# e-book The Computational Complexity of Differential and Integral Equations: An Information-Based Approach

In this method, all functions within D-bar integral equation are first expanded using the sinc basis functions. Then, the orthogonal properties of their products dissolve the integral operator of the D-bar equation and results a discrete convolution equation. That is, the new moment method leads to the equation solution without direct computation of the D-bar integral. The resulted discrete convolution equation maybe adapted to a suitable structure to be solved using fast Fourier transform.

This allows us to reduce the order of computational complexity to as low as O N 2 log N. Simulation results on solving D-bar equations arising in EIT problem show that the proposed method is accurate with an ultra-linear CR. Numerical solution to scattering equation is the key in solving a variety of inverse problems in engineering and science including electromagnetic inverse scattering,[ 1 ] quantum inverse scattering,[ 2 , 3 , 4 , 5 ] and medical imaging. There are several numerical methods for solving non-linear inverse scattering problem including direct and iterative approaches.

D-bar is one of the significant methods based on direct methodology introduced by Beals and Coifman[ 22 ] that recently received a lot of attentions due to circumventing highly complex iterative approaches. D-bar is based on formulating several non-linear inverse scattering integrals into some linear D-bar equations. Planar D-bar equation in partial differential form can be represented by:[ 22 ].

Here T k is a complex function. The -operator in 1. Convolving both sides of Eq. A couple of methods are introduced for the numerical solution of the D-bar equation including product integrals PI and multigrid MG. The PI method is introduced by Siltanen et al. In this approach, a uniform grid is used to discretize the Eq. Then, it computes the singular convolution integral in D-bar equation via separating it into smooth and singular parts.

The singular part of the integral is computed analytically and the smooth part may be computed using an interpolator polynomial. Then the discrete form of the equation may be solved by means of an iterative solver such as generalized minimal residual solver GMRES. In order to circumvent the huge computational complexity of PI methods, more recently, another class of algorithms is developed by Knudsen et al.

Note that, MG methodology was first introduced by Fedorenko as claimed[ 28 ] in Derivations of this methodology have shown good efficiency in solving partial differential equations PDEs.

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The first step in MG method is to obtain the periodic version of the D-bar integral equation. Next, the periodic equation is discretized using a special kind of grid mapping in which each function in 1. This type of mapping allows neglecting the singular computations of the convolution kernel; however, it imposes some error in approximating the solution of 1. Although fast implementations of MG approach gain a remarkable speed and decrease the computational burden from O N 6 to O N 4 log N via the use of fast Fourier transforms FFT on N-point grids, these methods suffer from low convergence rate CR , especially near discontinuities.

For example, the CR of the adaptation of MG in[ 27 ] is of O h where h denotes the spacing parameter of the grid, which is considered very low. Disadvantages of the MG method motivated us for a new computational method. We found out that sinc-based methods could guarantee ultra-linear CR s in solving integral equations.

Therefore, in this paper a method of moments with sinc basis functions speed moment method SMOM is used to numerically solve the D-bar Eq. This is based on sinc interpolation function; and it is able to achieve an ultra linear CR.

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In the proposed method, the nonpareil features of sinc functions are considered to solve the planar D-bar equation, efficiently. We show that the orthogonality of sinc basis functions may lead to discretize Eq. Thus, the closed form solution of 1. In the following section, the details of the proposed algorithm is introduced and discussed.

We start by expanding the product of. Assume the product is frequency band. Similarly, the Green's function can be expanded as. Inserting expansion 2. Remarkably, the orthogonality of sinc basis[ 37 , 38 ] helps us to get. Finally, Eq. The above equation is a discrete two-dimensional convolution equation. In the following section, we consider the computational complexity of solving Eq.

## Conscientia Beam || swalnafordoci.tk

We show that this assumption helps to reduce the computational complexity of the solution. Let us denote the domain of two-dimensional integral in Eq. That is, we may re-write the Eq.

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That is, the integral Eq. As a result, all we need to compute the convolution 2. Now, based on our own crucial observations, if sup T is assumed to be embedded in an open disc B 0 , R i. More specifically, as it is illustrated in Figure 1b , to compute the convolution integral in the right hand side of the Eq. Bounds of two-dimensional convolution in D-bar equation and required grid structure. That is, the computation grid is finite and as illustrated in Figure 1c may be embedded in the open disk B 0, 2R.

The consequence is that the two-dimensional discrete convolution operations in the right hand side of 2. Since the Eq. Thus, the Eq. Traditionally, the FFT algorithms are more effective when they are dealing with matrices that contain 2 m data points in each dimension. On the other hand, extending the number of data points in each dimension of the matrix of the two-dimensional functions, T k and g k , via zero padding may result in a more approximate computation of their DFTs. The number of double precision mathematical operations required to compute each two-dimensional FFT-based convolution computation is O N 2 log N.

Moreover, to solve Eq. Therefore, only O N 2 log N double-precision mathematical operations are required in each of the iterations of the Eq. That is, solving Eq. It is clear that solving Eq.

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Illustration of zero padding before fast Fourier transforms based implementation of the speed moment method. It is worth noting that, since FFT algorithms subdivide the computational data into small segments and combine them hierarchically, the effect of round-off error in their computations is considerably minimized. In this section, the proposed SMOM method is employed to solve the inverse conductivity problem. Another effect is the normal current distribution which can be measured through the same set of the electrodes.

The forward and inverse problems of EIT are related to this mapping. The inverse problem was first stated by Calderon. The significance of the forward EIT problem comes into account when a numerical test model is necessary for the solution evaluation of an inverse problem. In such a case, the Eq. Then, any intended inverse conductivity solution may be compared with this model.

For the inverse solution, Brown and Uhlman[ 49 ] proved that it is possible to solve such problems with a constructive solution and offered an algorithm. In the following, we review their constructive proof for inverse conductivity problem in C 1 conductivities. In this type of conductivity distribution, the induced potential is modeled with a continuous function.

In Brown and Uhlmann algorithm, two D-bar equations arises on the inverse conductivity problem defined on a body. There are some efforts to solve these two equations including Knudsen et al. In the following, we first review Brown and Uhlmann constructive description of the two dimension D-bar problem. Then a numerical model solution implying the cross-section of human chest during expiration, which is suggested in,[ 27 ] is formed to evaluate recent D-bar solution methods as well as our SMOM method and compare them together. The method of Brown and Uhlmann solves the inverse conductivity problem in two constructive steps:[ 49 ].