arXiv subjects

Barbara Wohlmuth

Publications and source records attributed to Barbara Wohlmuth.

At least 109 records · Page 6Linked to original sources

arXiv2016

Calibration to American Options: Numerical Investigation of the de-Americanization

American options are the reference instruments for the model calibration of a large and important class of single stocks. For this task, a fast and accurate pricing algorithm is indispensable. The literature mainly discusses pricing methods for American options that are based on Monte Carlo, tree and partial differential equation methods. We present an alternative approach that has become popular under the name de-Americanization in the financial industry. The method is easy to implement and enjoys fast run-times. Since it is based on ad hoc simplifications, however, theoretical results guaranteeing reliability are not available. To quantify the resulting methodological risk, we empirically test the performance of the de-Americanization method for calibration. We classify the scenarios in which de-Americanization performs very well. However, we also identify the cases where de-Americanization oversimplifies and can result in large errors.

q-fin.CP↗

arXiv2016

A two-scale approach for efficient on-the-fly operator assembly in massively parallel high performance multigrid codes

Matrix-free finite element implementations of massively parallel geometric multigrid save memory and are often significantly faster than implementations using classical sparse matrix techniques. They are especially well suited for hierarchical hybrid grids on polyhedral domains. In the case of constant coefficients all fine grid node stencils in the interior of a coarse macro element are equal. However, for non-polyhedral domains the situation changes. Then even for the Laplace operator, the non-linear element mapping leads to fine grid stencils that can vary from grid point to grid point. This observation motivates a new two-scale approach that exploits a piecewise polynomial approximation of the fine grid operator with respect to the coarse mesh size. The low-cost evaluation of these surrogate polynomials results in an efficient stencil assembly on-the-fly for non-polyhedral domains that can be significantly more efficient than matrix-free techniques that are based on an element-wise assembly. The performance analysis and additional hardware-aware code optimizations are based on the Execution-Cache-Memory model. Several aspects such as two-scale a priori error bounds and double discretization techniques are presented. Weak and strong scaling results illustrate the benefits of the new technique when used within large scale PDE solvers.

math.NA↗

arXiv2016

Scheduling massively parallel multigrid for multilevel Monte Carlo methods

The computational complexity of naive, sampling-based uncertainty quantification for 3D partial differential equations is extremely high. Multilevel approaches, such as multilevel Monte Carlo (MLMC), can reduce the complexity significantly, but to exploit them fully in a parallel environment, sophisticated scheduling strategies are needed. Often fast algorithms that are executed in parallel are essential to compute fine level samples in 3D, whereas to compute individual coarse level samples only moderate numbers of processors can be employed efficiently. We make use of multiple instances of a parallel multigrid solver combined with advanced load balancing techniques. In particular, we optimize the concurrent execution across the three layers of the MLMC method: parallelization across levels, across samples, and across the spatial grid. The overall efficiency and performance of these methods will be analyzed. Here the scalability window of the multigrid solver is revealed as being essential, i.e., the property that the solution can be computed with a range of process numbers while maintaining good parallel efficiency. We evaluate the new scheduling strategies in a series of numerical tests, and conclude the paper demonstrating large 3D scaling experiments.

cs.CE↗

arXiv2016

Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics

We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues.

math.NA↗

arXiv2016

Highly sparse surface couplings for subdomain-wise isoviscous Stokes finite element discretizations

The Stokes system with constant viscosity can be cast into different formulations by exploiting the incompressibility constraint. For instance the strain in the weak formulation can be replaced by the gradient to decouple the velocity components in the different coordinate directions. Thus the discretization of the simplified problem leads to fewer nonzero entries in the stiffness matrix. This is of particular interest in large scale simulations where a reduced memory bandwidth requirement can help to significantly accelerate the computations. In the case of a piecewise constant viscosity, as it typically arises in multi-phase flows, or when the boundary conditions involve traction, the situation is more complex, and one has to treat the cross derivatives in the original Stokes system with care. A naive application of the standard vectorial Laplacian results in a physically incorrect solution, while formulations based on the strain increase the computational effort everywhere, even when the inconsistencies arise only from an incorrect treatment in a small fraction of the computational domain. Here we propose a new approach that is consistent with the strain-based formulation and preserves the decoupling advantages of the gradient-based formulation in isoviscous subdomains. The modification is equivalent to locally changing the discretization stencils, hence the more expensive discretization is restricted to a lower dimensional interface, making the additional computational cost asymptotically negligible. We demonstrate the consistency and convergence properties of the method and show that in a massively parallel setup, the multigrid solution of the resulting discrete systems is faster than for the classical strain-based formulation. Moreover, we give an application example which is inspired by geophysical research.

