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In this paper, we consider the Poisson equation on a “long” domain which is the Cartesian product of a one-dimensional long interval with a (d − 1)-dimensional domain. The right-hand side is assumed to have a rank-1 tensor structure. We will present and compare methods to construct approximations of the solution which have tensor structure and the computational effort is governed by only solving elliptic problems on lower-dimensional domains. A zero-th order tensor approximation is derived by using tools from asymptotic analysis (method 1). The resulting approximation is an elementary tensor and, hence has a fixed error which turns out to be very close to the best possible approximation of zero-th order. This approximation can be used as a starting guess for the derivation of higher-order tensor approximations by a greedy-type method (method 2). Numerical experiments show that this method is converging towards the exact solution. Method 3 is based on the derivation of a tensor approximation via exponential sums applied to discretized differential operators and their inverses. It can be proved that this method converges exponentially with respect to the tensor rank. We present numerical experiments which compare the performance and sensitivity of these three methods.

A tensor v is the sum of at least rank⁡(v) elementary tensors. In addition, a ‘border rank’ is defined: rank_(w)=r holds if r is the minimum integer such that w is a limit of rank-r tensors. Usuall...

Various tensor formats exist which allow a data-sparse representation of tensors. Some of these formats are not closed. The consequences are (i) possible non-existence of best approximations and (ii) divergence of the representing parameters when a tensor within the format tends to a border tensor outside. The paper tries to describe the nature of this divergence. A particular question is whether the divergence is uniform for all border tensors.

The approximation of the function 1 / x by exponential sums has several interesting applications. It is well known that best approximations with respect to the maximum norm exist. Moreover, the error estimates exhibit exponential decay as the number of terms increases. Here we focus on the computation of the best approximations. In principle, the problem can be solved by the Remez algorithm, however, because of the very sensitive behaviour of the problem the standard approach fails for a larger number of terms. The remedy described in the paper is the use of other independent variables of the exponential sum. We discuss the approximation error of the computed exponential sums up to 63 terms and hint to a webpage containing the corresponding coefficients.

Tensors are in general large-scale data which require a special representation. These representations are also called a format. After mentioning the r-term and tensor subspace formats, we describe the hierarchical tensor format which is the most flexible one. Since operations with tensors often produce tensors of larger memory cost, truncation to reduced ranks is of utmost importance. The so-called higher-order singular-value decomposition (HOSVD) provides a save truncation with explicit error control. The paper explains in detail how the HOSVD procedure is performed within the hierarchical tensor format. Finally, we state special favourable properties of the HOSVD truncation.

A modification of standard linear iterative methods for the solution of linear equations is investigated aiming at improved data-sparsity with respect to a rank function. The convergence speed of the modified method is compared to the rank growth of its iterates for certain model cases. The considered general setup is common in the data-sparse treatment of high dimensional problems such as sparse approximation and low rank tensor calculus.

In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format., Comment: Version Improved

In order to treat high-dimensional problems, one has to find data-sparse representations. Starting with a six-dimensional problem, we first introduce the low-rank approximation of matrices. One purpose is the reduction of memory requirements, another advantage is that now vector operations instead of matrix operations can be applied. In the considered problem, the vectors correspond to grid functions defined on a three-dimensional grid. This leads to the next separation: these grid functions are tensors in $\mathbb {R}^{n}\otimes \mathbb {R}^{n}\otimes \mathbb {R}^{n}$ and can be represented by the hierarchical tensor format. Typical operations as the Hadamard product and the convolution are now reduced to operations between $\mathbb {R}^{n}$ vectors. Standard algorithms for operations with vectors from $\mathbb {R}^{n}$ are of order $\mathcal {O}(n)$ or larger. The tensorisation method is a representation method introducing additional data-sparsity. In many cases, the data size can be reduced from $\mathcal {O}(n)$ to $\mathcal {O}(\log n)$ . Even more important, operations as the convolution can be performed with a cost corresponding to these data sizes.

