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Financiado para publicación en acceso aberto: Universidade da Coruña/CISUG [Abstract] In this article we mainly extend to a multi-currency setting some previous works in the literature concerning total value adjustments in a single currency framework. The motivation comes from the fact that financial institutions operate in global markets, so that the financial derivatives in their portfolios involve different currencies. More precisely, in this multi-currency setting we pose the PDE formulations for pricing the total adjustment and the financial derivative with counterparty risk. Moreover, we also formulate the problem in terms of expectations, which allows the use of suitable Monte Carlo techniques that overcome the curse of dimensionality associated to the numerical solution of PDE formulation, when a high number of stochastic factors are involved. Finally, we present some examples to illustrate the performance of the formulations and the proposed numerical techniques. This work has been partially funded by EU H2020-MSCA-ITN-2018 (ABC-EU-XVA Grant Agreement 813261), Spanish Ministry of Science and Innovation (Grant PID2019-108584RB-I00) and by Galician Government (Grant ED431C2018/033), both including FEDER financial support. Authors also acknowledge the support received from the Centro de Investigacin de Galicia (CITIC), funded by Xunta de Galicia and the European Union (European Regional Development Fund- Galicia 2014-2020 Program), by grant ED431G 2019/01 Xunta de Galicia; ED431C2018/033 Xunta de Galicia; ED431G 2019/01

In this work, we analyze a nonlinear partial differential equation (PDE) model for the total value adjustment on European options in the presence of a counterparty risk. We transform the nonlinear PDE into an equivalent one, involving a sectorial operator, and prove the existence and uniqueness of a solution.

Recent observational and theoretical studies indicate that the damping of solar coronal loop oscillations depends on the oscillation amplitude. We consider two mechanisms, linear resonant absorption and a nonlinear damping model. We confront theoretical predictions from these models with observed data in the plane of observables defined by the damping ratio and the oscillation amplitude. The structure of the Bayesian evidence in this plane displays a clear separation between the regions where each model is more plausible relative to the other. There is qualitative agreement between the regions of high marginal likelihood and Bayes factor for the nonlinear damping model and the arrangement of observed data. A quantitative application to 101 loop oscillation cases observed with SDO/AIA results in the marginal likelihood for the nonlinear model being larger in the majority of them. The cases with conclusive evidence for the nonlinear damping model outnumber considerably those in favor of linear resonant absorption., 15 pages, 4 figures, 1 table

The heating of the solar chromosphere and corona to the observed high temperatures, imply the presence of ongoing heating that balances the strong radiative and thermal conduction losses expected in the solar atmosphere. It has been theorized for decades that the required heating mechanisms of the chromospheric and coronal parts of the active regions, quiet-Sun, and coronal holes are associated with the solar magnetic fields. However, the exact physical process that transport and dissipate the magnetic energy which ultimately leads to the solar plasma heating are not yet fully understood. The current understanding of coronal heating relies on two main mechanism: reconnection and MHD waves that may have various degrees of importance in different coronal regions. In this review we focus on recent advances in our understanding of MHD wave heating mechanisms. First, we focus on giving an overview of observational results, where we show that different wave modes have been discovered in the corona in the last decade, many of which are associated with a significant energy flux, either generated in situ or pumped from the lower solar atmosphere. Afterwards, we summarise the recent findings of numerical modelling of waves, motivated by the observational results. Despite the advances, only 3D MHD models with Alfv\'en wave heating in an unstructured corona can explain the observed coronal temperatures compatible with the quiet Sun, while 3D MHD wave heating models including cross-field density structuring are not yet able to account for the heating of coronal loops in active regions to their observed temperature., Comment: accepted in Space Science Reviews

This paper investigates the changes in spatial properties when magnetohydrodynamic (MHD) waves undergo resonant damping in the Alfvén continuum. The analysis is carried out for a 1D cylindrical pressure-less plasma with a straight magnetic field. The effect of the damping on the spatial wave variables is determined by using complex frequencies that arise as a result of the resonant damping. Compression and vorticity are used to characterise the spatial evolution of the MHD wave. The most striking result is the huge spatial variation in the vorticity component parallel to the magnetic field. Parallel vorticity vanishes in the uniform part of the equilibrium. However, when the MHD wave moves into the non-uniform part, parallel vorticity explodes to values that are orders of magnitude higher than those attained by the transverse components in planes normal to the straight magnetic field. In the non-uniform part of the equilibrium plasma, the MHD wave is controlled by parallel vorticity and resembles an Alfvén wave, with the unfamiliar property that it has pressure variations even in the linear regime.

