## Talks

Wednesday, July 14, 16:00 ~ 16:30 UTC-3

## Fast and Accurate Reconstruction of a Three Dimensional Axis-Symmetric Scatterer

### Carlos Borges

#### University of Central Florida, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakbf45f3a7af37c36062ae05ebb171e61b').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addybf45f3a7af37c36062ae05ebb171e61b = 'c&#97;rl&#111;s.b&#111;rg&#101;s' + '&#64;'; addybf45f3a7af37c36062ae05ebb171e61b = addybf45f3a7af37c36062ae05ebb171e61b + '&#117;cf' + '&#46;' + '&#101;d&#117;'; var addy_textbf45f3a7af37c36062ae05ebb171e61b = 'c&#97;rl&#111;s.b&#111;rg&#101;s' + '&#64;' + '&#117;cf' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakbf45f3a7af37c36062ae05ebb171e61b').innerHTML += '<a ' + path + '\'' + prefix + ':' + addybf45f3a7af37c36062ae05ebb171e61b + '\'>'+addy_textbf45f3a7af37c36062ae05ebb171e61b+'<\/a>';

We consider the problem of reconstructing the shape of a three dimensional impenetrable sound-soft axis-symmetric obstacle from measurements of the scattered field at multiple frequencies. This problem has important applications in locating and identifying obstacles with axial symmetry in general, such as, land mines. We will show a two-part framework for recovering the shape of the obstacle. In part 1, we introduce an algorithm to find the axis of symmetry of the obstacle by making use of the far field pattern. In part 2, we recover the shape of the obstacle by applying the recursive linearization algorithm (RLA) with multifrequency measurements of the scattered field. In the RLA, a sequence of inverse scattering problems using increasing single frequency measurements are solved. Each of those problems is ill-posed and nonlinear. The ill-posedness is treated by using a band-limited representation for the shape of the obstacle, while the nonlinearity is dealt with by applying the damped Gauss-Newton method. When using the RLA, a large number of forward scattering problems must be solved. Hence, it is paramount to have an efficient and accurate forward problem solver. For the forward problem, we apply separation of variables in the azimuthal coordinate and Fourier decompose the resulting problem, leaving us with a sequence of decoupled simpler forward scattering problems to solve. Numerical examples for the inverse problem are presented to show the feasibility of our two-part framework in different scenarios, particularly for objects with non-smooth boundaries.

Joint work with Jun Lai (Zhejiang University).

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Wednesday, July 14, 16:30 ~ 17:00 UTC-3

## A new toric Radon transform and its application in imaging.

### Marcela Morvidone

#### Centro de Matemática Aplicada - Universidad Nacional de San Martín, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak9f71028777e39d84901f3f8e78d660a1').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9f71028777e39d84901f3f8e78d660a1 = 'mm&#111;rv&#105;d&#111;n&#101;' + '&#64;'; addy9f71028777e39d84901f3f8e78d660a1 = addy9f71028777e39d84901f3f8e78d660a1 + '&#117;ns&#97;m' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_text9f71028777e39d84901f3f8e78d660a1 = 'mm&#111;rv&#105;d&#111;n&#101;' + '&#64;' + '&#117;ns&#97;m' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloak9f71028777e39d84901f3f8e78d660a1').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9f71028777e39d84901f3f8e78d660a1 + '\'>'+addy_text9f71028777e39d84901f3f8e78d660a1+'<\/a>';

In this talk we present a new toric Radon transform which models a Comtpon scattering tomography system in three dimensions. Compton scattering tomography is an emerging scanning technique with attractive applications in several fields such as non-destructive testing and medical imaging. The modality proposed here employs a fixed radiation source and a single detector moving on a spherical surface. The Radon transform modeling the data consists of integrals on toric surfaces. We show the invertibility of the toric Radon transform which ensures that the desired image of the object under study may be recovered from the measured data. We illustrate this through numerical reconstructions in three dimensions using a regularized approach.

Joint work with Javier Cebeiro (Universidad Nacional de San Martín, Argentina), Diana Rubio (Universidad Nacional de San Martín, Argentina), Cecilia Tarpau (Universite de Cergy-Pontoise, France) and Mai K. Nguyen (Universite de Cergy-Pontoise, France).

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Wednesday, July 14, 17:00 ~ 17:30 UTC-3

## Mechanical Characterization of Trabecular Bone

### María Gabriela Messineo

#### INTEMA - Universidad Nacional de Mar del Plata, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak00d9b505f4ecafa39e4769eeaa2d40ff').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy00d9b505f4ecafa39e4769eeaa2d40ff = 'gm&#101;ss&#105;n&#101;&#111;' + '&#64;'; addy00d9b505f4ecafa39e4769eeaa2d40ff = addy00d9b505f4ecafa39e4769eeaa2d40ff + 'f&#105;' + '&#46;' + 'mdp' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_text00d9b505f4ecafa39e4769eeaa2d40ff = 'gm&#101;ss&#105;n&#101;&#111;' + '&#64;' + 'f&#105;' + '&#46;' + 'mdp' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloak00d9b505f4ecafa39e4769eeaa2d40ff').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy00d9b505f4ecafa39e4769eeaa2d40ff + '\'>'+addy_text00d9b505f4ecafa39e4769eeaa2d40ff+'<\/a>';

