Option pricing with a direct adaptive sparse grid approach

This work proposes and analyzes an anisotropic sparse grid stochastic collocation
\nmethod for solving partial differential equations with random coefficients and forcing terms (input
\ndata of the model). The method consists of a Galerkin approximation in the space variables and a
\ncollocation, in probability space, on sparse tensor product grids utilizing either Clenshaw–Curtis or
\nGaussian knots. Even in the presence of nonlinearities, the collocation approach leads to the solution
\nof uncoupled deterministic problems, just as in the Monte Carlo method. This work includes a priori
\nand a posteriori procedures to adapt the anisotropy of the sparse grids to each given problem.
\nThese procedures seem to be very effective for the problems under study. The proposed method
\ncombines the advantages of isotropic sparse collocation with those of anisotropic full tensor product
\ncollocation: the first approach is effective for problems depending on random variables which weigh
\napproximately equally in the solution, while the benefits of the latter approach become apparent when
\nsolving highly anisotropic problems depending on a relatively small number of random variables, as
\nin the case where input random variables are Karhunen–Lo`eve truncations of “smooth” random
\nfields. This work also provides a rigorous convergence analysis of the fully discrete problem and
\ndemonstrates (sub)exponential convergence in the asymptotic regime and algebraic convergence in
\nthe preasymptotic regime, with respect to the total number of collocation points. It also shows
\nthat the anisotropic approximation breaks the curse of dimensionality for a wide set of problems.
\nNumerical examples illustrate the theoretical results and are used to compare this approach with
\nseveral others, including the standard Monte Carlo. In particular, for moderately large-dimensional
\nproblems, the sparse grid approach with a properly chosen anisotropy seems to be very efficient and
\nsuperior to all examined methods.

Adaptive sparse grid algorithms with applications to electromagnetic scattering under uncertainty

Sparse grids provide a very efficient method for the multilinear approximation of functions, especially in higher-dimensional spaces. In the d-dimensional space, the nodal multilinear basis on a grid with mesh size $h = 2^{ - n} $ consists of $O(2^{nd} )$ basis functions and leads to an $L\_2$-error of order $O(4^{ - n} )$ and an $H\_1$-error of order $O(2^{ - n} )$. -With sparse grids we get an $L\_2$-error of order $O(4^{ - n} n^{d - 1} )$ and an $H\_1$-error of order $O(2^{ - n} )$ with only $O(2^n n^{d - 1} )$ basis functions, if the function u fulfills the condition $\frac{{\partial ^{2d} }} {{\partial x_1^2 \partial x_2^2 \cdots \partial x_d^2 }}u < \infty $. Therefore, we can achieve much more accurate approximations with the same amount of storage. A data structure for the sparse grid representation of functions defined on cubes of arbitrary dimension and a finite element approach for the Helmholtz equation with sparse grid functions are introduced. Special emphasis is taken in the development of an efficient algorithm for the multiplication with the stiffness matrix. With an appropriate preconditioned conjugate gradient method (cg-method), the linear systems can be solved efficiently. Numerical experiments are presented for Helmholtz equations and eigenvalue problems for the Laplacian in two and three dimensions, and for a six-dimensional Poisson problem. The results support the assertion that the $L\_2$-error bounds for the sparse-grid approximation are also valid for sparse grid finite element solutions of elliptic differential equations. Problems with nonsmooth solutions are treated with adaptive sparse grids.

High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a 2d-dimensional problem equals that of a d-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.

Summary A sparse grid approach to orbit uncertainty propagation is presented. Efficient and accurate uncertainty propagation methods for nonlinear dynamic systems have been of enormous interest to space object tracking. Recent methods include those based on the time evolution of the probability density function, the statistical moments, the random samples, or a sum of Gaussian components. The idea of the sparse grid method for orbit uncertainty propagation is to represent the initial uncertainty by a sparse grid, propagate the sparse grid points individually through the nonlinear orbit dynamics, and compute the statistical moments from the propagated sparse grid points. The Smolyak rule is used to generate the sparse grid through linear combinations of low-level tensor products; the number of the resultant sparse grid points is polynomial in the dimension of the state. The sparse grid based uncertainty propagation may be considered a generalization of the Unscented Transformation (UT) of the well-known Unscented Kalman Filter but can achieve much higher accuracy levels than UT. Two examples of orbit propagation in different dynamic environments are used to show the efficacy of the sparse grid approach.