We use Coronal Multi-channel Polarimeter (CoMP) observations of propagating waves in the solar corona and Bayesian analysis to assess the evidence of models with resonant damping and foot-point wave power asymmetries. Two nested models are considered. The reduced model considers resonant damping as the sole cause of the measured discrepancy between outward and inward wave power. The larger model contemplates an extra source of asymmetry with origin at the foot-points. We first compute probability distributions of parameters conditional on the models and the observed data. The obtained constraints are then used to calculate the evidence for each model in view of data. We find that we need to consider the larger model to explain CoMP data and to accurately infer the damping ratio, hence, to better assess the possible contribution of the waves to coronal heating., Accepted in A&A

The Kelvin-Helmholtz instability has been proposed as a mechanism to extract energy from magnetohydrodynamic (MHD) kink waves in flux tubes, and to drive dissipation of this wave energy through turbulence. It is therefore a potentially important process in heating the solar corona. However, it is unclear how the instability is influenced by the oscillatory shear flow associated with an MHD wave. We investigate the linear stability of a discontinuous oscillatory shear flow in the presence of a horizontal magnetic field within a Cartesian framework that captures the essential features of MHD oscillations in flux tubes. We derive a Mathieu equation for the Lagrangian displacement of the interface and analyse its properties, identifying two different instabilities: a Kelvin-Helmholtz instability and a parametric instability involving resonance between the oscillatory shear flow and two surface Alfv\'{e}n waves. The latter occurs when the system is Kelvin-Helmholtz stable, thus favouring modes that vary along the flux tube, and as a consequence provides an important and additional mechanism to extract energy. When applied to flows with the characteristic properties of kink waves in the solar corona, both instabilities can grow, with the parametric instability capable of generating smaller scale disturbances along the magnetic field than possible via the Kelvin-Helmholtz instability. The characteristic time-scale for these instabilities is $\sim 100$ s, for wavelengths of $200$ km. The parametric instability is more likely to occur for smaller density contrasts and larger velocity shears, making its development more likely on coronal loops than on prominence threads., Comment: 12 pages, 4 figures, accepted for publication MNRAS

Some high-resolution observations have revealed that the active-region solar corona is filled with myriads of thin strands even in apparently uniform regions with no resolved loops. This fine structure can host collective oscillations involving a large portion of the corona due to the coupling of the motions of the neighbouring strands. We study these vibrations and the possible observational effects. Here we theoretically investigate the collective oscillations inherent to the fine structure of the corona. We have called them fundamental vibrations because they cannot exist in a uniform medium. We use the T-matrix technique to find the normal modes of random arrangements of parallel strands. We consider an increasing number of tubes to understand the vibrations of a huge number of tubes of a large portion of the corona. We additionally generate synthetic time-distance Doppler and line broadening diagrams of the vibrations of a coronal region to compare with observations. We have found that the fundamental vibrations are in the form of clusters of tubes where not all the tubes participate in the collective mode. The periods are distributed over a wide band of values. The width of the band increases with the number of strands but rapidly reaches an approximately constant value. The frequency band associated with the fine structure of the corona depends on the minimum separation between strands. We have found that the coupling between the strands is of large extent. The synthetic Dopplergrams and line-broadening maps show signatures of collective vibrations, not present in the case of purely random individual kink vibrations. We conclude that the fundamental vibrations of the corona can contribute to the energy budget of the corona and they may have an observational signature., Accepted for publication in the Astronomy and Astrophysics

This paper investigates the mixed properties of MHD waves in a non-uniform plasma. It starts with a short revision of MHD waves in a uniform plasma of infinite extent. In that case the MHD waves do not have mixed properties. They can be separated in Alfv\'{e}n waves and magneto-sonic waves. The Alfv\'{e}n waves propagate parallel vorticity and are incompressible. In addition they have no parallel displacement component. The magneto-sonic waves are compressible and in general do have a parallel component of displacement but do not propagate parallel vorticity. This clear separation has been the reason why there has been a strong inclination in the literature to use this classification in the study of MHD waves in non-uniform plasmas. The main part of this paper is concerned with MHD waves in a non-uniform plasma. It is shown that the MHD waves in that situation in general propagate both vorticity and compression and hence have mixed properties. Finally, the close connection between resonant absorption and MHD waves with mixed properties is discussed.

