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CONTACT INFORMATIONE-mail: dzubiaur@gmail.comPhone: +34 94 601 5947Address: Departamento de Matemática Aplicada, Estadística e Investigación Operativa,Barrio Sarriena S/N, Facultad de Ciencia y Tecnología,48940 Leioa (Bizkaia), Spain | SHORT BIOHis research interests include computational electromagnetics, petroleum-engineering applications (borehole simulations), adaptive finite-element and discontinuous Petrov-Galerkin methods, multigrid solvers, image restoration algorithms, and multiphysics and inverse problems. |

**MAIN RESEARCH INTERESTS**

**Numerical Analysis****Development of a novel goal-oriented adaptive method for wave propagation problems.**In collaboration with Ph.D. Student Vincent Darrigrand and Professor I. Muga (Pontificia Catholic University of Valparaíso, Chile), we are developing a new goal-oriented method for wave propagation problems that uses unconventional error representations. These algorithms are being generalized to non-linear quantities of interest (e.g., computation of the impedance matrix) and time-domain simulations in collaboration with Prof. Elisabete Alberdi (University of the Basque Country UPV/EHU, Spain).

**Development of a Dimensionally Adaptive Method (DAM) for Simulation and Inversion of Geophysical Measurements.**This gradient-based inversion algorithm employs a dimensional decomposition, i.e., an explicit separation of one-dimensional (1D), 1.5D, 1.75D, 2D, 2.5D, 2.75D, and 3D effects based on physical principles. This reduces dramatically the simulation and inversion time, making the method feasible to efficiently invert real multi-physics measurements. The resulting parallel software will combine fast implementations of semi-analytical electromagnetic, elastic, and elasto-acoustic 1D solutions, high-order finite element simulators equipped with goal-oriented adaptivity, a variety of mixed Fourier finite element discretizations, multiphysics H1, H(curl), H(div) and L2-discrete Finite Element spaces, and model reduction techniques based on physical principles. This work is being developed in collaboration with Postdoctoral Fellows Shaaban A. Bakr (University of the Basque Country UPV/EHU, Spain) and Ángel Rodríguez-Rozas (University of the Basque Country UPV/EHU, Spain).

**Development of new multi-physics data-adaptive inversion algorithms.**We employ gradient-based inversion algorithms with total variational (TV) regularization techniques and grid adaptive methods to invert a large variety of multi-physics measurements, such as those acquired in oil-engineering applications. The adaptivity is driven by the data misfit. We develop this work in collaboration with student J. Alvarez-Aramberri (University of the Basque Country UPV/EHU, Spain) and Professor C. Torres-Verdín (The University of Texas at Austin, USA).

**Development of a self-adaptive k-refinement strategies for Isogeometric Analysis (IGA).**In Isogeometric Analysis (IGA), high continuity has been employed to find robust solutions to physical problems with a reduced number of degrees of freedom (dof). However, highly continuous solutions are often unable to capture the true physics of rapidly varying physical properties. These situations occur in a large variety of applications, including geophysics, e.g., when exploring the Earth's subsurface to find and extract hydrocarbons. To handle spatial discontinuities and singularities on the domain, there exist various techniques within IGA. These often involve intensive h-refinements or a total reduction of the continuity, leading to a significant increase of the computational cost. Local h-refinements can also be employed using, for example, T-splines, PHT-splines and LR-splines. In here, we develop an automatic refining mechanism that consists of reducing the continuity of the solution over local areas of the domain, while keeping an optimal distribution of computational resources (number of dof). Notice that a reduction on the continuity of the discrete space involves an increase on the problem size. The research performed in collaboration with Prof. Victor M. Calo (KAUST, Saudi Arabia) and Ph.D. Student Daniel García (University of the Basque Country UPV/EHU, Spain) focuses on designing a k-refinement strategy, where k denotes the degree of continuity, and developing the corresponding computational framework.

**Development of new parallel direct and iterative solvers of linear equations for Isogeometric Analysis (IGA) and hp-refined grids.**In collaboration with Associate Professor M. Paszynski (AGH University of Sciences and Technology, Krakow, Poland), Prof. Victor M. Calo (KAUST, Saudi Arabia), and Ph.D. Student Daniel García (University of the Basque Country UPV/EHU, Spain), we develop fast parallel direct and iterative solvers of linear equations for Isogeometric Analysis and hp-refined grids. We have recently discovered that a proper combination of both methods that we call Enhanced Isogeometric Analysis (EIGA) produces dramatically better results that those attained with traditional IGA methods.

**Geophysical Applications****Simulation of marine Controlled Source Electromagnetic (CSEM) measurements.**We utilize a high-order Fourier-Finite Element method for the accurate simulation of marine CSEM measurements. Electromagnetic waves are sent by a transmitter that is typically located a few meters (10-100 m) above the sea-floor and is moved by a ship. This transmitter operates at frequencies in the range of 0.25Hz-1.25Hz in order for electromagnetic waves to reach long distances. Measurements are recorded by a set of receivers located along the sea-floor at a distance of up to 20 km from the source. Our objective is to accurately simulate marine CSEM measurements at the receiver antennas. These measurements are later used for inversion in order to characterize the resevoir. Simulations of marine CSEM measurements are being performed in collaboratin with Postdoctoral Fellow Shaaban A. Bakr and Prof Trond Mannseth (CIPR, Norway).