Life expectancy worldwide has been increasing during last decades, resulting in a significant proliferation of bone diseases such as osteoporosis [1]. Osteoporosis leads to the loss of bone mass, highly increasing the fracture risk. This entails an enormous cost for the social security system around the world. There is plenty research effort aimed at a better understanding of bone mechanics to improve diagnosis and treatments. A comprehensive characterization of bone structure is relevant to assess some aspects of its properties which are not yet understood. Quantitative ultrasound (QUS) methodologies have proven to be suitable for in vivo bone evaluation, as transmitted waves bring information about its mechanical properties. The most usual QUS techniques measure the speed of sound (SOS) and the broadband ultrasonic attenuation (BUA) [2-6]. In vitro SOS measurements are typically done via transmission tests in which a bone sample is placed in a water tank between a wave source and a wave receiver. Experimental studies have demonstrated that SOS is highly correlated with the bone volume-to-total volume ratio, BVTV, (volume of mineralised bone per unit volume of the sample), the bone mineral density (BMD) and bone strength. This suggests that SOS also might be a good candidate for fracture risk prediction. Cancellous bone is a poroelastic and biphasic medium composed of an elastic skeleton (trabecular network) filled with a viscous fluid (bone marrow). Theoretical models using the Biot theory [7,8] –initially developed in the context of geophysical applications– have been applied to model ultrasound in cancellous bone with some success. Biot's model predicts the existence of two longitudinal waves resulting from the coupling of the fluid and the solid. Experimental and theoretical works provide evidence that the propagation speed of both waves is influenced not only by BVTB, but also by the bone structural anisotropy and the orientation of the ultrasound beam relatively to the direction of trabeculae alignment [9].

The main objective of this work is to develop a procedure to compute the BVTV from SOS data by means of inverse analysis. The procedure is based on the analytical model developed by Nguyen et al. [10], which is a closed-form solution of the equations proposed by Biot and Willis [11-13]. Nguyen’s model allows for the computation of the SOS from the solution of an eigenvalue problem that is posed in terms of the bone elastic constants, the trabecular microstructure tortuosity, permeability, and mass densities of the solid, the fluid and the mixture. It is proposed here to relate the bone sample properties to the BVTV (using either correlations from the bibliography or by developing correlations ad hoc), in such a way that the SOS results a function of the BVTV only. Then, the estimation of the BVTV is found by minimizing the difference between the SOS measured in a transmission test and that calculated with the analytical model.

The talk addresses the work done to assess the performance of Nguyen’s model, to develop the correlations between the bone properties and BVTV, and to set-up the inverse problem.

Computed microtomographies were used to fully characterize 23 samples of bovine trabecular bone: they were processed with BoneJ [14] to measure BVTV, trabecular thickness and trabecular space; tortuosity was obtained with Dijkstra shortest path algorithm [15], and effective elastic tensors and permeabilities in the principal anisotropy directions were computed using micro-finite element models in combination with asymptotic homogenization schemes. The resulting data was used to correlate tortuosity, permeability and elastic tensor components with BVTV.

Three-dimensional ultrasonic transmission tests were simulated using the finite difference time derivative (FDTD) software SimSonic. The model geometries were directly obtained from microtomographies. Ultrasonic propagations were simulated in the three principal anisotropy directions for the 23 samples, and the corresponding SOS were estimated from the amplitude vs time data.

The SOS from the FDTD simulations were compared to those of Nguyen’s model with the elastic and geometric parameters of the bone samples. Very good agreement was found between the two sets of results. In most of the cases, SOS values predicted by Nguyen’s somewhat lower than of the simulations, with relative error around 4% on average. The ratio between the analytical model velocity and the simulated velocity is close to one for most of the samples. Based on these results, it was concluded that Nguyen’s model is accurate enough to perform reliable estimation of SOS of trabecular bone.

Finally, a sequential quadratic programming (SQP) method [16] implemented with fminimax Matlab function was used to solve the inverse problem to estimate BVTV from SOS results. Differences between the estimated and actual BVTV values of the 23 samples were around 18% in average with standard deviation of 25%.

It is concluded that the proposed procedure allows for acceptable estimations of the BVTV. Further developments of the procedure will address the effect of attenuation into the model in order to extend the analysis to incorporate BUA data.

Bibliography

1- https://www.osteoporosis.foundation/facts-statistics

2- Langton C.M., Palmer S.B. & Porter R. W., (1984) The measurement of broadband ultrasonic attenuation in cancellous bone, Eng. Med. 13, 89-91.

3- Fredfelt K. E., (1986) Sound velocity in the middle phalanges of the human hand, Acta Radiol. Diagn. (Stockh), 27, 95-6 8.

4- Otani, T. et al., (2009) Estimation of in vivo cancellous bone elasticity, Jpn. J. Appl. Phys. 48, 0-5.

5- Foldes, A. J., Rimon, A., Keinan, D. D. & Popovtzer, M. M., (1995) Quantitative ultrasound of the tibia: a novel approach for assessment of bone status, Bone 17, 363-367.

6- Barkmann, R. et al., (2008) In vivo measurements of ultrasound transmission through the human proximal femur, Ultrasound Med. Biol. 34, 1186-1190.

7- Biot M.A, (1956) Theory of propagation of elastic waves in fluid-saturated porous solid I. Low-frequency range, Jour. Acoust. Soc. Am., 28(2), 168-178.

8- Biot M.A, (1956) Theory of propagation of elastic waves in fluid-saturated porous solid II. Higher frequency range, Jour. Acoust. Soc. Am., 28(2), 179-191.

9- Nguyen, V.H., Naili, S., Sansalone, V., (2009) Simulation of ultrasonic wave propagation in anisotropic cancellous bone immersed in fluid, Wave Motion, 47 (2), 117-129.