The purpose of this work is to develop a procedure to obtain the normal modes of a coronal loop from time-dependent numerical simulations with the aim of better understanding observed transverse loop oscillations. To achieve this goal, in this paper we present a new method and test its performance with a problem for which the normal modes can be computed analytically. In a follow-up paper, the application to the simulations of \citet{rial2013} is tackled. The method proceeds iteratively and at each step consists of (i) a time-dependent numerical simulation followed by (ii) the Complex Empirical Orthogonal Function (CEOF) analysis of the simulation results. The CEOF analysis provides an approximation to the normal mode eigenfunctions that can be used to set up the initial conditions for the numerical simulation of the following iteration, in which an improved normal mode approximation is obtained. The iterative process is stopped once the global difference between successive approximate eigenfunctions is below a prescribed threshold. The equilibrium used in this paper contains material discontinuities that result in one eigenfunction with a jump across these discontinuities and two eigenfunctions whose normal derivatives are discontinuous there. After 6 iterations, the approximation to the frequency and eigenfunctions are accurate to $\lesssim 0.7$\% except for the eigenfunction with discontinuities, which displays a much larger error at these positions.

In this work we present a numerical method to approximate the solution of the steady-state compressible Reynolds equation with additional first-order slip flow terms. This equation models the hydrodynamic features of read/write processes in magnetic storage devices such as hard disks. The numerical scheme is based on the local discontinuous Galerkin method proposed by Cockburn and Shu (SIAM J Numer Anal 35:2440–2463, 1998), which shows good properties in the presence of internal layers appearing in convection-diffusion problems. Several test examples illustrate the good performance of the method.

In contrast to the situation in a laboratory, the study of the solar atmosphere has to be pursued without direct access to the physical conditions of interest. Information is therefore incomplete and uncertain and inference methods need to be employed to diagnose the physical conditions and processes. One of such methods, solar atmospheric seismology, makes use of observed and theoretically predicted properties of waves to infer plasma and magnetic field properties. A recent development in solar atmospheric seismology consists in the use of inversion and model comparison methods based on Bayesian analysis. In this paper, the philosophy and methodology of Bayesian analysis are first explained. Then, we provide an account of what has been achieved so far from the application of these techniques to solar atmospheric seismology and a prospect of possible future extensions., 19 pages, accepted in Advances in Space Research

In this work, the authors propose efficient numerical methods to solve mathematical models for different optimal investment problems with irreversible environmental effects. A relevant point is that both the benefits of the environment and the alternative project are uncertain. The cases with instantaneous and progressive transformation of the environment are addressed. In the first case, an augmented Lagrangian active set (ALAS) algorithm combined with finite element methods are proposed as a more efficient technique for the numerical solution to the obstacle problem associated with a degenerated elliptic PDE. In the second case, the mathematical model can be split into two subsequent steps: first we solve numerically a set of parameter dependent boundary value problems (the parameter being the level of progressive transformation), and secondly an evolutive nonstandard obstacle problem is discretized, thus leading to an obstacle problem at each time step. Also, an ALAS algorithm is proposed at each time step. Numerical solutions are validated through qualitative properties theoretically proven in the literature for different examples.

Recent observations show cool, oscillating prominence threads fading when observed in cool spectral lines and appearing in warm spectral lines. A proposed mechanism to explain this evolution is that the threads were heated by turbulence driven by the Kelvin--Helmholtz instability that developed as a result of wave-driven shear flows on the surface of the thread. As the Kelvin--Helmholtz instability is an instability that works to mix the fluids, in the solar corona it can be expected to work by mixing the cool prominence material with that of the hot corona to form a warm boundary layer. In this paper we develop a simple phenomenological model of nonlinear Kelvin--Helmholtz mixing, using it to determine the characteristic density and temperature of the mixing layer, which for the case under study with constant pressure across the two fluids are $\rho_{\rm mixed}=\sqrt{\rho_1\rho_2}$ and $T_{\rm mixed}=\sqrt{T_1T_2}$. One result from the model is that it provides an accurate, as determined by comparison with simulation results, determination of the kinetic energy in the mean velocity field. A consequence of this is that the magnitude of turbulence, and with it the energy that can be dissipated on fast time-scales, as driven by this instability can be determined. For the prominence-corona system, the mean temperature rise possible from turbulent heating is estimated to be less than 1\% of the characteristic temperature (which is found to be $10^5$\,K). These results highlight that mixing, and not heating, are likely to be the cause of the observed transition between cool to warm material in Okamoto et. al (2015). One consequence of this result is that the mixing creates a region with higher radiative loss rates on average than either of the original fluids, meaning that this instability could contribute a net loss of thermal energy from the corona, i.e. coronal cooling., Comment: 17 pages, 9 figures, accepted for publication in ApJ

