Inverse Operator Research Articles

Uncertainty quantification of multimodal surface wave inversion using artificial neural networks

An inversion of surface wave dispersion curves (DCs) is a nonunique and ill-conditioned problem. The inversion result has a probabilistic nature, which becomes apparent when simultaneously restoring the S-wave velocity and layer thickness. Therefore, the problem of uncertainty quantification is relevant. Existing methods through deterministic or global optimization approaches of uncertainty quantification via posterior probability density (PPD) of the model parameters are not computationally efficient because they demand multiple solutions to the inverse problem. We have developed an alternative method based on a multilayer fully connected artificial neural network (ANN). We improve the current unimodal approach, which is known from publications, to multimodal inversion. We use Cox’s and Teague’s algorithm to determine optimal parameterization (number of layers) and the ranges of possible model parameters. We uniformly draw training data sets within estimated ranges and train the ANN. Saved ANN weights map the phase velocity DCs to values of the S-wave velocity and layers thickness. To estimate the uncertainties, we adapt the Monte Carlo simulation strategy, according to which the frequency dependent data noise and the inverse operator errors are projected onto the resulting velocity model. The inverse operator errors are evaluated by the prediction of the training data set. The proposed combination of surface wave data processing methods, configured with each other, provides a novel surface wave multimodal dispersion data inversion and uncertainty quantification approach. We first test our approach on synthetic experiments for various velocity models: a positive gradient velocity, a low-velocity layer, and a high-velocity layer. This is done considering unimodal inversion at first and then compared with the multimodal inversion. Afterward, we apply our approach to field data and compare the resulting models with the body S-wave processing by the generalized reciprocal method. The experiments indicate high-potential results — using ANN yields the possibility to accurately estimate the PPD of restored model parameters without a significant computational effort. The PPD-based comparison demonstrates the advantages of a multimodal inversion over a unimodal inversion. The trained ANN provides reasonable model parameters predictions and related uncertainties in real time.

Neural Inverse Operators for solving PDE Inverse Problems

Neural Inverse Operators for solving PDE Inverse Problems

Neural Inverse Operators for solving PDE Inverse Problems

Neural Inverse Operators for solving PDE Inverse Problems

A Symbolic Method for Solving a Class of Convolution-Type Volterra–Fredholm–Hammerstein Integro-Differential Equations under Nonlocal Boundary Conditions

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial and boundary conditions. Closed-form solutions are always desirable, not only because they are more efficient, but also because they can be valuable benchmarks for validating approximate and numerical procedures. This paper presents a direct operator method for solving, in closed form, a class of Volterra–Fredholm–Hammerstein-type integro-differential equations under nonlocal boundary conditions when the inverse operator of the associated Volterra integro-differential operator exists and can be found explicitly. A technique for constructing inverse operators of convolution-type Volterra integro-differential operators (VIDEs) under multipoint and integral conditions is provided. The proposed methods are suitable for integration into any computer algebra system. Several linear and nonlinear examples are solved to demonstrate the effectiveness of the method.

Операторное уравнение дифракции на отрезке кругового цилиндра

A new, mathematically efficient method for solving the vector equation of diffraction on an open surface of rotation is proposed. The method is based on the allocation of the main operator, the definition of functional spaces and the reduction of the operator equation to the Fredholm equation of the second kind. Sobolev spaces are used as ones that take into account the Meixner condition on the edge. In the selected spaces, the main operator is bounded and invertible; the inverse operator is also bounded. A projection method for solving operator equations has been developed.

On orthogonally additive band operators and orthogonally additive disjointness preserving operators

Let $M$ and $N$ be Archimedean vector lattices. We introduce orthogonally additive band operators and orthogonally additive inverse band operators from $M$ to $N$ and examine their properties. We investigate the relationship between orthogonally additive band operators and orthogonally additive disjointness preserving operators and show that under some assumptions on vector lattices $M$ or $N$, these two classes are the same. By using this relation, we show that if ${\mu }$ is a bijective orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator) from $M$ into $N$ then ${\mu }^{-1}$:$N$${\rightarrow}$$M$ is an orthogonally additive band operator (resp. orthogonally additive disjointness preserving operator).