math.NA↗

arXiv2016

Simultaneous Reduced Basis Approximation of Parameterized Elliptic Eigenvalue Problems

The focus is on a model reduction framework for parameterized elliptic eigenvalue problems by a reduced basis method. In contrast to the standard single output case, one is interested in approximating several outputs simultaneously, namely a certain number of the smallest eigenvalues. For a fast and reliable evaluation of these input-output relations, we analyze a posteriori error estimators for eigenvalues. Moreover, we present different greedy strategies and study systematically their performance. Special attention needs to be paid to multiple eigenvalues whose appearance is parameter-dependent. Our methods are of particular interest for applications in vibro-acoustics.

math.NA↗

arXiv2015

Solution Techniques for the Stokes System: A priori and a posteriori modifications, resilient algorithms

This article proposes modifications to standard low order finite element approximations of the Stokes system with the goal of improving both the approximation quality and the parallel algebraic solution process. Different from standard finite element techniques, we do not modify or enrich the approximation spaces but modify the operator itself to ensure fundamental physical properties such as mass and energy conservation. Special local a~priori correction techniques at re-entrant corners lead to an improved representation of the energy in the discrete system and can suppress the global pollution effect. Local mass conservation can be achieved by an a~posteriori correction to the finite element flux. This avoids artifacts in coupled multi-physics transport problems. Finally, hardware failures in large supercomputers may lead to a loss of data in solution subdomains. Within parallel multigrid, this can be compensated by the accelerated solution of local subproblems. These resilient algorithms will gain importance on future extreme scale computing systems.

math.NA↗

arXiv2015

A quantitative performance analysis for Stokes solvers at the extreme scale

This article presents a systematic quantitative performance analysis for large finite element computations on extreme scale computing systems. Three parallel iterative solvers for the Stokes system, discretized by low order tetrahedral elements, are compared with respect to their numerical efficiency and their scalability running on up to $786\,432$ parallel threads. A genuine multigrid method for the saddle point system using an Uzawa-type smoother provides the best overall performance with respect to memory consumption and time-to-solution. The largest system solved on a Blue Gene/Q system has more than ten trillion ($1.1 \cdot 10 ^{13}$) unknowns and requires about 13 minutes compute time. Despite the matrix free and highly optimized implementation, the memory requirement for the solution vector and the auxiliary vectors is about 200 TByte. Brandt's notion of "textbook multigrid efficiency" is employed to study the algorithmic performance of iterative solvers. A recent extension of this paradigm to "parallel textbook multigrid efficiency" makes it possible to assess also the efficiency of parallel iterative solvers for a given hardware architecture in absolute terms. The efficiency of the method is demonstrated for simulating incompressible fluid flow in a pipe filled with spherical obstacles.

cs.CE↗

arXiv2015

Pore-scale lattice Boltzmann simulation of laminar and turbulent flow through a sphere pack

The lattice Boltzmann method can be used to simulate flow through porous media with full geometrical resolution. With such a direct numerical simulation, it becomes possible to study fundamental effects which are difficult to assess either by developing macroscopic mathematical models or experiments. We first evaluate the lattice Boltzmann method with various boundary handling of the solid-wall and various collision operators to assess their suitability for large scale direct numerical simulation of porous media flow. A periodic pressure drop boundary condition is used to mimic the pressure driven flow through the simple sphere pack in a periodic domain. The evaluation of the method is done in the Darcy regime and the results are compared to a semi-analytic solution. Taking into account computational cost and accuracy, we choose the most efficient combination of the solid boundary condition and collision operator. We apply this method to perform simulations for a wide range of Reynolds numbers from Stokes flow over seven orders of magnitude to turbulent flow. Contours and streamlines of the flow field are presented to show the flow behavior in different flow regimes. Moreover, unknown parameters of the Forchheimer, the Barree--Conway and friction factor models are evaluated numerically for the considered flow regimes.