A classic application of coronal seismology uses transverse oscillations of waveguides to obtain estimates of the magnetic field strength. The procedure requires information on the density of the structures. Often, it ignores the damping of the oscillations. We computed marginal posteriors for parameters such as the waveguide density; the density contrast; the transverse inhomogeneity length-scale; and the magnetic field strength, under the assumption that the oscillations can be modelled as standing magnetohydrodynamic (MHD) kink modes damped by resonant absorption. Our results show that the magnetic field strength can be properly inferred, even if the densities inside and outside the structure are largely unknown. Incorporating observational estimates of plasma density further constrains the obtained posteriors. The amount of information one is willing to include (a priori) for the density and the density contrast influences their corresponding posteriors, but very little the inferred magnetic field strength. The decision to include or leave out the information on the damping and the damping time-scales have a minimal impact on the obtained magnetic field strength. In contrast to the classic method which provides with numerical estimates with error bars or possible ranges of variation for the magnetic field strength, Bayesian methods offer the full distribution of plausibility over the considered range of possible values. The methods are applied to available datasets of observed transverse loop oscillations, can be extended to prominence fine structures or chromospheric spicules and implemented to propagating waves in addition to standing oscillations., Comment: 16 pages, A&A accepted

The final publication is available via https://doi.org/10.1016/j.cam.2016.01.056 [Abstract] The massive use of Internet in the last twenty years has created a huge demand for telecommunications networks capacity. In this work, financial option pricing methods are applied to the problem of network investment decision timing. The main innovative aspect is the consideration of two uncertain factors: the capacity demand and the bandwidth price, the evolution of which is modeled by suitable stochastic processes. Thus, we consider the optimal decision problem of upgrading a line in terms of the (highly volatile) uncertain demand for capacity and the price. By using real options pricing methodology, a set of partial differential equation problems are posed and appropriate numerical methods based on characteristics methods combined with finite elements to approximate the solution are proposed. The combination with a dynamic programming strategy gives rise to a global algorithm to help in the decision of optimizing the value of the line. Ministerio de Economía y Competitividad; MTM2010 21135-C0-01 Xunta de Galicia; GRC2014/044

High-resolution observations provide evidence about the existence of small-amplitude transverse oscillations in solar filament fine structures. These oscillations are believed to represent fast magnetohydrodynamic (MHD) waves and the disturbances are seen to be damped in short timescales of the order of 1 to 4 periods. In this Letter we propose that, due to the highly inhomogeneous nature of the filament plasma at the fine structure spatial scale, the phenomenon of resonant absorption is likely to operate in the temporal attenuation of fast MHD oscillations. By considering transverse inhomogeneity in a straight flux tube model we find that, for density inhomogeneities typical of filament threads, the decay times are of a few oscillatory periods only., 11 pages, 2 figures, ApJL accepted

Recent observations with the Hinode Solar Optical Telescope display an active region prominence whose fine threads oscillate in the vertical direction as they move along a path parallel to the photosphere. A seismological analysis of this event is carried out by taking advantage of the small radius of these structures compared to the total length of magnetic field lines, i.e. by using the thin tube approximation. This analysis reveals that the oscillatory period is only slightly modified by the existence of the flow and that the difference between the period of a flowing thread and a static one is below the error bars of these observations. Moreover, although it is not possible to obtain values of the physical parameters, a lower bound for the Alfv\'en speed (ranging between 120 km s$^{-1}$ and 350 km s$^{-1}$) is obtained for each of the threads. Such Alfv\'en speeds agree with the intense magnetic fields and large densities usually found in active region prominences.

Solar tornadoes are dark vertical filamentary structures observed in the extreme ultraviolet associated with prominence legs and filament barbs. Their true nature and relationship to prominences requires understanding their magnetic structure and dynamic properties. Recently, a controversy has arisen: is the magnetic field organized forming vertical, helical structures or is it dominantly horizontal? And concerning their dynamics, are tornadoes really rotating or is it just a visual illusion? Here, we analyze four consecutive spectropolarimetric scans of a prominence hosting tornadoes on its legs which help us shed some light on their magnetic and dynamical properties. We show that the magnetic field is very smooth in all the prominence, probably an intrinsic property of the coronal field. The prominence legs have vertical helical fields that show slow temporal variation probably related to the motion of the fibrils. Concerning the dynamics, we argue that 1) if rotation exists, it is intermittent, lasting no more than one hour, and 2) the observed velocity pattern is also consistent with an oscillatory velocity pattern (waves)., Comment: accepted for publication in ApJ

