Quasi-inverse Method Research Articles

Construction of the differential equations system of the program motion in Lagrangian variables in the presence of random perturbations

The classification of inverse problems of dynamics in the class of ordinary differential equations is given in the Galiullin’s monograph. The problem studied in this paper belongs to the main inverse problem of dynamics, but already in the class of second-order stochastic differential equations of the Ito type. Stochastic equations of the Lagrangian structure are constructed according to the given properties of motion under the assumption that the random perturbing forces belong to the class of processes with independent increments. The problem is solved as follows: First, a second-order Ito differential equation is constructed so that the properties of motion are the integral manifold of the constructed stochastic equation. At this stage, the quasi-inversion method, Erugin’s method and Ito’s rule of stochastic differentiation of a complex function are used. Then, by applying the constructed Ito equation, an equivalent stochastic equation of the Lagrangian structure is constructed. The necessary and sufficient conditions for the solvability of the problem of constructing the stochastic equation of the Lagrangian structure are illustrated by the example of the problem of constructing the Lagrange function from a motion property of an artificial Earth satellite under the action of gravitational forces and aerodynamic forces.

ОБ ОБРАТНОЙ СТОХАСТИЧЕСКОЙ ЗАДАЧЕ ВОССТАНОВЛЕНИЯ

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

ОБ ОБРАТНОЙ СТОХАСТИЧЕСКОЙ ЗАДАЧЕ ВОССТАНОВЛЕНИЯ

In this paper, general reconstruction problem in the class of second-order stochastic differential equations of the Ito type is considered for given properties of motion, when the control is included in the drift coefficient. And the form of control parameters is determined by the quasi-inversion method, which provides necessary and sufficient conditions for existence of a given integral manifold. Further, the solution of the Meshchersky’s stochastic problem is given, which is one of the inverse problems of dynamics and, according to the well-known Galiullin’s classification, refers to the restoration problem. It is assumed that random perturbations belong to the class of processes with independent increments. To solve the posed problem an equation of perturbed motion is drawn up by the Ito rule of stochastic differentiation. And, further, the Erugin method in combination with the quasi-inversion method is used to construct: 1) a set of control vector functions and 2) a set of diffusion matrices that provide necessary and sufficient conditions for a given second-order differential equation of Ito type to have a given integral manifold. The linear case of a stochastic problem with drift control is considered separately. In the linear setting, in contrast to the nonlinear formulation, the conditions of solvability in the presence of random perturbations from the class of processes with independent increments coincide with the conditions of solvability in a similar linear case in the presence of random perturbations from the class of independent Wiener processes. Also considered is the scalar case of the recovery problem with drift controls.

Two Point Problem for the Abstract Differential Equation of the Second Order

The work is about the study of the boundary value problem for the second-order differential operator equation. The problem is investigated for conditional correctness. Theorems on uniqueness and conditional stability are proved. An approximate solution of the problem by the quasi-inversion method is constructed, an estimate of the norm between the exact and approximate solutions is obtained, and a formula for the regularization parameter is found.

On the Solvability of the Main Inverse Problem for Stochastic Differential Systems

By using the quasiinversion method, we establish necessary and sufficient conditions for the solvability of the main (according to Galiullin’s classification) inverse problem in the class of first-order Ito stochastic differential equations with random perturbations from the class of processes with independent increments with diffusion degenerated with respect to a part of variables and given properties depending on a part of variables.

An Efficient Spectral Method to Solve Multi-Dimensional Linear Partial Different Equations Using Chebyshev Polynomials

We present a new method to efficiently solve a multi-dimensional linear Partial Differential Equation (PDE) called the quasi-inverse matrix diagonalization method. In the proposed method, the Chebyshev-Galerkin method is used to solve multi-dimensional PDEs spectrally. Efficient calculations are conducted by converting dense equations of systems sparse using the quasi-inverse technique and by separating coupled spectral modes using the matrix diagonalization method. When we applied the proposed method to 2-D and 3-D Poisson equations and coupled Helmholtz equations in 2-D and a Stokes problem in 3-D, the proposed method showed higher efficiency in all cases than other current methods such as the quasi-inverse method and the matrix diagonalization method in solving the multi-dimensional PDEs. Due to this efficiency of the proposed method, we believe it can be applied in various fields where multi-dimensional PDEs must be solved.

