Low-Latitude Reduction-to-the-Pole and Upward Continuation between Arbitrary Surfaces based on the Partial Differential Equation Framework

Summary Magnetic surveys conducted in complex conditions, such as low magnetic latitudes, uneven observation surfaces, or above high-susceptibility sources, pose significant challenges for obtaining stable solutions for reduction-to-the-pole (RTP) and upward-continuation processing on arbitrary surfaces. To tackle these challenges, in this study, we propose constructing an equivalent-susceptibility model based on the partial differential equation (PDE) framework in the space domain. A multilayer equivalent-susceptibility method was employed for RTP and upward-continuation operations, thus allowing for application on undulating observation surfaces and strong self-demagnetisation effect in a non-uniform mesh. A novel positivity constraint is introduced to improve the accuracy and efficiency of the inversion. We analysed the effect of the depth-weighting function in the inversion of equivalent susceptibility for RTP and upward-continuation reproduction. Iterative and direct solvers were utilised and compared in solving the large, sparse, nonsymmetric, and ill-conditioned system of linear equations produced by PDE-based equivalent-source construction. Two synthetic models were used to illustrate the efficiency and accuracy of the proposed method in processing both ground and airborne magnetic data. Aeromagnetic, ground data, and prior magnetic orebody information collected in Brazil at a low magnetic latitude region were used to validate the proposed method for processing RTP and upward-continuation operations on magnetic data sets with strong self-demagnetisation.

Free surface models of partially molten rock with visco-elasto–plastic rheology: numerical solutions using a staggered-grid finite-difference discretisation

<p>It is broadly accepted that magmatism plays a key dynamic role in continental and oceanic rifting. However, these dynamics remain poorly studied, largely due to the difficulty of consistently modelling liquid/solid interaction across the lithosphere. The RIFT-O-MAT project seeks to quantify the role of magma in rifting by using models that build upon the two-phase flow theory of magma/rock interaction. A key challenge is to extend the theory to account for the non-linear rheological behaviour of the host rocks, and investigate processes such as diking, faulting and their interaction. Here we present our progress in consistent numerical modelling of poro-viscoelastic–plastic modelling of deformation with a free surface.</p><p>Failure of rocks (plasticity) is an essential ingredient in geodynamics models because Earth materials cannot sustain unbounded stresses. However, plasticity represents a non-trivial problem even for single-phase flow formulations (Spiegelman et al. 2016). The elastic deformation of rocks can also affect the propagation of internal failure. Furthermore, deformation and plastic failure drives topographic change, which imposes a significant static stress field. Robustly solving a discretised model that includes this physics presents severe challenges, and many questions remain as to effective solvers for these strongly nonlinear systems. </p><p>We present a new finite difference staggered grid framework for solving partial differential equations (FD-PDE) for single-/two-phase flow magma dynamics (Pusok et al., 2020). Staggered grid finite-difference methods are mimetic, conservative, inf-sup stable and with small stencil — thus they are well suited to address these problems. The FD-PDE framework uses PETSc (Balay et al., 2020) and aims to separate the user input from the discretization of governing equations. The core goals for the FD-PDE framework is to allow for extensible development and implement a framework for rigorous code validation. Here, we present simplified model problems using the FD-PDE framework for two-phase flow visco-elasto-plastic models designed to characterise the solution quality and assess both the discretisation and solver robustness. We also present results obtained using the phase-field method (Sun and Beckermann, 2007) for representing the free surface. Verification of the phase-field approach will be shown via simplified problems previously examined in the geodynamics community (Crameri et al, 2012).</p><p>Balay et al. (2020), PETSc Users Manual, ANL-95/11 - Revision 3.13.</p><p>Pusok et al. (2020) https://doi.org/10.5194/egusphere-egu2020-18690 </p><p>Spiegelman et al. (2016) https://doi.org/10.1002/2015GC006228</p><p>Sun and Beckermann (2007) https://doi.org/10.1016/j.jcp.2006.05.025</p><p>Crameri et al. (2012) https://doi.org/10.1111/j.1365-246X.2012.05388.x</p><div> <div><span></span><div></div> </div> <div></div> </div><div> <div><span></span><div></div> </div> <div></div> </div>

Total Variation and Mean Curvature PDEs on the Homogeneous Space of Positions and Orientations

Two key ideas have greatly improved techniques for image enhancement and denoising: the lifting of image data to multi-orientation distributions and the application of nonlinear PDEs such as total variation flow (TVF) and mean curvature flow (MCF). These two ideas were recently combined by Chambolle and Pock (for TVF) and Citti et al. (for MCF) for two-dimensional images. In this work, we extend their approach to enhance and denoise images of arbitrary dimension, creating a unified geometric and algorithmic PDE framework, relying on (sub-)Riemannian geometry. In particular, we follow a different numerical approach, for which we prove convergence in the case of TVF by an application of Brezis–Komura gradient flow theory. Our framework also allows for additional data adaptation through the use of locally adaptive frames and coherence enhancement techniques. We apply TVF and MCF to the enhancement and denoising of elongated structures in 2D images via orientation scores and compare the results to Perona–Malik diffusion and BM3D. We also demonstrate our techniques in 3D in the denoising and enhancement of crossing fiber bundles in DW-MRI. In comparison with data-driven diffusions, we see a better preservation of bundle boundaries and angular sharpness in fiber orientation densities at crossings.

Homotopy Analysis Method for Boundary-Value Problem of Turbo Warrant Pricing under Stochastic Volatility

Turbo warrants are liquidly traded financial derivative securities in over-the-counter and exchange markets in Asia and Europe. The structure of turbo warrants is similar to barrier options, but a lookback rebate will be paid if the barrier is crossed by the underlying asset price. Therefore, the turbo warrant price satisfies a partial differential equation (PDE) with a boundary condition that depends on another boundary-value problem (BVP) of PDE. Due to the highly complicated structure of turbo warrants, their valuation presents a challenging problem in the field of financial mathematics. This paper applies the homotopy analysis method to construct an analytic pricing formula for turbo warrants under stochastic volatility in a PDE framework.

ANALYTICAL APPROXIMATION FOR NON-LINEAR FBSDEs WITH PERTURBATION SCHEME

In this work, we have presented a simple analytical approximation scheme for generic non-linear FBSDEs. By treating the interested system as the linear decoupled FBSDE perturbed with non-linear generator and feedback terms, we have shown that it is possible to carry out a recursive approximation to an arbitrarily higher order, where the required calculations in each order are equivalent to those for standard European contingent claims. We have also applied the perturbative method to the PDE framework following the so-called Four Step Scheme. The method is found to render the original non-linear PDE into a series of standard parabolic linear PDEs. Due to the equivalence of the two approaches, it is also possible to derive approximate analytic solution for the non-linear PDE by applying the asymptotic expansion to the corresponding probabilistic model. Two simple examples are provided to demonstrate how the perturbation works and show its accuracy relative to known numerical techniques. The method presented in this paper may be useful for various important problems which have eluded analytical treatment so far.

A Coupled Approach for Fluid Dynamic Problems Using the PDE Framework Peano

We couple different flow models, i.e. a finite element solver for the Navier-Stokes equations and a Lattice Boltzmann automaton, using the framework Peano as a common base. The new coupling strategy between the meso- and macroscopic solver is presented and validated in a 2D channel flow scenario. The results are in good agreement with theory and results obtained in similar works by Latt et al. In addition, the test scenarios show an improved stability of the coupled method compared to pure Lattice Boltzmann simulations.