10- Nguyen, V.H., Naili, S., Sansalone, V., (2010) A closed-form solution for in vitro transient ultrasonic wave propagation in cancellous bone, Mech. Res. Com., 37 (4), 377-383.

11- M.A. Biot, General theory of three-dimensional consolidation (1941), J. Appl. Phys. 12 (2) 155–164.

12- M.A. Biot, Theory of elasticity and consolidation for a porous anisotropic solid (1955), J. Appl. Phys. 26 (2) 182–185.

13- M.A. Biot, D.G. Willis (1957), The elastic coefficients of the theory of consolidation, J. Appl. Mech. 79 594–601.

14- https://bonej.org/

15- Dijkstra E.W, (1959) A Note on Two Problems in Connexion with Graphs, Num. Math. Vol. 1, 269-271.

16- Brayton, R. K., Director, S. W., Hachtel, G. D. and Vidigal, L., (1979) A New Algorithm for Statistical Circuit Design Based on Quasi-Newton Methods and Function Splitting, IEEE Trans. Circuits and Systems, Vol. CAS-26, pp. 784-794.

Joint work with Ing. Joaquín García Zárate (INTEMA - Universidad Nacional de Mar del Plata) and Prof. Adrián Cisilino (INTEMA - Universidad Nacional de Mar del Plata).

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Wednesday, July 14, 17:50 ~ 18:20 UTC-3

## Some results on discrete inverse problems for elliptic equations

### Jaime Ortega

#### DIM & CMM - IRL 2807 CNRS, Universidad de Chile, Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak7bc8e11aee0d08400dcf0c869d75944e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy7bc8e11aee0d08400dcf0c869d75944e = 'j&#111;rt&#101;g&#97;' + '&#64;'; addy7bc8e11aee0d08400dcf0c869d75944e = addy7bc8e11aee0d08400dcf0c869d75944e + 'd&#105;m' + '&#46;' + '&#117;ch&#105;l&#101;' + '&#46;' + 'cl'; var addy_text7bc8e11aee0d08400dcf0c869d75944e = 'j&#111;rt&#101;g&#97;' + '&#64;' + 'd&#105;m' + '&#46;' + '&#117;ch&#105;l&#101;' + '&#46;' + 'cl';document.getElementById('cloak7bc8e11aee0d08400dcf0c869d75944e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy7bc8e11aee0d08400dcf0c869d75944e + '\'>'+addy_text7bc8e11aee0d08400dcf0c869d75944e+'<\/a>';

In this work, we are interested in the study of discrete inverse problems. We note that motivated by the relation of Unique Continuation Properties (UCP) with controllability and stabilization, and more recently with inverse problems, unique continuation properties have attracted a great attention from researchers. In this talk we will present some results of UCP for discrete problems, to do this , we derive a discrete quantitative propagation of smallness for discrete harmonic functions, in particular we obtain a three sphere inequality for harmonic functions. The proof of these results are based on a Carleman estimates for a finite difference approximation of Laplace operator in arbitrary dimension with boundary terms, in which the large parameter is connected to the mesh size.

Joint work with Luz de Teresa (Universidad Autónoma de México, México), Rodrigo Lecaros (Universidad Técnica Federico Santa María, Chile) and Ariel Pérez (Universidad de Chile).

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Wednesday, July 14, 18:20 ~ 18:50 UTC-3

## Identification of the source for full parabolic equations

### Guillermo Federico Umbricht

#### Universidad Nacional de San Martín / Universidad Nacional de General Sarmiento, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak587e63c2620117413e5d6c4e0d0d8049').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy587e63c2620117413e5d6c4e0d0d8049 = 'g&#117;&#105;ll&#101;&#117;ngs' + '&#64;'; addy587e63c2620117413e5d6c4e0d0d8049 = addy587e63c2620117413e5d6c4e0d0d8049 + 'y&#97;h&#111;&#111;' + '&#46;' + 'c&#111;m' + '&#46;' + '&#97;r'; var addy_text587e63c2620117413e5d6c4e0d0d8049 = 'g&#117;&#105;ll&#101;&#117;ngs' + '&#64;' + 'y&#97;h&#111;&#111;' + '&#46;' + 'c&#111;m' + '&#46;' + '&#97;r';document.getElementById('cloak587e63c2620117413e5d6c4e0d0d8049').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy587e63c2620117413e5d6c4e0d0d8049 + '\'>'+addy_text587e63c2620117413e5d6c4e0d0d8049+'<\/a>';

In this work, we consider the problem of identifying the time independent source for full parabolic equations in $\mathbb{R}^n$ from noisy data. This is an ill-posed problem in the sense of Hadamard. To compensate the factor that causes the instability, a family of parametric regularization operators is introduced, where the rule to select the value of the regularization parameter is included. This rule, known as regularization parameter choice rule, depends on the data noise level and the degree of smoothness that it is assumed for the source. The proof for the stability and convergence of the regularization criteria is presented and a Hölder type bound is obtained for the estimation error. Numerical examples are included to illustrate the effectiveness of this regularization approach.

Important: The presentation will be in English but the talk will be in Spanish.

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Wednesday, July 14, 18:50 ~ 19:20 UTC-3

## A review on Procrustes problems for matrix inverse eigenvalue problems

### Silvia Gigola

#### Facultad de Ingeniería, Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak050213beb20baff357ee7d2f68acefef').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy050213beb20baff357ee7d2f68acefef = 'sg&#105;g&#111;l&#97;' + '&#64;'; addy050213beb20baff357ee7d2f68acefef = addy050213beb20baff357ee7d2f68acefef + 'f&#105;' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_text050213beb20baff357ee7d2f68acefef = 'sg&#105;g&#111;l&#97;' + '&#64;' + 'f&#105;' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloak050213beb20baff357ee7d2f68acefef').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy050213beb20baff357ee7d2f68acefef + '\'>'+addy_text050213beb20baff357ee7d2f68acefef+'<\/a>';

The inverse eigenvalue problem consists of the reconstruction of a matrix from given spectral data. This kind of problems occurs in different engineering areas and arises in various applications.