The main goal of this paper is to extend the so-called Dirac-Frenkel Variational Principle in the framework of tensor Banach spaces. To this end we observe that a tensor product of normed spaces can be described as a union of disjoint connected components. Then we show that each of these connected components, composed by tensors in Tucker format with a fixed rank, is a Banach manifold modelled in a particular Banach space, for which we provide local charts. The description of the local charts of these manifolds is crucial for an algorithmic treatment of high-dimensional partial differential equations and minimization problems. In order to describe the relationship between these manifolds and the natural ambient space we prove under natural conditions that each connected component can be immersed in a particular ambient Banach space. This fact allows us to finally extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces., Comment: arXiv admin note: substantial text overlap with arXiv:1505.03027

The term ‘matrix-product state’ (MPS) is introduced in quantum physics (see, e.g., Verstraete-Cirac [190], [105, Eq. (2)]). The related tensor representation can be found already in Vidal [191] without a special naming of the representation. The method has been reinvented by Oseledets and Tyrtyshnikov ([152], [155], [159]) and called ‘TT decomposition’.

The discussion of topological tensor spaces has been started by Schatten [167] and Grothendieck [79, 80]. In Sect. 4.2 we discuss the question how the norms of V and W are related to the norm of \(V \otimes W.\)

Since tensor spaces are in particular vector spaces, we start in Sect. 3.1 with vector spaces. Here, we introduce the free vector space (§3.1.2) and the quotient vector space (§3.1.3) which are needed later. Furthermore, the spaces of linear mappings and dual mappings are discussed in §3.1.4. The core of this chapter is Sect. 3.2 containing the definition of the tensor space. Section 3.3 is devoted to linear and multilinear mappings as well as to tensor spaces of linear mappings. Algebra structures are discussed in Sect. 3.4. Finally, symmetric and antisymmetric tensors are defined in Sect. 3.5.

The exact representation of \({\rm v} \in {\rm V} = {\bigotimes}_{j=1}^{d} V_{j}\) by a tensor subspace representation (8.6b) may be too expensive because of the high dimensions of the involved subspaces or even impossible since v is a topological tensor admitting no finite representation. In such cases we must be satisfied with an approximation u ≈ v which is easier to handle. We require that \(\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}}\), i.e., there are bases \(\{{b}_{1}^{(j)},\ldots,{b}_{{r}_{j}}^{(j)}\} \subset {V }_{j}\) such that $$\mathbf{u} ={ \sum \nolimits }_{{i}_{1}=1}^{{r}_{1} }\cdots {\sum \nolimits }_{{i}_{d}=1}^{{r}_{d} }\mathbf{a}[{i}_{1}\cdots {i}_{d}]{\bigotimes}_{j=1}^{d}{b}_{{ i}_{j}}^{(j)}.$$ (1) The basic task of this chapter is the following problem: $$\text{ Given }\mathbf{v} \in \mathbf{V}\text{, find a suitable approximation }\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}} \subset \mathbf{V,}$$ (2) where \(\mathbf{r} = \left ({r}_{1},\ldots,{r}_{d}\right ) \in {\mathbb{N}}^{d}.\) Finding \(\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}}\) means finding coefficients \(\mathbf{a}[{i}_{1}\cdots {i}_{d}]\) as well as basis vectors b i (j) ∈ V j in (10.1). Problem (10.2) is formulated rather vaguely. If an accuracy e > 0 is prescribed, r ∈ ℕ d as well as \(\mathbf{u} \in {\mathcal{T}}_{\mathbf{r}}\) are to be determined. The strict minimisation of \(\left \Vert \mathbf{v} -\mathbf{u}\right \Vert\) is often replaced by an appropriate approximation \(\mathbf{u}\) requiring low computational cost. Instead of e > 0, we may prescribe the rank vector r ∈ ℕ d in (10.2).Optimal approximations (so-called ‘best approximations’) will be studied in Sect. 10.2. While best approximations require an iterative computation, quasi-optimal approximations can be determined explicitly using the HOSVD basis introduced in Sect. 8.3. The latter approach is explained in Sect. 10.1.