cs.CE↗

arXiv2015

Non-Isothermal, Multi-phase, Multi-component Flows through Deformable Methane Hydrate Reservoirs

We present a hydro-geomechanical model for subsurface methane hydrate systems. Our model considers kinetic hydrate phase change and non-isothermal, multi-phase, multi-component flow in elastically deforming soils. The model accounts for the effects of hydrate phase change and pore pressure changes on the mechanical properties of the soil, and also for the effect of soil deformation on the fluid-solid interaction properties relevant to reaction and transport processes (e.g., permeability, capillary pressure, reaction surface area). We discuss a 'cause-effect' based decoupling strategy for the model and present our numerical discretization and solution scheme. We then identify the important model components and couplings which are most vital for a hydro-geomechanical hydrate simulator, namely, 1) dissociation kinetics, 2) hydrate phase change coupled with non-isothermal two phase two component flow, 3) two phase flow coupled with linear elasticity (poroelasticity coupling), and finally 4) hydrate phase change coupled with poroelasticity (kinetics-poroelasticity coupling) and present numerical examples where, for each example, one of the aforementioned model components/couplings is isolated. A special emphasis is laid on the kinetics-poroelasticity coupling. We also present a more complex 3D example based on a subsurface hydrate reservoir which is destabilized through depressurization using a low pressure gas well. In this example, we simulate the melting of hydrate, methane gas generation, and the resulting ground subsidence and stress build-up in the vicinity of the well.

math.NA↗

arXiv2015

Large scale lattice Boltzmann simulation for the coupling of free and porous media flow

In this work, we investigate the interaction of free and porous media flow by large scale lattice Boltzmann simulations. We study the transport phenomena at the porous interface on multiple scales, i.e., we consider both, computationally generated pore-scale geometries and homogenized models at a macroscopic scale. The pore-scale results are compared to those obtained by using different transmission models. Two-domain approaches with sharp interface conditions, e.g., of Beavers--Joseph--Saffman type, as well as a single-domain approach with a porosity depending viscosity are taken into account. For the pore-scale simulations, we use a highly scalable communication-reducing scheme with a robust second order boundary handling. We comment on computational aspects of the pore-scale simulation and on how to generate pore-scale geometries. The two-domain approaches depend sensitively on the choice of the exact position of the interface, whereas a well-designed single-domain approach can significantly better recover the averaged pore-scale results.

cs.CE↗

arXiv2015

Resilience for Multigrid Software at the Extreme Scale

Fault tolerant algorithms for the numerical approximation of elliptic partial differential equations on modern supercomputers play a more and more important role in the future design of exa-scale enabled iterative solvers. Here, we combine domain partitioning with highly scalable geometric multigrid schemes to obtain fast and fault-robust solvers in three dimensions. The recovery strategy is based on a hierarchical hybrid concept where the values on lower dimensional primitives such as faces are stored redundantly and thus can be recovered easily in case of a failure. The lost volume unknowns in the faulty region are re-computed approximately with multigrid cycles by solving a local Dirichlet problem on the faulty subdomain. Different strategies are compared and evaluated with respect to performance, computational cost, and speed up. Especially effective are strategies in which the local recovery in the faulty region is executed in parallel with global solves and when the local recovery is additionally accelerated. This results in an asynchronous multigrid iteration that can fully compensate faults. Excellent parallel performance on a current peta-scale system is demonstrated.