Context. The geometry and physical conditions in solar coronal loops are complicated and still not understood well. Recent high-resolution observations obtained with TRACE indicate the existence of sub-resolution transverse structuring not accessible to direct observation. This ingredient has not yet been taken into account in the previous theoretical models used for the study of transversal coronal loop oscillations and of their damping due to resonant conversion of energy. Aims. This study aims to assess the effect of the possibly unresolved internal structure of a coronal loop on the properties of its transverse oscillations and on the efficiency of resonant absorption as a damping mechanism of these oscillations. Methods. The equilibrium configuration of a single coronal loop with internal density structuring is modelled by considering the loop as composed of two very close, parallel, identical coronal slabs in Cartesian geometry. The period of the oscillation and the damping time are computed for the resonantly damped fundamental kink mode. These quantities are then compared to those obtained for two models for a single equivalent slab without internal density structuring. Results. We find that the period and the damping time of a coronal loop with internal density structuring change by less than 15%, when compared to the same oscillatory properties of a single coronal loop with either the same density contrast or a single coronal loop with the same total mass. Conclusions. Therefore the internal density structuring of a coronal loop does not affect its oscillatory properties very much. However, the sub-resolution structuring of a coronal loop with different densities in its components or with different widths could vary these results.

In this work, a new elastohydrodynamic model governing a head-tape magnetic reading device is proposed. The coupled model is mainly based on a nonlinear compressible Reynolds equation and a Koiter model for shells. After its mathematical analysis, a set of numerical techniques is presented to solve the different difficulties associated to the problem. Finally, numerical examples illustrate the good performance of the algorithm.

The spatial variation of physical quantities, such as the mass density, across solar atmospheric waveguides governs the timescales and spatial scales for wave damping and energy dissipation. The direct measurement of the spatial distribution of density, however, is difficult and indirect seismology inversion methods have been suggested as an alternative. We applied Bayesian inference, model comparison, and model-averaging techniques to the inference of the cross-field density structuring in solar magnetic waveguides using information on periods and damping times for resonantly damped magnetohydrodynamic (MHD) transverse kink oscillations. Three commonly employed alternative profiles were used to model the variation of the mass density across the waveguide boundary. Parameter inference enabled us to obtain information on physical quantities such as the Alfv\'en travel time, the density contrast, and the transverse inhomogeneity length scale. The inference results from alternative density models were compared and their differences quantified. Then, the relative plausibility of the considered models was assessed by performing model comparison. Our results indicate that the evidence in favor of any of the three models is minimal, unless the oscillations are strongly damped. In such a circumstance, the application of model-averaging techniques enables the computation of an evidence-weighted inference that takes into account the plausibility of each model in the calculation of a combined inversion for the unknown physical parameters., Comment: 12 pages, 9 figures, 3 tables

Aims. We combine the magnetohydrodynamic (MHD) theory of resonantly damped quasi-mode kink oscillations with observational estimates of the period and damping of transverse coronal loop oscillations to extract information on physical parameters in oscillating loops. Methods. A numerical study of the quasi-mode period and damping, in one-dimensional fully non-uniform flux tubes, is used to obtain equilibrium models that reproduce the observed periods and damping rates. This scheme is applied to 11 loop oscillation events. Results. When only the damping rate is used. the valid equilibrium models form a one-dimensional solution curve in the two-dimensional parameter space (density contrast, transverse inhomogeneity length-scale). Lower limits to the transverse inhomogeneity are obtained in the limit of high contrast loops. When both the period and the damping rate are used, the equilibrium Alfven speed (or Alfven travel time) comes into play. The valid equilibrium models then form a one-dimensional solution curve in the three-dimensional parameter space (density contrast, transverse inhomogeneity length-scale, Alfven speed or Alfven travel time). The projection of these solutions onto the Alfven speed axis is found to be constrained to a rather limited interval. Upper limits to the internal Alfven speed are derived for 9 of the 1 1 analysed events.

In the last decade we have been overwhelmed by an avalanche of discoveries of magnetohydrodynamic (MHD) waves by the Solar and Heliospheric Observatory and Transition Region and Coronal Explorer observatories. Both standing and propagating versions of fast magnetoacoustic and slow magnetoacoustic MHD waves have been detected. Information on the damping times and damping distances of these waves is less detailed and less accurate than that on periods and amplitudes. Nevertheless, observations show the damping times and damping lengths are often short. Also, different types of MHD waves in different types of magnetic structures likely require different damping mechanisms. The phenomenon of fast damping is well documented for the standing fast magnetosonic kink waves in coronal loops. This paper concentrates on standing fast magnetosonic waves. It reports on results on periods and damping times due to resonant absorption in one-dimensional and two-dimensional models of coronal loops. Special attention is given to multiple modes.