Given a matrix $X$ and a diagonal matrix $D$, we are looking for solutions of the equation $AX =XD$, where $A$ is a matrix with a prescribed structure and a predefined spectrum. Based on these restrictions on matrix $A$, a variety of inverse eigenvalue problems arises.

The Procrustes problem, or the best approximation problem, associated to the inverse eigenvalue one can be described synthetically as follows: given an experimentally obtained matrix, the problem consists on finding a matrix from the problem solution set, such that it is the best approximation to the data matrix.

We will show the existence of the solutions of the inverse eigenvalue problem and the associated Procrustes problem for three kind of matrices: Hermitian reflexive matrices with respect to a normal and ${k+1}$-potent matrix, normal $J$-Hamiltonian matrices, and normal $J$-skew Hamiltonian matrices.

Joint work with Néstor Thome (Universitat Politècnica de València, Spain) and Leila Lebtahi (Universitat de València, Spain).

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Thursday, July 15, 16:00 ~ 16:30 UTC-3

## New Inverse Problem Approach to Thin Soil Layer Identification Applications in Earthquake Engineering

### Eileen Martin

#### Virginia Tech, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak664edf1e5dc6012e7099e1ccc9ca164c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy664edf1e5dc6012e7099e1ccc9ca164c = '&#101;&#105;l&#101;&#101;nrm&#97;rt&#105;n' + '&#64;'; addy664edf1e5dc6012e7099e1ccc9ca164c = addy664edf1e5dc6012e7099e1ccc9ca164c + 'vt' + '&#46;' + '&#101;d&#117;'; var addy_text664edf1e5dc6012e7099e1ccc9ca164c = '&#101;&#105;l&#101;&#101;nrm&#97;rt&#105;n' + '&#64;' + 'vt' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak664edf1e5dc6012e7099e1ccc9ca164c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy664edf1e5dc6012e7099e1ccc9ca164c + '\'>'+addy_text664edf1e5dc6012e7099e1ccc9ca164c+'<\/a>';

A common field-testing method to characterize soil properties in geotechnical engineering is the cone penetration test (CPT), in which a probe with a conical-shaped tip is hydraulically pushed into the ground at a constant rate, and the tip resistance is recorded at depth intervals of 1 to 2 cm, typically. This results in a tip resistance depth profile. These tip resistance profiles might initially appear to be comprised of depth-specific in-situ measurements, but the observed profiles appear blurred when compared to the true layered soil stratigraphy. This is because the measured penetration resistance reflects a stress bulb that forms around the penetrometer tip as it advances and the stress bulb may extend into neighboring layers. This can reduce the ability to detect thin layers and directly impacts the accuracy of the predictions of earthquake liquefaction potential of the soil. Previous methods (including one prior inverse problem approach) to correct these tip resistance profiles to remove this blurring have, in some scenarios, had limited efficacy in the presence of thin layers or multiple layers. We posed this deblurring as an inverse problem constrained to the space of layered soil profiles (with a thickness and corrected tip resistance at each depth). We propose a new, efficient correction algorithm that incorporates new techniques for initial guess generation, particle swarm optimization, and the addition or removal of layers throughout the optimization process. We compare the performance of two objective functions with modifications of this algorithm, one of which tends to yield more accurate results but takes longer to converge. In addition to proposing and testing this new algorithm, we have released open-source code so others may apply and improve upon our methods. This work focuses on the inversion algorithm, but our analyses indicate further improvement may be possible with more accurate forward modeling procedures.

Joint work with Jon Cooper (Virginia Tech), Kaleigh Yost (Virginia Tech), Alba Yerro (Virginia Tech) and Russell A. Green (Virginia Tech).

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Thursday, July 15, 16:30 ~ 17:00 UTC-3

## Estimation of physical properties of a multilayer material

### Diana Rubio

#### ECyT- Universidad Nacional de San Martin, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak9cf36610cebc6a8d8e521e4b59a311a9').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy9cf36610cebc6a8d8e521e4b59a311a9 = 'dr&#117;b&#105;&#111;' + '&#64;'; addy9cf36610cebc6a8d8e521e4b59a311a9 = addy9cf36610cebc6a8d8e521e4b59a311a9 + '&#117;ns&#97;m' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_text9cf36610cebc6a8d8e521e4b59a311a9 = 'dr&#117;b&#105;&#111;' + '&#64;' + '&#117;ns&#97;m' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloak9cf36610cebc6a8d8e521e4b59a311a9').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy9cf36610cebc6a8d8e521e4b59a311a9 + '\'>'+addy_text9cf36610cebc6a8d8e521e4b59a311a9+'<\/a>';

Composite materials are widely used in engineering for various purposes. In particular, multilayer materials are used in the aeronautical industry for the possibility of obtaining new materials with particular properties such as durability and resistance to high temperatures. Non-destructive evaluation methods are needed to detect and characterize the degradation of these materials due, for instance, to exposure to high temperatures or stress for a long period of time. In this work we deal with estimating the 'effective' permittivity of multilayer materials based on material reflectance data. For the estimation it is assumed that the permittivity can be described using a probability measure that is determined by minimizing the error between the estimated reflectance and the data for a given frequency range. Sensitivity and elasticity analysis are included to better understand the influence of the modelling parameters in the estimation.

