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Full waveform inversion (FWI) is a process in which seismic numerical simulations are fit to observed data by changing the wave velocity model of the medium under investigation. The problem is nonlinear, and therefore optimization techniques have been used to find a reasonable solution to the problem. The main problem in fitting the data is the lack of low spatial frequencies. This deficiency often leads to a local minimum and to nonplausible solutions. In this work we explore how to obtain low-frequency information for FWI. Our approach involves augmenting FWI with travel time tomography, which has low-frequency features. By jointly inverting these two problems we enrich FWI with information that can replace low-frequency data. In addition, we use high-order regularization, in a preliminary inversion stage, to prevent high-frequency features from polluting our model in the initial stages of the reconstruction. This regularization also promotes the nondominant low-frequency modes that exist in the FWI sensitivity. By applying a joint FWI and travel time inversion we are able to obtain a smooth model than can later be used to recover a good approximation for the true model. A second contribution of this paper involves the acceleration of the main computational bottleneck in FWI--the solution of the Helmholtz equation. We show that the solution time can be reduced by solving the equation for multiple right-hand sides using block multigrid preconditioned Krylov methods. [ABSTRACT FROM AUTHOR]

This paper concerns kernel-based interpolation methods for vector data with correlated components. It gives conditions for a matrix kernel to be conditionally positive definite in an appropriate sense. The conditions allow construction of matrix kernels from nonsymmetric mixtures and scalings of scalar kernels. In particular the kernel used to model the influence of component i on component j can be different from that used to model the influence of component j on component i. The vector modeling techniques considered are particularly appropriate when there are relatively few measurements of one quantity and relatively many of another "correlated" quantity. The paper concludes with some numerical tests on model problems. [ABSTRACT FROM AUTHOR]

Linear advection of a scalar quantity by a specified velocity field arises in a number of different applications. Of particular interest here is the transport of species and energy in low Mach number models for combustion, atmospheric flows, and astrophysics, as well as contaminant transport in Darcy models of saturated subsurface flow. An important characteristic of these problems is that the velocity field is not known analytically. Instead, an auxiliary equation is solved to compute averages of the velocities over faces in a finite volume discretization. In this paper, we present a customized three-dimensional finite volume advection scheme for this class of problems that provides accurate resolution for smooth problems while avoiding undershoot and overshoot for nonsmooth profiles. The method is an extension of an algorithm by Bell, Dawson, and Shubin (BDS), which was developed for a class of scalar conservation laws arising in porous media flows in two dimensions. The original BDS algorithm is a variant of unsplit. higher-order Godunov methods based on construction of a limited bilinear profile within each computational cell. Here we present a three-dimensional extension of the original BDS algorithm that is based on a limited trilinear profile within each cell. We compare this new method to several other unsplit approaches, including piecewise linear methods, piecewise parabolic methods, and wave propagation schemes. [ABSTRACT FROM AUTHOR]

Many complex real-world systems can be accurately modeled by simulations. However, high-fidelity simulations may take hours or even days to compute. Because this can be impractical, a surrogate model is often used to approximate the dynamic behavior of the original simulator. This model can then be used as a cheap, drop-in replacement for the simulator. Because simulations can be very expensive, the data points, which are required to build the model, must be chosen as optimally as possible. Sequential design strategies offer a huge advantage over one-shot experimental designs because they can use information gathered from previous data points in order to determine I lie location of new data points. Each sequential design strategy must perform a trade-off between exploration and exploitation, where the former involves selecting data points in unexplored regions of the design space, while the latter suggests adding data points in regions which were previously identified to be interesting (for example, highly nonlinear regions). In this paper, a novel hybrid sequential design strategy is proposed which uses a Monte Carlo-based approximation of a Voronoi tessellation for exploration and local linear approximations of the simulator for exploitation. The advantage of this method over other sequential design methods is that it is independent of the model type, and can therefore be used in heterogeneous modeling environments, where multiple model types are used at the same time. The new method is demonstrated on a number of test problems, showing that it is a robust, competitive, and efficient sequential design strategy. [ABSTRACT FROM AUTHOR]

Discrete duality finite volume (DDFV) schemes have recently been developed in two dimensions to approximate nonlinear diffusion problems on general meshes. In this paper, a three-dimensional extension of these schemes is proposed. The construction of this extension is detailed and its main properties are proved: a priori bounds, well-posedness, and error estimates. The practical implementation of this scheme is easy. Numerical experiments are presented to illustrate its good behavior. [ABSTRACT FROM AUTHOR]

In [F. Li, C.-W. Shu, Y.-T. Zhang, H. Zhao, J. Comput. Phys., 227 (2008) pp. 8191-8208], we developed a fast sweeping method based on a hybrid local solver which is a combination of a discontinuous Galerkin (DG) finite element solver and a first order finite difference solver for Eikonal equations. The method has second order accuracy in the L¹ norm and a very fast convergence speed, but only first order accuracy in the L∞ norm for the general cases. This is an obstacle to the design of higher order DG fast sweeping methods. In this paper, we overcome this problem by developing uniformly accurate DG fast sweeping methods for solving Eikonal equations. We design novel causality indicators which guide the information flow directions for the DG local solver. The values of these indicators are initially provided by the first order finite difference fast sweeping method, and they are updated during iterations along with the solution. We observe both a uniform second order accuracy in the L∞ norm (in smooth regions) and the fast convergence speed (linear computational complexity) in the numerical examples. [ABSTRACT FROM AUTHOR]