Stochastic inverse problem with indirect control

In the class of stochastic differential systems of equations of Ito type with indirect control by the first or second derivative, we consider the inverse dynamic problem of constructing a system controller such that a given manifold is an integral manifold of the system. In both cases, the quasi-inversion method is used to construct the whole set of equations of controllers providing the solution of this problem.

Simultaneous EUV Flare Variation Minimization and CMP Control by Coupling-Aware Dummification

Extreme ultraviolet (EUV) flare and post-chemical mechanical polishing (CMP) metal thickness are two main manufacturability concerns that introduce critical dimension distortions in nanometer process technology. Dummification, the addition of dummy patterns, is an effective technique to address the two concerns. However, while the two dummification objectives are competing in nature, existing works only tackle them separately, leading to problem-prone solutions because optimizing one would unavoidably deteriorate the other. This paper presents a new and effective method that simultaneously considers both concerns during dummification. Manufacturing sensitivity toward the two concerns are taken into account with a user-specified evaluation model adaptive to the adopted technology. Given the point spread function of a system and the evaluation model, our proposed two-stage method is able to find at the first stage an initial dummy assignment with better EUV flare uniformity than that obtained by a previous quasi-inverse method. With the initial assignment as the starting point, the gradient-guided optimization is then adopted to iteratively refine dummy distribution toward improved CMP quality. Experimental results on industrial test cases show the effectiveness of our method.

Quasi-inversion method for an evolutionary equation of fractional order

Let B be a linear closed densely defined operator on a Banach space having no resolvent in general. The paper studies the Cauchy-type problem for the evolutionary fractional-order equation D α Bu(t), t > 0, where 0 < α ≤ 1, D α u(t) is the Riemann–Liouville fractional derivative of order α.

Reconstruction of boundary regimes in the inverse problem of thermal convection of a high-viscosity fluid

A problem of reconstruction of boundary regimes in a model for free convection of a high-viscosity fluid is considered. A variational method and a quasi-inversion method are suggested for solving the problem in question. The variational method is based on the reduction of the original inverse problem to some equivalent variational minimum problem for an appropriate objective functional and solving this problem by a gradient method. When realizing the gradient method for finding a minimizing element of the objective functional, an iterative process actually reducing the original problem to a series of direct well-posed problems is organized. For the quasi-inversion method, the original differential model is modified by means of introducing special additional differential terms of higher order with small parameters as coefficients. The new perturbed problem is well-posed; this allows one to solve this problem by standard methods. An appropriate choice of small parameters gives an opportunity to obtain acceptable qualitative and quantitative results in solving the inverse problem. A comparison of the methods suggested for solving the inverse problem is made with the use of model examples.

Application of the Quasi-Inverse Method to Data Assimilation

Four-dimensional variational data assimilation (4D-Var) seeks to find an optimal initial field that minimizes a cost function defined as the squared distance between model solutions and observations within an assimilation window. For a perfect linear model, Lorenc showed that the 4D-Var forecast at the end of the window coincides with a Kalman filter analysis if two conditions are fulfilled: (a) addition to the cost function of a term that measures the distance to the background at the beginning of the assimilation window, and (b) use of the Kalman filter background error covariance in this term. The standard 4D-Var requires minimization algorithms along with adjoint models to compute gradient information needed for the minimization. In this study, an alternative method is suggested based on the use of the quasi-inverse model that, for certain applications, may help accelerate the solution of problems close to 4D-Var. The quasi-inverse approach for the forecast sensitivity problem is introduced, and then a closely related variational assimilation problem using the quasi-inverse model is formulated (i.e., the model is integrated backward but changing the sign of the dissipation terms). It is shown that if the cost function has no background term, and has a complete set of observations (as assumed in many classical 4D-Var papers), the new method solves the 4D-Var-minimization problem efficiently, and is in fact equivalent to the Newton algorithm but without having to compute a Hessian. If the background term is included but computed at the end of the interval, allowing the use of observations that are not complete, the minimization can still be carried out very efficiently. In this case, however, the method is much closer to a 3D-Var formulation in which the analysis is attained through a model integration. For this reason, the method is called ‘‘inverse 3D-Var’’ (I3D-Var). The I3D-Var method was applied to simple models (viscous Burgers’ equation and Lorenz model), and it was found that when the background term is ignored and complete fields of noisy observations are available at multiple times, the inverse 3D-Var method minimizes the same cost function as 4D-Var but converges much faster. Tests with the Advanced Regional Prediction System (ARPS) indicate that I3D-Var is about twice as fast as the adjoint Newton method and many times faster than the quasi-Newton LBFGS algorithm, which uses the adjoint model. Potential problems (including the growth of random errors during the integration back in time) and possible applications to preconditioning, and to problems such as storm-scale data assimilation and reanalysis are also discussed.