This is part of the project partially supported by SOARD/AFOSR under Grant FA9550-18-1-0523.

Joint work with Marcela Morvidone (Universidad Nacional de San Martin, Argentina) and Nicolas Saintier (Universidad de Buenos Aires, Argentina).

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Thursday, July 15, 17:00 ~ 17:30 UTC-3

## Characterization of Pipelines During Flow Transients

### HELCIO ORLANDE

#### Department of Mechanical Engineering, Politécnica/COPPE, Federal University of Rio de Janeiro, UFRJ, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak987c2b7d89e97467213af53fe3f30f02').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy987c2b7d89e97467213af53fe3f30f02 = 'h&#101;lc&#105;&#111;' + '&#64;'; addy987c2b7d89e97467213af53fe3f30f02 = addy987c2b7d89e97467213af53fe3f30f02 + 'm&#101;c&#97;n&#105;c&#97;' + '&#46;' + 'c&#111;pp&#101;' + '&#46;' + '&#117;frj' + '&#46;' + 'br'; var addy_text987c2b7d89e97467213af53fe3f30f02 = 'h&#101;lc&#105;&#111;' + '&#64;' + 'm&#101;c&#97;n&#105;c&#97;' + '&#46;' + 'c&#111;pp&#101;' + '&#46;' + '&#117;frj' + '&#46;' + 'br';document.getElementById('cloak987c2b7d89e97467213af53fe3f30f02').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy987c2b7d89e97467213af53fe3f30f02 + '\'>'+addy_text987c2b7d89e97467213af53fe3f30f02+'<\/a>';

In nowadays complex hydraulic systems, transients in pumps and valves may induce significant variations in pressures and flow rates. For example, if the liquid velocity is suddenly reduced by the closure of a valve, pressure waves with large magnitudes and large velocities propagate through the pipelines. Such a phenomenon is known as water hammer and can seriously affect the pipeline integrity. This work deals with the estimation of parameters of a water hammer model, with focus on a transient friction factor and an empirical parameter related to the pipeline elasticity. The hyperbolic water hammer model was solved with a TVD (Total Variation Diminishing) version of the WAF (Weighted Average Flux) finite volume scheme. Pressure and flow rate measurements, taken near the inlet and the outlet of a hydraulic circuit, were used for the solution of the parameter estimation problem. A recent parallel computation version of the Metropolis-Hastings algorithm was applied for the parameter estimation.

Joint work with Raphael C. Carvalho (COPPE, Federal University of Rio de Janeiro, UFRJ), Marcelo J. Colaço (Politécnica/COPPE, Federal University of Rio de Janeiro, UFRJ) and Italo M. Madeira (CENPES Research and Development Center, PETROBRAS).

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Thursday, July 15, 17:50 ~ 18:20 UTC-3

## Anomalous Diffusion with Caputo-Fabrizio Time Derivative: an Inverse Problem

### Silvia Seminara

#### Facultad de Ingeniería - Universidad de Buenos Aires, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakb6d8f96c0353ca9fc20d2c5adbc1599c').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyb6d8f96c0353ca9fc20d2c5adbc1599c = 'ss&#101;m&#105;n&#97;r&#97;' + '&#64;'; addyb6d8f96c0353ca9fc20d2c5adbc1599c = addyb6d8f96c0353ca9fc20d2c5adbc1599c + 'f&#105;' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r'; var addy_textb6d8f96c0353ca9fc20d2c5adbc1599c = 'ss&#101;m&#105;n&#97;r&#97;' + '&#64;' + 'f&#105;' + '&#46;' + '&#117;b&#97;' + '&#46;' + '&#97;r';document.getElementById('cloakb6d8f96c0353ca9fc20d2c5adbc1599c').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyb6d8f96c0353ca9fc20d2c5adbc1599c + '\'>'+addy_textb6d8f96c0353ca9fc20d2c5adbc1599c+'<\/a>';

The motion of microscopic particles in a fluid has been investigated for a long time. The "Brownian motion'' consists of random displacements of the suspended particles within a medium. The molecules of the fluid, much smaller than the solid particles, knock them incessantly, both driving and damping their movement; macroscopically, those knockings give rise to the "fluid viscosity''.

The traditional mathematical model for this diffusion phenomenon is based on Einstein's theory by which the mean square displacement of diffusion particles is proportional to time. This idea leads to the classical diffusion equation, $$u'_t({\textbf{x}},t)-k\nabla^2 u({\textbf{x}},t)=0,$$ where $u({\textbf{x}},t)$ is the quantity of particles per unit volume at position ${\textbf{x}}$ and time $t$, and $k$ is a diffusion coefficient. If there is also a source of additional incoming particles, the equation becomes non-homogeneous: $$u'_t({\textbf{x}},t)-k\nabla^2 u({\textbf{x}},t)=s({\textbf{x}},t).$$

But there are experimental results - like anomalous diffusion of particles in porous or fractal media, biological media, turbulent plasmas, polymers, etc. - that show that, in some cases, the mean square displacement of particles must be considered to be proportional not to the time, but to a fractional power of time, to fit the empirical data. This fractional order may be less than unity (subdiffusion) or greater than one (superdiffusion).

Fractional differential equations have been proposed to model this anomalous diffusion phenomenon: $${\cal{D}}_t^\alpha u({\textbf{x}},t)-k\nabla^2 u({\textbf{x}},t)=s({\textbf{x}},t),$$ where $\alpha$ is a not integer order of derivation with respect to time; $0<\alpha<1$ for subdiffusion, $\alpha>1$ for superdiffusion.