We consider elliptic partial differential equations in d variables and their discretisation in a product grid \(\mathbf{I} = \times^{d}_{j=1}I_{j}\). The solution of the discrete system is a grid function, which can directly be viewed as a tensor in \(\mathbf{V} = {\bigotimes}^{d}_{j=1}\mathbb{K}^{I_{j}}\). In Sect. 16.1 we compare the standard strategy of local refinement with the tensor approach involving regular grids. It turns out that the tensor approach can be more efficient. In Sect. 16.2 the solution of boundary value problems is discussed. A related problem is the eigenvalue problem discussed in Sect. 16.3.

We use the term ‘tensor subspace’ for the tensor product \(\mathbf{U}:=a{\bigotimes}^{d}_{j=1}\,\,\rm U_{j}\) of subspaces \(\rm U_{j}{\subset}\rm V_{j}\). Obviously, U is a subspace of \(\mathbf{V}:=a{\bigotimes}^{d}_{j=1}\,\,\rm V_{j}\), but not any subspace of V is a tensor subspace. For d = 2, r-term and tensor subspace representations (also called Tucker representation) are identical. Therefore, both approaches can be viewed as extensions of the concept of rank-r matrices to the tensor case d ≥ 3. The resulting set \(\mathcal{T}_{{\rm r}}\) introduced in Sect. 8.1 will be characterised by a vector-valued rank r = (r 1, …, r d). Since by definition, tensors \({\rm v} \in \mathcal{T}_{{\rm r}}\) are closely related to subspaces, their descriptions by means of frames or bases is of interest (see Sect. 8.2). Differently from the r-term format, algebraic tools like the singular value decomposition can be applied and lead to a higher order singular value decomposition (HOSVD), which is a quite important feature of the tensor subspace representation (cf. Sect. 8.3). Moreover, HOSVD yields a connection to the minimal subspaces from Chap. 6. In Sect. 8.5 we compare the formats discussed so far and describe conversions between the formats. In a natural way, a hybrid format appears using the r-term format for the coefficient tensor of the tensor subspace representation (cf. §8.2.4). Section 8.6 deals with the problem of joining two representation systems, as it is needed when we add two tensors involving different tensor subspaces.

The computation of a six-dimensional density matrix is the crucial step for the evaluation of kinetic energy in electronic structure calculations. For molecules with heavy nuclei, one has to consider a very refined mesh in order to deal with the nuclear cusps. This leads to high computational time and needs huge memory for the computation of the density matrix. To reduce the computational complexity and avoid discretization errors in the approximation, we use mesh-free canonical tensor products in electronic structure calculations. In this paper, we approximate the six-dimensional density matrix in an efficient way and then compute the kinetic energy. Accuracy is examined by comparing our computed kinetic energy with the exact computation of the kinetic energy.

Scientific computations or measurements may result in huge volumes of high-dimensional data, for instance 1020 or 100300 elements. Often these can be thought of representing a real-valued function on a high-dimensional domain. In this and also in most other cases the data can be conceptually arranged in the format of a tensor of high degree, and stored in some truncated or lossy compressed format. We look at some common post-processing tasks which are too time and storage consuming in the uncompressed data format and not obvious in the compressed format, as such huge data sets can not be stored in their entirety, and the value of an element is not readily accessible through simple look-up. The tasks we consider are finding the location of maximum or minimum, or finding all elements in some interval — i.e. level sets, or the number of elements with a value in such a level set, the probability of an element being in a particular level set, and the mean and variance of the total collection. The algorithms to be described are fixed point iterations of particular point-wise functions of the data, which will then exhibit the desired result. To formulate these algorithms, the data is considered as an element of a commutative algebra, and in an abstract sense, the algorithms, which only use these algebraic operations, are independent of the representation. We allow the actual computational representation to be a lossy compression, and we allow the algebra operations to be performed in an approximate fashion, so as to maintain a high compression level. One such example format which is addressed explicitly and described in some detail is the representation of the data as a tensor with compression in the form of a low-rank representation.