Evaluating two sparse grid surrogates and two adaptation criteria for groundwater Bayesian uncertainty quantification

We consider the problem of multidimensional adaptive hierarchical interpolation. We use sparse grids points and functions that are induced from a one dimensional hierarchical rule via tensor products. The classical locally adaptive sparse grid algorithm uses an isotropic refinement from the coarser to the denser levels of the hierarchy. However, the multidimensional hierarchy provides a more complex structure that allows for various anisotropic and hierarchy selective refinement techniques. We consider the more advanced refinement techniques and apply them to a number of simple test functions chosen to demonstrate the various advantages and disadvantages of each method. While there is no refinement scheme that is optimal for all functions, the fully adaptive family-direction-selective technique is usually more stable and requires fewer samples.

An adaptive sparse grid local discontinuous Galerkin method for Hamilton-Jacobi equations in high dimensions

These days sparse grids are of increasing interest in numerical simulations. Based upon hierarchical tensor product bases, the sparse grid approach is a very efficient one improving the ratio of invested storage and computing time to the achieved accuracy for many problems in the area of numerical solution of differential equations, for instance in numerical fluid mechanics. The particle tracing algorithms that are available so far cannot cope with sparse grids. Now we present an approach that directly works on sparse grids. As a second aspect in this paper, we suggest to use sparse girds as a data compression method in order to visualize huge data sets even on small workstations. Because the size of data sets used in numerical simulations is still growing, this feature makes it possible that workstations can continue to handle these data sets.

The sparse grid approach uses basis functions which are tensor products of univariate basis functions. This gives a connection between sparse grids and the tensor space setting. On the other hand, in the tensor calculus one is interested in suitable representations of tensors (cf. Hackbusch, Tensor spaces and numerical tensor calculus, Monograph in preparation). Then the storage size is in particular described by certain rank parameters. Limited rank requires an approximation of the tensors (functions). In particular, function approximations like polynomial interpolation may lead to such representations. Here, we exploit the sparse grid approximation. We show that the involved rank is not the sparse grid complexity \(O(N{\log }^{d-1}N),\) but \(O({N{}^{(d-1)/d}\log }^{d-3}N).\) KeywordsSparse GridTensor SpaceNumerical Tensor CalculusUnivariate Basis FunctionsTucker FormatThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present the parallelization of a sparse grid finite element discretization of the Black-Scholes equation, which is commonly used for option pricing. Sparse grids allow to handle higher dimensional options than classical approaches on full grids, and can be extended to a fully adaptive discretization method. We introduce the algorithmical structure of efficient algorithms operating on sparse grids, and demonstrate how they can be used to derive an efficient parallelization with OpenMP of the Black-Scholes solver. We show results on different commodity hardware systems based on multi-core architectures with up to 8 cores, and discuss the parallel performance using Intel and AMD CPUs.

The sparse grid combination technique provides a framework to solve high-dimensional numerical problems with standard solvers by assembling a sparse grid from many coarse and anisotropic full grids called component grids. Hierarchization is one of the most fundamental tasks for sparse grids. It describes the transformation from the nodal basis to the hierarchical basis. In settings where the component grids have to be frequently combined and distributed in a massively parallel compute environment, hierarchization on component grids is relevant to minimize communication overhead.

Investigating the interplay between muscular activity and motion is the basis to improve our understanding of healthy or diseased musculoskeletal systems. To be able to analyze the musculoskeletal systems, computational models are used. Albeit some severe modeling assumptions, almost all existing musculoskeletal system simulations appeal to multibody simulation frameworks. Although continuum-mechanical musculoskeletal system models can compensate for some of these limitations, they are essentially not considered because of their computational complexity and cost. The proposed framework is the first activation-driven musculoskeletal system model, in which the exerted skeletal muscle forces are computed using 3-dimensional, continuum-mechanical skeletal muscle models and in which muscle activations are determined based on a constraint optimization problem. Numerical feasibility is achieved by computing sparse grid surrogates with hierarchical B-splines, and adaptive sparse grid refinement further reduces the computational effort. The choice of B-splines allows the use of all existing gradient-based optimization techniques without further numerical approximation. This paper demonstrates that the resulting surrogates have low relative errors (less than 0.76%) and can be used within forward simulations that are subject to constraint optimization. To demonstrate this, we set up several different test scenarios in which an upper limb model consisting of the elbow joint, the biceps and triceps brachii, and an external load is subjected to different optimization criteria. Even though this novel method has only been demonstrated for a 2-muscle system, it can easily be extended to musculoskeletal systems with 3 or more muscles.

A proof of convergence for the combination technique for the Laplace equation using tools of symbolic computation

An adaptive sparse grid method for elliptic PDEs with stochastic coefficients