An Example of a Bounded Potential <i>q</i>(<i>x</i>) on the Half-Line, for Estimates of <i>A</i>(<i>α</i>) Amplitude

We show an example of a bounded potential on the half-line obtained as the image of an Inverse Transformation Operator of the Bessel singular potential of the Reduced Radial Schrödinger Equation, and show us the Estimates of the A(α) amplitude.

Inverse Differential Operators in Time and Space

Inverse Differential Operators in Time and Space

Avoiding Small Denominator Problems by Means of the Homotopy Analysis Method

The so-called ``small denominator problem'' was a fundamental problem of dynamics, as pointed out by Poincar\'{e}. Small denominators appear most commonly in perturbative theory. The Duffing equation is the simplest example of a non-integrable system exhibiting all problems due to small denominators. In this paper, using the forced Duffing equation as an example, we illustrate that the famous ``small denominator problems'' never appear if a non-perturbative approach based on the homotopy analysis method (HAM), namely ``the method of directly defining inverse mapping'' (MDDiM), is used. The HAM-based MDDiM provides us great freedom to directly define the inverse operator of an undetermined linear operator so that all small denominators can be completely avoided and besides the convergent series of multiple limit-cycles of the forced Duffing equation with high nonlinearity are successfully obtained. So, from the viewpoint of the HAM, the famous ``small denominator problems'' are only artifacts of perturbation methods. Therefore, completely abandoning perturbation methods but using the HAM-based MDDiM, one would be never troubled by ``small denominators''. The HAM-based MDDiM has general meanings in mathematics and thus can be used to attack many open problems related to the so-called ``small denominators''.

A derivative-type operator and its application to the solvability of a nonlinear three point boundary value problem

In this paper we introduce an operator that can be thought as a derivative of variable order, i.e. the order of the derivative is a function. We prove several properties of this operator, for instance, we obtain a generalized Leibniz’s formula, Rolle and Cauchy’s mean theorems and a Taylor type polynomial. Moreover, we obtain its inverse operator. Also, with this derivative we analyze the existence of solutions of a nonlinear three-point boundary value problem of “variable order”.

Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems

We study the problem of reconstructing solutions of inverse problems when only noisy measurements are available. We assume that the problem can be modeled with an infinite-dimensional forward operator that is not continuously invertible. Then, we restrict this forward operator to finite-dimensional spaces so that the inverse is Lipschitz continuous. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the operator. In addition, we show that these neural networks can be learned from appropriately perturbed training data. We demonstrate the admissibility of this approach to a wide range of inverse problems of practical interest. Numerical examples are given that support the theoretical findings.

Estimates of singular numbers (s$$ s $$‐numbers) and eigenvalues of a mixed elliptic‐hyperbolic type operator with parabolic degeneration

This paper is concerned with a mixed type differential operator which is initially defined with , where and is a set of infinitely differentiable functions with compact support with respect to the variable and satisfying the conditions: Regarding the coefficient , with supposition that satisfies the condition: is a piecewise continuous and bounded function in . The coefficients and are continuous functions in and can be unbounded at infinity. The operator admits closure in the space , and the closure is also denoted by . Taking into consideration certain constraints on the coefficients , apart from the above‐mentioned conditions, the existence of a bounded inverse operator is proved in this paper; a condition guaranteeing compactness of the resolvent kernel is found; and we also obtained two‐sided estimates for singular numbers ( ‐numbers). Here, we note that the estimate of singular numbers ( ‐numbers) shows the rate of approximation of the resolvent of the operator by linear finite‐dimensional operators. It is given an example of how the obtained estimates for the ‐numbers enable to identify the estimates for the eigenvalues of the operator . We note that the above results are apparently obtained for the first time for a mixed‐type operator in the case of an unbounded domain with rapidly oscillating and greatly growing coefficients at infinity.

Singular boundary behaviour and large solutions for fractional elliptic equations.

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equationsposed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace-Poisson equationwith zero boundarydata. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equationsdriven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Greenfunction.

A unified precision matrix estimation framework via sparse column-wise inverse operator under weak sparsity

In this paper, we estimate the high-dimensional precision matrix under the weak sparsity condition where many entries are nearly zero. We revisit the sparse column-wise inverse operator estimator and derive its general error bounds under the weak sparsity condition. A unified framework is established to deal with various cases including the heavy-tailed data, the non-paranormal data, and the matrix variate data. These new methods can achieve the same convergence rates as the existing methods and can be implemented efficiently.

