizing objective functional combining with Tikhonov regularization. The L-curve method is suggested for choosing the regularization parameter.

Electrical impedance tomography is a recently established technique by which impedance of an object (medical or nonmedical applications) is measured data from the surface of the object, and a numerically simulated reconstruction of the object internal shape of the image can be obtained. This imaging technique based on boundary or surface voltage is measured when the different current pattern is injected into it. For current pulse, we are creating a voltage controlled current source, which is based on the different RC circuits, according to current amplitude and frequency values. The current source used in inject the current pulse of the various phantoms. The current position and measuring voltage is controlled by the created control unit or programmable system on chip (PSOC) of the proposed EIT system. After that image reconstruction of the cross-sectional image of resistivity requires sufficient data collection from used phantoms, which is based on finite element method (FEM) method and Tikhonov regularization method with helps of graphical user interface (GUI) on MatLab. The objective of the GUI was to produce an image (2D/3D), impedance distribution graph, and the FEM mesh model according to used electrode combinations from the various phantoms. EIT system has a great potential for imaging modality, is non-invasive, radiation-free, and inexpensive for medical applications.

By a new concept and method we shall give practical and numerical solutions of linear singular integral equations by combining the two theories of the Tikhonov regularization and reproducing kernels.

We shall give very natural, analytical, numerical and approximate real inversion formulas of the Laplace transform for natural reproducing kernel Hilbert spaces by using the ideas of best approximations, generalized inverses and the theory of reproducing kernels having a good connection with the Tikhonov regularization. These approximate real inversion formulas may be expected to be practical to calculate the inverses of the Laplace transform by computers when the real data contain noises or errors. We shall illustrate examples, by using computers.

We give a new algorithm constructing harmonic functions from data on a part of a boundary. Our approach is based on a general concept and we can apply our methodology to many problems, but here we numerically deal with an ill-posed Cauchy problem which appears in many applications. On some part Γ of a boundary, for suitably given functions f and g, we look for a constructing formula of an approximate harmonic function u satisfying u = f and , the outer normal derivative. Our method is based on the Dirichlet principle by combinations with generalized inverses, Tikhonov's regularization and the theory of reproducing kernels.

We shall give a new inversion formula for a linear system based on physical experimental data and by using reproducing kernels and Tikhonov regularization. In particular, we will not make any analytical assumption on the linear system, but will use physical experimental data for obtaining an approximate inversion formula for the linear system.

Inverse problems is a vibrant and expanding branch of mathematics that has found numerous applications. In this work, we will survey some of the main techniques available in the literature to solve the inverse problem of identifying variable parameters in partial differential equations. Besides carefully defining the problem, we give simple examples to depict some of the difficulties associated with the study of this inverse problem. We will discuss in sufficient detail twelve different methods available to solve this particular class of inverse problems. We also analyze some of their most exciting applications. We will also point out some of the research directions which can be pursued in this fascinating branch of applied and industrial mathematics.