There are several definitions of ${\cal{D}}_t^\alpha$ (Riemann-Liouville's, Caputo's, Atangana-Baleanu's, Caputo-Fabrizio's, etc.), all of them involving an integral operator which takes account of the "past history'' of the function.

In this work, we have chosen the Caputo-Fabrizio fractional derivative to describe a 1-dimensional model of subdiffusion of the form $${\cal{D}}_t^\alpha u(x,t)-k\frac{\partial^2}{\partial x^2} u(x,t)=s(x)h(t),$$ where $x\in (0,1)$ and $t\in (0,T)$ and the source is supposed to be a separable function of variables $x$ and $t$.

We are interested in solving the inverse problem that consists in finding $s(x)$ and $u(x,t)$, for a known $h(t)$ and additional data: measurements $u(x_i,T), \, x_i\in (0,1)$, for $i=1, \cdots, N$.

We obtained an approximate solution by separating variables and solving the resulting fractional differential equation. We present some numerical examples that show the good performance of the proposed scheme and state some conclusions.

Joint work with María Inés Troparevsky (Universidad de Buenos Aires), Marcela Fabio (Universidad de San Martín) and Guillermo La Mura (Universidad de San Martín).

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Thursday, July 15, 18:20 ~ 18:50 UTC-3

## Combination of SEM and Light Scattering Data for the Inverse Estimation of Particle Size Distribution using a Bayesian Approach

### Fernando Otero

#### Universidad Nacional de Mar del Plata, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak3deb1119574eccdf7f06108289b82587').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy3deb1119574eccdf7f06108289b82587 = 'f&#111;t&#101;r&#111;v&#101;g&#97;' + '&#64;'; addy3deb1119574eccdf7f06108289b82587 = addy3deb1119574eccdf7f06108289b82587 + 'f&#105;' + '&#46;' + 'mdp' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r'; var addy_text3deb1119574eccdf7f06108289b82587 = 'f&#111;t&#101;r&#111;v&#101;g&#97;' + '&#64;' + 'f&#105;' + '&#46;' + 'mdp' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#97;r';document.getElementById('cloak3deb1119574eccdf7f06108289b82587').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy3deb1119574eccdf7f06108289b82587 + '\'>'+addy_text3deb1119574eccdf7f06108289b82587+'<\/a>';

Particle systems are morphologically characterized through their particle size distribution (PSD). It can define various final properties of the material, which is why it is of fundamental importance to obtain an estimate as accurate and precise as possible. However, there is no single experimental technique capable of providing us with a complete description of PSD . This reason motivates the present talk, which is to analyze the optimization of combining measurements from different techniques in a process that is known as data fusion. We analyze various elements of the process of data fusion to optimize the performance of the PSD estimation applied to the Combination of SEM (Scanning Electron Microscopy) micrographs with SLS (Static Light Scattering) scattered light measurements.

The scheme applied here is to solve an inverse problem using a Bayesian inference where the prior information is deduced from the SEM micrographs and measurements are data from SLS. For this purpose we propose two Bayesian schemes (one parametric and another non-parametric) to solve the stated light scattering problem and take advantage of the obtained results to summarize some features of the Bayesian approach within the context of inverse problems.

The features presented in this talk include the improvement of the results when some useful prior information from an alternative experiment is considered instead of a non-informative prior as it occurs in a deterministic maximum likelihood estimation. This improvement will be shown in terms of accuracy and precision in the corresponding results and also in terms of minimizing the effect of multiple minima by including significant information in the optimization. Both Bayesian schemes are implemented using Markov Chain Monte Carlo methods. They have been developed on the basis of the Metropolis– Hastings (MH) algorithm using Matlab® and are tested with the analysis of simulated and experimental examples of concentrated and semi-concentrated particles. In the simulated examples, SLS measurements were generated using a rigorous model, while the inversion stage was solved using an approximate model in both schemes and also using the rigorous model in the parametric scheme. Priors from SEM micrographs were also simulated and experimental, where the simulated ones were obtained using a Monte Carlo routine and Monte Carlo-based statistical tools are also employed to assess the quality of these priors.

In addition to the presentation of these features of the Bayesian approach, some other topics will be discussed, such as regularization and some implementation issues of the proposed schemes, among which we remark the selection of the parameters used in the MH algorithm.

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Friday, July 23, 16:00 ~ 16:30 UTC-3

## Bilevel learning for inverse problems

### Juan Carlos De los Reyes

#### MODEMAT, Escuela Politécnica Nacional, Ecuador   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakef6ebd99fd700d47edfc0029cf9c3c80').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyef6ebd99fd700d47edfc0029cf9c3c80 = 'j&#117;&#97;n.d&#101;l&#111;sr&#101;y&#101;s' + '&#64;'; addyef6ebd99fd700d47edfc0029cf9c3c80 = addyef6ebd99fd700d47edfc0029cf9c3c80 + '&#101;pn' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#101;c'; var addy_textef6ebd99fd700d47edfc0029cf9c3c80 = 'j&#117;&#97;n.d&#101;l&#111;sr&#101;y&#101;s' + '&#64;' + '&#101;pn' + '&#46;' + '&#101;d&#117;' + '&#46;' + '&#101;c';document.getElementById('cloakef6ebd99fd700d47edfc0029cf9c3c80').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyef6ebd99fd700d47edfc0029cf9c3c80 + '\'>'+addy_textef6ebd99fd700d47edfc0029cf9c3c80+'<\/a>';