The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for obtaining a representation of tensors with either bounded or fixed tree-based rank in the underlying algebraic tensor space. We provide a new characterisation of minimal subspaces which extends the existing characterisations. We also introduce a definition of topological tensor spaces in tree-based format, with the introduction of a norm at each vertex of the tree, and prove the existence of best approximations from sets of tensors with bounded tree-based rank, under some assumptions on the norms weaker than in the existing results., arXiv admin note: substantial text overlap with arXiv:1505.03027 Some misprints are corrected

Various tensor formats are used for the data-sparse representation of large-scale tensors. Here we investigate how symmetric or antisymmetric tensors can be represented. We mainly investigate the hierarchical format, but also the use of the canonical format is mentioned.

Usually, one avoids numerical algorithms involving operations with large, fully populated matrices. Instead, one tries to reduce all algorithms to matrix-vector multiplications involving only sparse matrices. The reason is the large number of floating point operations; e.g., $\mathcal {O}(n^{3})$ for the multiplication of two general n × n matrices. The hierarchical matrix ( $\mathcal {H}$ -matrix) technique provides tools to perform the matrix operations approximately in almost linear work $\mathcal {O}(n\log ^{\ast }n)$ . The approximation errors are nevertheless acceptable, since large-scale matrices are usually obtained from discretisations which anyway contain a discretisation error. Adjusting the approximation error to the discretisation error yields the factor $\mathcal {O}(\log ^{\ast }n).$ The operations enabled by the $\mathcal {H}$ -matrix technique are not only the matrix addition and multiplication but also the matrix inversion and the LU or Cholesky decomposition. The positive statements from above do not hold for all matrices, but they are valid for the important class of matrices originating from standard discretisations of elliptic partial differential equations or the related integral equations. An important aspect is the fact that the algorithms can be applied in a black-box fashion. Having all matrix operations available, a much larger class of problems can be treated than by the restriction to matrix-vector multiplications. The LU decomposition can be used to construct fast iterations for solving linear systems. Also eigenvalue problems can be treated. The computation of matrix-valued functions is possible (e.g., the matrix exponential function) as well as the solution of matrix equations (e.g., of the Riccati equation).

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead one may use a block decomposition. Approximating the small submatrices by low-rank matrices and agglomerating them into a new, coarser block decomposition, one obtains a recursive method. The required computation work is $$O(rnm)$$O(rnm) where $$r$$r is the desired rank and $$n\times m$$n×m is the size of the matrix. The paper discusses the new error estimates for $$A-B$$A-B and $$M-A$$M-A where $$A$$A is the result of the recursive truncation applied to $$M$$M, while $$B$$B is the best approximation.

Section 5.1 introduces the general elliptic linear differential equation of second order together with the Dirichlet boundary values. An important statement is the maximum-minimum principle in §5.1.2. In §5.1.3 sufficient conditions for the uniqueness of the solution and the continuous dependence on the data are proved. The discretisation of the general differential equation in a square is described in §5.1.4. Section 5.2 treats alternative boundary conditions replacing the Dirichlet data. Examples are the Neumann condition, the conormal derivative and the Robin boundary condition. Their discretisation (cf. §5.2.2) for general domains is rather laborious. Section 5.3 discusses differential equations of higher order. In particular, the biharmonic equation of fourth order is described in §5.3.1 followed by equations of order 2m in §5.3.2. The discretisation of the biharmonic equation is in §5.3.3.

