Numerische Mathematik最新文献

We propose a sparse algebra for samplet compressed kernel matrices to enable efficient scattered data analysis. We show that the compression of kernel matrices by means of samplets produces optimally sparse matrices in a certain S-format. The compression can be performed in cost and memory that scale essentially linearly with the number of data points for kernels of finite differentiability. The same holds true for the addition and multiplication of S-formatted matrices. We prove that the inverse of a kernel matrix, given that it exists, is compressible in the S-format as well. The use of selected inversion allows to directly compute the entries in the corresponding sparsity pattern. Moreover, S-formatted matrix operations enable the efficient, approximate computation of more complicated matrix functions such as ({varvec{A}}^alpha ) or (exp ({varvec{A}})) of a matrix ({varvec{A}}). The matrix algebra is justified mathematically by pseudo differential calculus. As an application, we consider Gaussian process learning algorithms for implicit surfaces. Numerical results are presented to illustrate and quantify our findings.

We analyze the local accuracy of the virtual element method. More precisely, we prove an error bound similar to the one holding for the finite element method, namely, that the local (H^1) error in a interior subdomain is bounded by a term behaving like the best approximation allowed by the local smoothness of the solution in a larger interior subdomain plus the global error measured in a negative norm.

The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter (C_{text {st},1}) that is found not robust as the polynomial degree p increases. This is related to the (H^1) stability bound of the (L^{2}) projection onto polynomials of degree at most p and its growth (C_{textrm{st, 1}}propto (p+1)^{1/2}) as (p rightarrow infty ). A similar estimate for the Galerkin projection holds with a p-robust constant (C_{text {st},2}) and (C_{text {st},2} le 2) for right-isosceles triangles. This paper utilizes the new inequality with the constant (C_{text {st},2}) to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved (L^{2}) error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.

This paper proposes a new numerical method for a fully-coupled, quasi-static thermo-poroelasticity model in a unified enriched Galerkin (EG) method framework. In our method, the mechanics sub-problem is solved using a locking-free EG method, and the flow and heat sub-problems are solved using a locally-conservative EG method. The proposed method offers mass and energy conservation properties with much lower costs than other methods with the same properties, including discontinuous Galerkin methods and mixed finite element methods. The well-posedness and optimal a priori error estimates are carefully derived. Several numerical tests confirm the theoretical optimal convergence rates and the mass and energy conservation properties of the new method.

Solutions to many important partial differential equations satisfy bounds constraints, but approximations computed by finite element or finite difference methods typically fail to respect the same conditions. Chang and Nakshatrala (Comput Methods Appl Mech Eng 320:287–334, 2017) enforce such bounds in finite element methods through the solution of variational inequalities rather than linear variational problems. Here, we provide a theoretical justification for this method, including higher-order discretizations. We prove an abstract best approximation result for the linear variational inequality and estimates showing that bounds-constrained polynomials provide comparable approximation power to standard spaces. For any unconstrained approximation to a function, there exists a constrained approximation which is comparable in the (W^{1,p}) norm. In practice, one cannot efficiently represent and manipulate the entire family of bounds-constrained polynomials, but applying bounds constraints to the coefficients of a polynomial in the Bernstein basis guarantees those constraints on the polynomial. Although our theoretical results do not guaruntee high accuracy for this subset of bounds-constrained polynomials, numerical results indicate optimal orders of accuracy for smooth solutions and sharp resolution of features in convection–diffusion problems, all subject to bounds constraints.

We introduce a new discretization based on a polynomial Trefftz-DG method for solving the Stokes equations. Discrete solutions of this method fulfill the Stokes equations pointwise within each element and yield element-wise divergence-free solutions. Compared to standard DG methods, a strong reduction of the degrees of freedom is achieved, especially for higher polynomial degrees. In addition, in contrast to many other Trefftz-DG methods, our approach allows us to easily incorporate inhomogeneous right-hand sides (driving forces) by using the concept of the embedded Trefftz-DG method. On top of a detailed a priori error analysis, we further compare our approach to other (hybrid) discontinuous Galerkin Stokes discretizations and present numerical examples.

Abstract

This paper deals with the equation (-varDelta u+mu u=f) on high-dimensional spaces ({mathbb {R}}^m) , where the right-hand side (f(x)=F(Tx)) is composed of a separable function F with an integrable Fourier transform on a space of a dimension (n>m) and a linear mapping given by a matrix T of full rank and (mu ge 0) is a constant. For example, the right-hand side can explicitly depend on differences (x_i-x_j) of components of x. Following our publication (Yserentant in Numer Math 146:219–238, 2020), we show that the solution of this equation can be expanded into sums of functions of the same structure and develop in this framework an equally simple and fast iterative method for its computation. The method is based on the observation that in almost all cases and for large problem classes the expression (Vert T^tyVert ^2) deviates on the unit sphere (Vert yVert =1) the less from its mean value the higher the dimension m is, a concentration of measure effect. The higher the dimension m, the faster the iteration converges.

We consider a generalization of local density of states which is “windowed” with respect to position and energy, called the windowed local density of states (wLDOS). This definition generalizes the usual LDOS in the sense that the usual LDOS is recovered in the limit where the position window captures individual sites and the energy window is a delta distribution. We prove that the wLDOS is local in the sense that it can be computed up to arbitrarily small error using spatial truncations of the system Hamiltonian. Using this result we prove that the wLDOS is well-defined and computable for infinite systems satisfying some natural assumptions. We finally present numerical computations of the wLDOS at the edge and in the bulk of a “Fibonacci SSH model”, a one-dimensional non-periodic model with topological edge states.

The Fourier-cosine expansion (COS) method is used to price European options numerically in a very efficient way. To apply the COS method, one has to specify two parameters: a truncation range for the density of the log-returns and a number of terms N to approximate the truncated density by a cosine series. How to choose the truncation range is already known. Here, we are able to find an explicit and useful bound for N as well for pricing and for the sensitivities, i.e., the Greeks Delta and Gamma, provided the density of the log-returns is smooth. We further show that the COS method has an exponential order of convergence when the density is smooth and decays exponentially. However, when the density is smooth and has heavy tails, as in the Finite Moment Log Stable model, the COS method does not have exponential order of convergence. Numerical experiments confirm the theoretical results.