In recent years, novel optimization ideas have been applied to several inverse problems in combination with machine learning approaches, to improve the inversion by optimally choosing different quantities/functions of interest. A fruitful approach in this sense is bilevel optimization, where the inverse problems are considered as lower-level constraints, while on the upper-level a loss function based on a training set is used. When confronted with inverse problems with nonsmooth regularizers, however, the bilevel optimization problem structure becomes quite involved to be analyzed, as classical nonlinear or bilevel programming results cannot be directly utilized. As a remedy, tools from nonsmooth variational analysis have to be employed to cope with the difficulties related with the lack of differentiability of the solution mapping or the failure of standard constraint qualifications. In this talk, I will discuss on the different challenges that these problems pose, and provide some analytical results of the bilevel problems, as well as solutions algorithms that may be devised based on the nonsmooth properties of the problems at hand.

Joint work with David Villacís (MODEMAT, Escuela Politécnica Nacional, Ecuador).

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Friday, July 23, 16:30 ~ 17:00 UTC-3

## Efficient Edge-Preserving Methods for Large-Scale Dynamic Inverse Problems.

### Mirjeta Pasha

#### Arizona State University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakf0cb2bf032f8641259823c15a2f7e08a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyf0cb2bf032f8641259823c15a2f7e08a = 'mp&#97;sh&#97;3' + '&#64;'; addyf0cb2bf032f8641259823c15a2f7e08a = addyf0cb2bf032f8641259823c15a2f7e08a + '&#97;s&#117;' + '&#46;' + '&#101;d&#117;'; var addy_textf0cb2bf032f8641259823c15a2f7e08a = 'mp&#97;sh&#97;3' + '&#64;' + '&#97;s&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakf0cb2bf032f8641259823c15a2f7e08a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyf0cb2bf032f8641259823c15a2f7e08a + '\'>'+addy_textf0cb2bf032f8641259823c15a2f7e08a+'<\/a>';

In this talk we consider numerical methods for computing the maximum a posteriori (MAP) estimator for time dependent inverse problems, where the target of interest changes during the measurement process as well as the operator that models the forward problem. For example, solving the limited angle tomography problem is challenging because of both its dynamic nature and the limited amount of data available. Moreover, incorporating spatial and temporal information about the prior and providing properties like edge-preserving can be computationally not attractive. Hence, we propose efficient and effective edge-preserving and sparsity promoting iterative regularization methods that incorporate spatial and temporal information by introducing regularizers that enforce simultaneous regularization in space and time, and that typically enhance edges (at each time instant) and enforce proximity (at consecutive time instants) by total variation (TV) and group sparsity. The methods that we develop here are iterative methods based on majorization minimization strategy with quadratic tangent majorant that allow the resulting least squares problem to be solved with a generalized Krylov subspace method for large scale problems. The regularization parameter can be defined automatically and at a low cost in the projected subspaces of a relatively small dimension. Numerical examples from a wide range of applications like limited angle computerized tomography (CT), space-time deblurring, and photoacoustic tomography (PAT) illustrate the effectiveness of the described approaches.

Joint work with Malena Espanol (Arizona State University, USA), Silvia Gazzola (Universoty of Bath, UK), Arvind Saibaba (North Carolina State University, USA ) and Eric de Sturler (Virginia Tech, USA).

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Friday, July 23, 17:00 ~ 17:30 UTC-3

## Inversion from a random sampling of the Radon transform in the context of LCA groups

### Erika Porten

#### Universidad Nacional de San Martín , Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloakdc3498214b66c5999019e1c55b74b721').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addydc3498214b66c5999019e1c55b74b721 = '&#101;r&#105;k&#97;p&#111;rt&#101;n' + '&#64;'; addydc3498214b66c5999019e1c55b74b721 = addydc3498214b66c5999019e1c55b74b721 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_textdc3498214b66c5999019e1c55b74b721 = '&#101;r&#105;k&#97;p&#111;rt&#101;n' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloakdc3498214b66c5999019e1c55b74b721').innerHTML += '<a ' + path + '\'' + prefix + ':' + addydc3498214b66c5999019e1c55b74b721 + '\'>'+addy_textdc3498214b66c5999019e1c55b74b721+'<\/a>';

Based on the definition of the Fourier transform on a locally compact abelian group $G$, we consider the problem of reconstructing a measurable function over $G$ from a countable subset of random samples. The sampling is carried out according to a Poisson random point process. For the interpolation step, we propose an iterative procedure in which a resampling is performed at each iteration, and we present error estimates of the approximation obtained. We apply these results to the problem of approximating the inverse Radon transform of a function. Finally, we present numerical simulations that support the effectiveness of the method.

Joint work with Marcela Morvidone (Universidad de San Martin, Argentina) and Juan Miguel Medina (Universidad de Buenos Aires, Argentina).