In Section 2.1 the simplest but prototypical elliptic differential equation of second order is presented. The solutions of this equation are called harmonic. Together with a boundary condition, one obtains a boundary-value problem. An important tool is the singularity function, which is defined in Section 2.2. The Green formulae allow a representation of the solution in Theorem 2.8. In Section 2.3 functions with mean-value property are introduced. It is shown that these functions coincide with harmonic functions. The mean-value property implies the maximum minimum principle: non-constant functions have no local extrema. An important conclusion is the uniqueness of the solution (Theorem 2.18). Finally, in Section 2.4, it is shown that the solution depends continuously on the boundary data.

The difference method replaces the derivatives by difference quotients. Section 4.1 describes the difference method applied to the one-dimensional Poisson equation \(-u^{\prime\prime}\;=\;f\). This simple example is chosen to show the generation of the discrete system of equations. The difference equations are complemented by the Dirichlet boundary condition. The equations of the resulting linear system correspond to the inner grid points, while the boundary data appear in the right– hand side of the system.

Here we collect those definitions and statements which are needed in the next chapters. Section 6.1 introduces the normed spaces, Banach and Hilbert spaces as well as the operators as linear and bounded mappings between these spaces. In most of the later applications these spaces will be function spaces, containing for instance the solutions of the differential equations. It will turn out that the Sobolev spaces from Section 6.2 are well suited for the solutions of boundary value problems. The Sobolev space \(H^{k}(\it\Omega)\) and \(H^{k}_{0}(\it\Omega)\) for nonnegative integers k as well as \(H^{s}(\it\Omega)\) for real \(s\geq 0\) are introduced. The definition of the trace (restriction to the boundary Γ) will be essential in §6.2.5 for the interpretation of boundary values. To this end the Sobolev spaces \(H^{s}(\it\Gamma)\) of functions on the boundary Γ must be defined (cf. Theorem 6.57). Sobolev’s Embedding Theorem 6.48 connects Sobolev spaces and classical spaces. Section 6.3 introduces dual spaces and dual mappings. Compactness properties are important for statements about the unique solvability. Compact operators and the Riesz–Schauder theory are presented in Section 6.4. The weak formulation \(H^{s}(\it\Gamma)\) of the boundary-value problem is based on bilinear forms described in Section 6.5. The inf-sup condition in Lemma 6.94 is a necessary and sufficient criterion for the solvability of the weak formulation.

In Chapter 7 the variational formulation has been introduced to prove the existence of a (weak) solution. Now it will turn out that the variational formulation is extremely important for numerical purposes. It establishes a new, very flexible discretisation method. After historical remarks in Section 8.1 we introduce the Ritz–Galerkin method in Section 8.2. The basic principle is the replacement of the function space V in the variational formulation by an N-dimensional space. This leads to a system of N linear equations (§8.2.1). As described in §8.2.2, the theory from Chapter 7 can be applied. In §8.2.3 two criteria, the inf-sup condition and V-ellipticity are described which are sufficient for solvability. §8.2.4 contains numerical examples. Error estimates are discussed in Section 8.3. The quasioptimality of the Ritz–Galerkin method proved in §8.3.1 shifts the discussion to the approximation properties of the subspace (§8.3.2). The finite elements introduced in Section 8.4 form a special finite-dimensional subspace offering many practical advantages. The corresponding error estimates are given in Section 8.5. Generalisations to differential equations of higher order and to non-polygonal domains are investigated in Section 8.6. An important practical subject are a-posteriori error estimates discussed in Section 8.7. When solving the arising system of linear equations, the properties of the system matrix is of interest which are investigated in Section 8.8. Several other topics are sketched in the final Section 8.9.