Quantum Monte Carlo (QMC) methods are often used to calculate properties of many body quantum systems. The main cost of many QMC methods, for example, the variational Monte Carlo (VMC) method, is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators so that the cost of constructing Slater matrices in these systems is now linear in the number of particles, whereas computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing the determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution of this paper is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of incomplete LU decomposition with threshold and pivoting (ILUTP) preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of QMC algorithms from O(n³) per sweep to roughly O(n²), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors. Our results show that preconditioned iterative solvers can dramatically reduce the cost of VMC for large(r) systems. [ABSTRACT FROM AUTHOR]

In the image processing community, there have recently been many restoration and reconstruction problems that can be reformulated into linearly constrained convex programming models whose objective functions have separable structures. These favorable reformulations have promoted impressive applications of the alternating direction method (ADM) in the field of image processing. At each iteration, the computation of ADM is dominated by solving two subproblems exactly. However, in many restoration and reconstruction applications, it is cither impossible or extremely expensive to obtain exact solutions of these ADM subproblems. This fact urges the development on inexact versions of ADM, which allow the generated ADM subproblems to be solved approximately subject to certain inexactness criteria. In this paper, we develop some truly implementable inexact ADMs whose inexactness criteria controlling the accuracy of the ADM subproblems are easily implementable. The convergence of the new inexact ADMs will be proved. Numerical results on several image processing problems will be given to illustrate the effectiveness of the proposed inexact ADMs. [ABSTRACT FROM AUTHOR]

This paper presents a method for simulating compressible two-phase flow by combining the best features of a front tracking method (FT) and a ghost fluid method (GFM). In contrast to GFM, a Riemann problem is solved to find the ghost states. And in contrast to FT, the front states used in the Riemann problem are not dynamic variables but are obtained by extrapolation from the interior (grid) states. Pressure jumps associated with surface tension forces are modeled in the Riemann problem. This method handles surface tension forces in a sharp way and avoids artificially spreading surface tension forces over the computational grid as used in continuous surface models. To handle the topological bifurcations of a three-dimensional (3D) surface mesh in the FT, an improved locally grid-based method (LGB) is proposed. The method is robust and minimizes the numerical mass diffusion due to interface reconstruction. The performance of the method is assessed from a broad set of test problems including compressible Kelvin-Helmholtz instabilities, parasitic currents, drop oscillation, bubble-shock interaction, and Rayleigh instabilities. The proposed new method is shown to be comparable to either of its constituent methods by themselves. The advantage of the new method lies in the removal of late time instabilities associated with both of the constituent methods when applied to the 3D simulation of a high speed jet. [ABSTRACT FROM AUTHOR]

This paper presents an efficient multiscale stochastic framework for uncertainty quantification in modeling of flow through porous media with multiple rock types. The governing equations are based on Darcy's law with nonstationary stochastic permeability represented as a sum of local Karhunen-Loève expansions. The approximation uses stochastic collocation on either a tensor product or a sparse grid, coupled with a domain decomposition algorithm known as the multiscale mortar mixed finite element method. The latter method requires solving a coarse scale mortar interface problem via an iterative procedure. The traditional implementation requires the solution of local fine scale linear systems on each iteration. We employ a recently developed modification of this method that precomputes a multiscale flux basis to avoid the need for subdomain solves on each iteration. In the stochastic setting, the basis is further reused over multiple realizations, leading to collocation algorithms that are more efficient than the traditional implementation by orders of magnitude. Error analysis and numerical experiments are presented. [ABSTRACT FROM AUTHOR]

In this paper we show that recently introduced quantics tensor train (QTT) decomposition call be considered as an algebraic wavelet transform with adaptively determined filters. The main algorithm for obtaining QTT decomposition can be reformulated as a method seeking "good subspaces" or "good bases" and considered as a parameterized transformation of an initial tensor into a sparse tensor. This interpretation allows us to introduce a modification of the tensor train-SVD (TT-SVD) algorithm to make it work in cases where the original algorithm does not work; it results in the new wavelet-like transforms called wavelet tensor train (WTT) transform. Properties of WTT transforms are studied numerically, and a theoretical conjecture on the number of vanishing moments is proposed. It is shown that WTT transforms are orthogonal by construction, and the efficiency of WTT is compared with and often outperforms Daubechies wavelet transforms on certain classes of function-related vectors and matrices. [ABSTRACT FROM AUTHOR]