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Friday, July 23, 17:50 ~ 18:20 UTC-3

## An $\ell_p$ Variable Projection Method for Large-Scale Separable Nonlinear Inverse Problems

### Malena Espanol

#### Arizona State University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak270308b6ea867bdb77a5d67034522585').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy270308b6ea867bdb77a5d67034522585 = 'm&#97;l&#101;n&#97;.&#101;sp&#97;n&#111;l' + '&#64;'; addy270308b6ea867bdb77a5d67034522585 = addy270308b6ea867bdb77a5d67034522585 + '&#97;s&#117;' + '&#46;' + '&#101;d&#117;'; var addy_text270308b6ea867bdb77a5d67034522585 = 'm&#97;l&#101;n&#97;.&#101;sp&#97;n&#111;l' + '&#64;' + '&#97;s&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak270308b6ea867bdb77a5d67034522585').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy270308b6ea867bdb77a5d67034522585 + '\'>'+addy_text270308b6ea867bdb77a5d67034522585+'<\/a>';

Variable projection methods are among the classical and efficient methods to solve separable nonlinear least squares problems such as blind deconvolution, system identification, and machine learning. In this talk, we present a modified variable projection method for large-scale separable nonlinear inverse problems, that promotes edge-preserving and sparsity properties on the desired solution, and enhances the convergence of the parameters that define the forward problem. Specifically, we adopt a majorization minimization method that relies on constructing quadratic tangent majorants to approximate an $\ell_p$ regularization term, by a sequence of $\ell_2$ problems that can be solved by the aid of generalized Krylov subspace methods at a relatively low cost compared to the original unprojected problem. In addition, more potential generalized regularizers including total variation (TV), framelet, and wavelet operators can be used, and the regularization parameter can be defined automatically at each iteration with the aid of generalized cross validation. Numerical examples on large-scale two-dimensional imaging problems arising from blind deconvolution are used to highlight the performance of the proposed method in both quality of the reconstructed image as well as the reconstructed forward operator.

Joint work with Mirjeta Pasha (Arizona State University, USA).

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Friday, July 23, 18:20 ~ 18:50 UTC-3

## A new method for gridding passive microwave data with mixed measurements and spatial correlation

### Juan Lucas Bali

#### Universidad Nacional de San Martín, CONICET, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak518d254f7f45194f740cd50efc258a1a').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy518d254f7f45194f740cd50efc258a1a = 'j&#117;&#97;n.l&#117;c&#97;s.b&#97;l&#105;' + '&#64;'; addy518d254f7f45194f740cd50efc258a1a = addy518d254f7f45194f740cd50efc258a1a + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text518d254f7f45194f740cd50efc258a1a = 'j&#117;&#97;n.l&#117;c&#97;s.b&#97;l&#105;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak518d254f7f45194f740cd50efc258a1a').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy518d254f7f45194f740cd50efc258a1a + '\'>'+addy_text518d254f7f45194f740cd50efc258a1a+'<\/a>';

In this article we develop a new method to grid passive microwave data in the presence of spatial correlation patterns. Our proposal combines a Tychonov inverse method with a generalized cross validation procedure to grid the observations over a discrete retrieval grid. To build this grid, the study region is partitioned into objects following an object-based image analysis procedure. Then, this partition is refined into small cells of similar size. Two cells from the same object are considered of the same type. The procedure to estimate the brightness temperature of each cell is based on a least-squares estimation with a cell-type aware Tychonov regularization method. This method assumes that the brightness temperature heterogeneity within each cell can be neglected and that adjacent cells of the same type have similar brightness temperature. In other words, spatial correlations are considered within each object in the scene. The Tychonov regularization parameter is found using a fast generalized cross validation procedure that makes it possible to solve the inverse problem when the observational error variance is not known. We evaluate the proposed method using different synthetic scenarios and compare it with other methods. The evaluation shows an excellent performance of the proposed method when the brightness temperature field varies smoothly over each object in the scene. But it also shows that the method is competitive when the brightness temperature field does not present spatial correlations. We conclude that the proposed algorithm provides a fast and robust method to solve the original inverse problem.

Joint work with Manuela Cerdeiro (IC, FCEyN, UBA, Argentina), Mariela Rajngewerc (IIIA, UNSAM-CONICET) and Rafael Grimson (IIIA, UNSAM-CONICET).

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## Solution of an integro-differential equation with conditions of Dirichlet using techniques of the inverse moments problem.

### María Beatriz Pintarelli

#### Universidad Nacional de La Plata, Argentina   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak989ae58cb62a99008745e677cddf8ed0').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy989ae58cb62a99008745e677cddf8ed0 = 'm&#97;r&#105;&#97;bp&#105;nt&#97;r&#101;ll&#105;' + '&#64;'; addy989ae58cb62a99008745e677cddf8ed0 = addy989ae58cb62a99008745e677cddf8ed0 + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m'; var addy_text989ae58cb62a99008745e677cddf8ed0 = 'm&#97;r&#105;&#97;bp&#105;nt&#97;r&#101;ll&#105;' + '&#64;' + 'gm&#97;&#105;l' + '&#46;' + 'c&#111;m';document.getElementById('cloak989ae58cb62a99008745e677cddf8ed0').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy989ae58cb62a99008745e677cddf8ed0 + '\'>'+addy_text989ae58cb62a99008745e677cddf8ed0+'<\/a>';

We want to find $w(x,t)$ such that $w_{t}= \int_{0}^{t}k(t-s)w_{xx}(x,s)ds+f(x,t)$ about a domain $E= \left\lbrace (x,t),\quad 00 \right\rbrace$, with conditions $w(x,0)=h_{1}(x); \qquad w(0,t)=k_{1}(t); \qquad w(L,t)=k_{2}(t)$ where $k(x)$ has continuous derivative on $x=0$, the value of $k(0)$ is known, $f(x,t)$ known and derivative with respect to $t$ continuous, using the problem generalized moments techniques.

We will see that an approximate solution of the equation integro-differential can be found using the techniques of generalized inverse moments problem and bounds for the error of the estimated solution.

First the problem is reduced to solving a hyperbolic or parabolic partial derivative equation considering the unknown source. The method consists of two steps. In each one an integral equation is solved numerically using the two-dimensional inverse moments problem techniques. We illustrate the different cases with examples.

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