Modified Adomian Method through Efficient Inverse Integral Operators to Solve Nonlinear Initial-Value Problems for Ordinary Differential Equations

The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable.

A fast inversion approach for the identification of highly transient surface heat flux based on the generative adversarial network

The efficient identification of surface heat flux during highly transient pool boiling process is an essential task for engineering production and manufacture, which can be mathematically defined as nonlinear inverse heat transfer problems (IHTP) of Neumann boundary conditions. Conventional regularization-based inversion approaches for these types of problems are computationally expensive since packs of nonlinear partial differential equations (PDE) constrained optimization problems in complex computational domains need to be solved. In this paper, an operator-learning method is proposed for the efficient solution of complex three-dimensional (3D) transient nonlinear IHTP. FluxNet, a generated adversarial network composed of a discrete wavelet transform (DWT) layer, a “generator” and a “discriminator”, is constructed to significantly improve the computational efficiency, reaching sub-second for the first time. The DWT layer is capable of extracting detailed information of the temperature distribution while the “generator” approximates the inverse operator of the governing PDE system with its convolution kernels. A minimax optimization strategy is adopted through adversarial training between the “generator” and the “discriminator” to encourage deeper understanding of data distribution. Compared with other types of deep learning methods, the proposed method shows its superiority when dealing with highly transient experimental data and micro-nano porous structured geometry. Based on the developed sub-second light-weight numerical solver with operator learning, the development of advanced real-time soft sensor techniques for a class of 3D transient IHTP in practical engineering applications is possible.

Application of the laws to compose fractal computation algorithms when modeling and analyzing their growth

Generalized analysis of the methods to construct fractals let to find algebraic patterns of fractal algorithm, fractal operators fi formalizing basic operations to construct the fractal structures of a specific type being the major concept. The fi are introduced with multiplication and addition operations together with the concepts of singular, zero and inverse operator that allow defining periodic and quasi-periodic fractal structures. An endless fractal length - basic operator which can give a possibility to obtain a limit fractal synthesis idea as a structure without self-similarity but with a regularly decreasing scale. The connection between fractal growth computations and widely examples of application in the nature is considered. Methods to analyze growth processes are offered, their basic characteristics are shown. The mathematical description of the fractal growth characteristics and their dependence on specifically defined parameters of algorithm are given. The analysis of fractal point growth trajectory is made. The algorithms applied and the methods to analyze growth processes are shown to be modernized and adapted for wide spectrum of tasks.

Conic Optimization with Spectral Functions on Euclidean Jordan Algebras

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones—the root-determinant cones and the matrix monotone derivative cones—we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open-source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support.Funding: This work was supported by Office of Naval Research [Grant N00014-18-1-2079] and the National Science Foundation [Grant OAC-1835443].

Super-localization of elliptic multiscale problems

Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a d d -dimensional domain. The application of the inverse operator to some standard finite element space defines an approximation space with uniform algebraic approximation rates with respect to the mesh size parameter H H . This holds even for under-resolved rough coefficients. However, the true challenge of numerical homogenization is the localized computation of a localized basis for such an operator-dependent approximation space. This paper presents a novel localization technique that leads to a super-exponential decay of its basis relative to H H . This suggests that basis functions with supports of width O ( H | log ⁡ H | ( d − 1 ) / d ) \mathcal O(H|\log H|^{(d-1)/d}) are sufficient to preserve the optimal algebraic rates of convergence in H H without pre-asymptotic effects. A sequence of numerical experiments illustrates the significance of the new localization technique when compared to the so far best localization to supports of width O ( H | log ⁡ H | ) \mathcal O(H|\log H|) .

Helicity, chirality, and spin of optical fields without vector potentials

Helicity $H$, chirality $C$, and spin angular momentum $\mathbf{S}$ are three physical observables that play an important role in the study of optical fields. These quantities are closely related, but their connection is hidden by the use of four different vector fields for their representation, namely, the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$, and the two transverse potential vectors $\mathbf{C}^\perp$ and $\mathbf{A}^\perp$. Helmholtz's decomposition theorem restricted to solenoidal vector fields, entails the introduction of a bona fide inverse curl operator, which permits one to express the above three quantities in terms of the observable electric and magnetic fields only. This yields clear expressions for $H, C$, and $\mathbf{S}$, which are automatically gauge-invariant and display electric-magnetic democracy.