The aim of the paper is to construct a conservative domain decomposition procedure for solving nonlinear diffusion equations. In this procedure, the underlying discretization consists of a cell-centered finite volume scheme on arbitrary quadrilateral grids, which is first linearized by the usual Picard nonlinear iteration. Then in each nonlinear iteration step a domain decomposition algorithm for solving the linearized problem is presented, in which Dirichlet boundary data at inner interfaces for subdomain problems are given by the adjacent cell-centered values obtained in the previous nonlinear iteration step. After the Picard nonlinear iteration converges, discrete flux boundary data are constructed at inner interfaces of subdomains. Finally, they serve as Neumann boundary conditions at inner interfaces to solve subdomain problems once more. The procedure is globally conservative since it is locally conservative both in the subdomains and across inner interfaces. Numerical results are presented to examine the performance of the conservative domain decomposition method, in terms of stability, accuracy, conservative error, and parallel speedup. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical simulations for the dynamics of the molecular beam epitaxy (MBE) model. The numerical simulations of the MBE models require long time computations, and therefore large time-stepping methods become necessary. In this work, we consider some unconditionally energy stable finite difference schemes, which will be used in the time adaptivity strategies. It is found that the use of the time adaptivity cannot only resolve the steady-state solutions but also the dynamical changes of the solution accurately and efficiently. The adaptive time step is selected based on the energy variation or the change of the roughness of the solution. The numerical experiments demonstrated that the CPU time is significantly saved for long time simulations. [ABSTRACT FROM AUTHOR]

Projection onto polynomial chaos (PC) basis functions is often used to reformulate a system of ordinary differential equations (ODEs) with uncertain parameters and initial conditions as a deterministic ODE system that describes tide evolution of the PC modes. The deterministic Jacobian of this projected system is different and typically much larger than the random Jacobian of tide original ODE system. This paper shows that the location of the eigenvalues of the projected Jacobian is largely determined by the eigenvalues of the original Jacobian, regardless of PC order or choice of orthogonal polynomials. Specifically, the eigenvalues of the projected Jacobian always lie in tide convex hull of the numerical range of the Jacobian of the original system. [ABSTRACT FROM AUTHOR]

In this paper, we propose a semi-implicit coupling scheme for the numerical simulation of fluid-structure interaction systems involving a viscous incompressible fluid. The scheme is stable irrespective of the so-called added-mass effect and allows for conservative time-stepping within the structure. The efficiency of the scheme is based on the explicit splitting of the viscous effects and geometrical/convective nonlinearities through the use of the Chorin-Temam projection scheme within the fluid. Stability comes from the implicit pressure-solid coupling and a specific Robin treatment of the explicit viscous-solid coupling, derived from Nitsche's method. [ABSTRACT FROM AUTHOR]

In this paper, we develop, analyze, and test a local discontinuous Galerkin (LDG) method for solving the Hunter-Saxton (HS) equation and its zero-viscosity and zero-dispersion limits. The energy stability for general solutions are proved, and numerical simulation results for different types of solutions of the nonlinear HS equation are provided to illustrate the accuracy and capability of the LDG method. The zero-viscosity and zero-dispersion properties of the HS equation are studied in a numerical simulation. [ABSTRACT FROM AUTHOR]

The problem of maintaining consistent representations of solids in computer-aided design and of giving rigorous proofs of error bounds for operations such as regularized Boolean intersection has been widely studied for at least two decades. One of the major difficulties is that the representations used in practice not only are in error but are fundamentally inconsistent. Such inconsistency is one of the main bottlenecks in downstream applications. This paper provides a framework for error analysis in the context of solid modeling, in the case where the data is represented using the standard representational method, and where the data may be uncertain. Included are discussions of ill-condition, error measurement, stability of algorithms, inconsistency of defining data, and the question of when we should invoke methods outside the scope of numerical analysis. A solution to the inconsistency problem is proposed and supported by theorems: it is based on the use of Whitney extension to define sets, called Quasi-NURBS sets, which are viewed as realizations of the inconsistent data provided to the numerical method. A detailed example illustrating the problem of regularized Boolean intersection is also given. [ABSTRACT FROM AUTHOR]

The time-dependent Ginzburg-Landau model is used extensively in studying the nonequilibrium state of superconductivity. The computer simulation of this model requires highperformance computing power and reliable and efficient numerical methods to solve the GinzburgLandau equations. In this paper, a linearized Crank-Nicolson-Galerkin method is proposed for solving these nonlinear and coupled partial differential equations. The method uses the Galerkin finite element approximation in spatial discretization and the Crank-Nicolson implicit scheme in time discretization, together with certain techniques which linearize and decouple the Ginzburg-Landau equations. While retaining the stability and accuracy of the Crank-Nicolson scheme, the proposed approach results in symmetric and positive definite matrix problems, thus substantially improving the computational efficiency. Furthermore, and even more important, the proposed approach is suitable for large-scale parallel computation. Numerical results from simulating the vortex dynamics of superconductivity by using the linearized Crank-Nicolson-Galerkin method are presented. [ABSTRACT FROM AUTHOR]