cs.MS↗

arXiv2015

The Influence of Quadrature Errors on Isogeometric Mortar Methods

Mortar methods have recently been shown to be well suited for isogeometric analysis. We review the recent mathematical analysis and then investigate the variational crime introduced by quadrature formulas for the coupling integrals. Motivated by finite element observations, we consider a quadrature rule purely based on the slave mesh as well as a method using quadrature rules based on the slave mesh and on the master mesh, resulting in a non-symmetric saddle point problem. While in the first case reduced convergence rates can be observed, in the second case the influence of the variational crime is less significant.

math.NA↗

arXiv2015

Simultaneous quasi-optimal convergence in FEM-BEM coupling

We consider the symmetric FEM-BEM coupling that connects two linear elliptic second order partial differential equations posed in a bounded domain $Ω$ and its complement, where the exterior problem is restated by an integral equation on the coupling boundary $Γ=\partialΩ$. We assume that the corresponding transmission problem admits a shift theorem for data in $H^{-1+s}$, $s \in [-1,-1+s_0]$, $s_0 > 1/2$. We analyze the discretization by piecewise polynomials of degree $k$ for the domain variable and piecewise polynomials of degree $k-1$ for the flux variable on the coupling boundary. Given sufficient regularity we show that (up to logarithmic factors) the optimal convergence $O(h^{k+1/2})$ in the $H^{-1/2}(Γ)$-norm is obtained for the flux variable, while classical arguments by Céa-type quasi-optimality and standard approximation results provide only $O(h^k)$ for the overall error in the natural product norm on $H^1(Ω)\times H^{-1/2}(Γ)$.

math.NA↗

arXiv2015

Resilience for Exascale Enabled Multigrid Methods

With the increasing number of components and further miniaturization the mean time between faults in supercomputers will decrease. System level fault tolerance techniques are expensive and cost energy, since they are often based on redundancy. Also classical check-point-restart techniques reach their limits when the time for storing the system state to backup memory becomes excessive. Therefore, algorithm-based fault tolerance mechanisms can become an attractive alternative. This article investigates the solution process for elliptic partial differential equations that are discretized by finite elements. Faults that occur in the parallel geometric multigrid solver are studied in various model scenarios. In a standard domain partitioning approach, the impact of a failure of a core or a node will affect one or several subdomains. Different strategies are developed to compensate the effect of such a failure algorithmically. The recovery is achieved by solving a local subproblem with Dirichlet boundary conditions using local multigrid cycling algorithms. Additionally, we propose a superman strategy where extra compute power is employed to minimize the time of the recovery process.

cs.CE↗

arXiv2014

Trace and flux a priori error estimates in finite element approximations of Signorni-type problems

Variational inequalities play in many applications an important role and are an active research area. Optimal a priori error estimates in the natural energy norm do exist but only very few results in other norms exist. Here we consider as prototype a simple Signorini problem and provide new optimal order a priori error estimates for the trace and the flux on the Signorini boundary. The a priori analysis is based on the exact and a mesh-dependent Steklov-Poincaré operator as well as on duality in Aubin-Nitsche type arguments. Numerical results illustrate the convergence rates of the finite element approach.

math.NA↗

arXiv2014

Reduced basis methods for pricing options with the Black-Scholes and Heston model

In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined POD-Greedy and Angle-Greedy procedure for the construction of the primal and dual reduced spaces. Analytically, we prove the reproduction property of the reduced scheme and derive a posteriori error estimators. Numerical examples are provided, illustrating the approximation quality and convergence of our approach for the different option pricing models. Also, we investigate the reliability and effectivity of the error estimators.

math.NA↗

arXiv2014

Isogeometric mortar methods

The application of mortar methods in the framework of isogeometric analysis is investigated theoretically as well as numerically. For the Lagrange multiplier two choices of uniformly stable spaces are presented, both of them are spline spaces but of a different degree. In one case, we consider an equal order pairing for which a cross point modification based on a local degree reduction is required. In the other case, the degree of the dual space is reduced by two compared to the primal. This pairing is proven to be inf-sup stable without any necessary cross point modification. Several numerical examples confirm the theoretical results and illustrate additional aspects. Keywords: isogeometric analysis, mortar methods, inf-sup stability, cross point modification.

math.NA↗