**Simulation and inversion of Magnetotelluric (MT) measurements.**We utilize a goal-oriented hp-adaptive Finite Element method to simulate and invert MT measurements. Movements of charges on the ionosphere produce a plane wave of unknown amplitude that penetrates on the Earth´s subsurface. In this application, the quantity of interest for the inverse problem is the impedance matrix, whose entries are quotients of a component of the electric field divided by another component of the magnetic field. In this project, we discovered that the impedance matrix is a quantity that converges significantly faster than the electromagnetic fields themselves, and therefore, any mesh-refinement strategy should be based on the direct minimization of the impedance error rather than on accurately representing the electromagnetic fields. We are developing this work in collaboration with Ph.D. Student Julen Alvarez-Aramberri and Prof. Helene Barucq (INRIA and University of Pau, France).

**Fast 1D Inversion of Logging-While-Drilling Resistivity Measurements.**In collaboration with Professor C. Torres-Verdín (The University of Texas at Austin, USA) we introduced an efficient inversion method to estimate layer-by-layer electrical resistivity from logging-while-drilling (LWD) electromagnetic induction measurements. The method assumes a one-dimensional (1D) model based on planarly layered transversely isotropic (TI) formations with known bed boundaries, penetrated by arbitrary well trajectories. Forward simulations are based on a 1D reduced model where both borehole and mandrel effects are assumed negligible.

**Simulation of resistivity logging measurements.**Borehole resistivity measurements have been routinely acquired by oil-companies since 1927, when the Schlumberger brothers recorded resistivity data for the first time in Pechelbronn, France. In order to properly interpret resistivity measurements, it is essential the use of computer-aided simulations, which are also employed for the design with Professor C. Torres-Verdín (The University of Texas at Austin, USA), we perform a variety of simulations in complex borehole environments such as those encountered in deviated wells when the logging instrument is possibly-borehole eccentered, and we include the presence of anisotropy. We also consider the latest logging-while-drilling instruments as well as through-casing measurements.

**In collaboration with Professor C. Torres-Verdín and Dr. P. Matuszyk (The University of Texas at Austin, USA) we work on the simulation of sonic logging instruments both in the frequency and time-domain (via an inverse Fourier transform). These multiphysics problems (acoustics in the borehole and elasticity in the logging instrument and formation) are simulated with a self-adaptive high-order finite element method, and results are properly post-processed to determine the arrival time of different waves (P- and S- waves) at each receiver.**

Simulation of sonic logging measurements.

Simulation of sonic logging measurements.

**Other Research Areas**

**Underwater Image Restoration.**Underwater images suffer from low contrast as a result of the exponential decay that light suffers as it travels. Additionally, they exhibit a particular colour distortion associated to different wavelengths having different attenuation rates, being the red wavelength the one that attenuates the fastest. To overcome this unbalanced loss of contrast and colour distortion, we have developed a new approach, based on a modified Dark Channel prior. We call this technique the Red Channel method. The Red Channel method is designed to restore the lost contrast while recovering colours associated to short wavelengths. We are developing this work in collaboration wtih Ph.D. Student Adrián Galdrán and Tecnalia Expert Dr. Artzai Picón.

**Enhanced Variation Image Dehazing (EVID).**Images obtained under adverse weather conditions, such as haze or fog, typically exhibit low contrast and faded colors, which may severely limit the visibility within the scene. This parallels in some sense the process suffered by underwater images. In the same way, unveiling the image structure under the haze layer and recovering vivid colors out of a single image remains a challenging task, since the degradation is depth-dependent and conventional methods are unable to overcome this problem. We present experimental results demonstrating that the developed EVID method outperforms other state-of-the-art methods both qualitatively and quantitatively. In particular, when the illuminant is uneven, the EVID method is the only one that recovers realistic colors, avoiding the appearance of strong chromatic artifacts. We are developing this work in collaboration wtih Ph.D. Student Adrián Galdrán, Tecnalia Expert Dr. Artzai Picón, and Prof. Marcelo Bertalmio (University Pompeu Fabra, Spain).

**Improvement on the development of mega- and tera- chains.**The proper design and efficient construction of mega- and tera- chains constitutes a challenging task that requires from advanced knowledge supported on R&D activities in order to minimize the costs while maximizing the robustness and reliability of the final product. Specifically, all parameters of the machine involved in the welding process need to be adjusted, quantifying the allowed admissible tolerances. Until now, this adjustment of the welding machines has been mostly performed in smaller chains by using a trial-and-error approach based on previous results. For larger chains where welding failures are expected to be more frequent and expensive, the combination of more advanced data-mining techniques together with finite element simulations seem now necessary to adjust of the welding machines that goes beyond the traditional trial-and-error approach. We are developing this work in collaboration wtih Professors M. Lezaun, E. Sainz de la Maza and C. Gorría (Univ. of the Basque Country UPV/EHU, Spain) and the company Vicinay Cadenas.

**Numerical modeling of Cold Crucible Induction Melting (CCIM).**The CCIM is a process to melt extremely reactive alloys in molten conditions when purity is required. An example of such an alloy is the use of titanium on medical implants. In collaboration with Professor Z. Azpilgain and Ph.D. student J. Kintana (Univ. of Mondragón, Basque Country, Spain), a computer simulation model has been developed using the software COMSOL Multiphysics in order to understand how each parameter of the process affects in the overheating of the liquid.