Some direct and inverse problems in land subsidence theory: V. Dimova, in: Land subsidence. Proc. international symposium, The Hague, 1995, ed. F. B. J. Barends & others, (IAHS; Publication, 234), 1995, pp 269–275

One of the effects of the underground human activities (geotechnological or civil engineering works) is the occurrence of land subsidence. The latter creates unfavourable conditions for the functioning of surface items (buildings, equipment, bands, etc.). In connection with this, the following two problems are considered and solved in this paper. (a) Direct problem the mining system is given; determine the equation of the land subsidence. This problem is reduced to the Dirichlet problem for Fourier's equation, (b) Inverse problem the equation of the mining subsidence is given, determine the underground operations from which the given subsidence is realized. This problem is of extreme importance in the cases when mining has to be done under built-up areas. In these cases the building standards dictate the land subsidence equation. This necessitates a more rigorous formulation of a new inverse problem in the land subsidence theory. The problem is reduced to an inverse Dirichlet problem for Fourier's equation. This is an incorrectly posed problem of mathematical physics in the sense of Hadamard. Its solution is obtained by Lions' quasi-inversion method. Some generalizations of the posed problems are discussed. The posing and solving of the unique inverse problem in the applied geosciences allows to speak about laying the foundations of a modern land protection geo-engineering field.

On the uniform approximation of the solutions of a certain inverse problem by the quasiinversion method

In accordance with [1], the quasiinversion type methods include algorithms for solving inverse problems for partial differential equations, related to the introduction of additional terms with small parameters at the highest derivatives into the equations. In connection with the quasiinversion method, the problem for the heat conduction equation with inverse time direction, investigated first by A. N. Tikhonov [2], has obtained considerable popularity. As shown in [3], under definite conditions, the quasiinversion method is a regularizing algorithm. The convergence problems in the uniform metric have been investigated separately [4]. Below we present a method for the approximate solution of the inverse problem, connected with the restoration of the boundary condition in the boundary formulation for the one-dimensional heat conduction equation with a coefficient that is piecewise constant with respect to the space variable, from additional information on the solution, given on the other part of the boundary. The method is based on the replacement of the initial equation by a hyperbolic one with a small parameter at the highest derivative. We prove the convergence of the obtained approximation to the exact solution of the inverse problem in the uniform metric when the number, characterizing the error of the initial information, tends to zero. Assume that a function u(x, t) of two variables satisfies the equation and the boundary conditions that form the boundary value problem

REGULARIZED DIFFERENCE SCHEMES FOR EVOLUTIONARY SECOND ORDER EQUATIONS

The questions of approximate solution of unstable problems for evolutionary second order equations are discussed in this paper. The classical Cauchy problem for elliptic type equation is a significant example of such problem. Incorrectness of this problem (the Hadamard example) is due to instability of the solution towards small perturbations of the initial conditions. The extension problem of the solutions of well-posed elliptic problems beyond the calculation region boundary is also discussed. The stability of corresponding difference schemes is investigated by basing on general theory of ρ-stability. The principle of the regularization of three-layer difference schemes is developed for the unstable problems. It is shown that the regularized difference schemes correspond to some modification of quasi-inversion method.

Stability of the quasi-inversion method in the solution of incorrect evolutionary problems

Stability of the quasi-inversion method in the solution of incorrect evolutionary problems