If \((L-\lambda I)u\;=\;f\) is not solvable, the Riesz–Schauder theory states that there are eigenvalues and eigenvectors. The weak formulation of the eigenvalue problem and some basic terms are discussed in Section 11.1. Since we do not require the system to be symmetric, also the adjoint problem must be treated. Section 11.2 is devoted to the finite-element discretisation by a family \(\{V_h\;:\;h \in\;H\}\) of subspaces. Theorems 11.13 and 11.15 state an important result: Each eigenvalue λ 0 of L is associated to a sequence of discrete eigenvalues converging to λ 0, and vice versa. The corresponding error estimates are given in §11.2.3 for the case of simple eigenvalues. A related estimate for the eigenfunctions is provided by Lemma 11.23. Finally, Theorem 11.24 presents an improved error estimate of the eigenvalues by means of the eigenfunctions. The Riesz–Schauder theory also states that the equation \((L-\lambda I)u\;=\;f\)> can even be solved for eigenvalues λ if f satisfies suitable side conditions. These equations are treated in §11.2.4. Section 11.3 discusses the discretisation by difference schemes. Also in this case similar results can be obtained.

A higher-order tensor allows several possible matricizations (reshapes into matrices). The simultaneous decay of singular values of such matricizations has crucial implications on the low-rank approximability of the tensor via higher-order singular value decomposition. It is therefore an interesting question which simultaneous properties the singular values of different tensor matricizations actually can have, but it has not received the deserved attention so far. In this paper, preliminary investigations in this direction are conducted. While it is clear that the singular values in different matricizations cannot be prescribed completely independent from each other, numerical experiments suggest that sufficiently small, but otherwise arbitrary perturbations preserve feasibility. An alternating projection heuristic is proposed for constructing tensors with prescribed singular values (assuming their feasibility). Regarding the related problem of characterising sets of tensors having the same singular values in specified matricizations, it is noted that orthogonal equivalence under multilinear matrix multiplication is a sufficient condition for two tensors to have the same singular values in all principal, Tucker-type matricizations, but, in contrast to the matrix case, not necessary. An explicit example of this phenomenon is given.

The higher-order singular values for a tensor of order $d$ are defined as the singular values of the $d$ different matricizations associated with the multilinear rank. When $d \ge 3$, the singular values are generally different for different matricizations but not completely independent. Characterizing the set of feasible singular values turns out to be difficult. In this work, we contribute to this question by investigating which first-order perturbations of the singular values for a given tensor are possible. We prove that, except for trivial restrictions, any perturbation of the singular values can be achieved for almost every tensor with identical mode sizes. This settles a conjecture from [W. Hackbusch and A. Uschmajew, Numer. Math., 135 (2017), pp. 875--894] for the case of identical mode sizes. Our theoretical results are used to develop and analyze a variant of the Newton method for constructing a tensor with specified higher-order singular values or, more generally, with specified Gramians for th...

The usual large-scale discretizations are applied to two or three spatial dimensions. The standard methods fail for higher dimensions because the data size increases exponentially with the dimension. In the case of a regular grid withngrid points per direction, a spatial dimensiondyieldsndgrid points. A grid function defined on such a grid is an example of a tensor of orderd. Here, suitable tensor formats help, since they try to approximate these huge objects by a much smaller number of parameters, which increases only linearly ind. In this way, data of sizend= 10001000can also be treated.This paper introduces the algebraic and analytical aspects of tensor spaces. The main part concerns the numerical representation of tensors and the numerical performance of tensor operations.

In this proof-of-principle study, we discuss the application of various tensor representation formats and their implications on memory requirements and computational effort for tensor manipulations as they occur in typical post-Hartree–Fock (post-HF) methods. A successive tensor decomposition/rank reduction scheme in the matrix product state (MPS) format for the two-electron integrals in the AO and MO bases and an estimate of the t 2 amplitudes as obtained from second-order many-body perturbation theory (MP2) are described. Furthermore, the AO–MO integral transformation, the calculation of the MP2 energy and the potential usage of tensors in low-rank MPS representation for the tensor contractions in coupled cluster theory are discussed in detail. We are able to show that the overall scaling of the memory requirements is reduced from the conventional N 4 scaling to approximately N 3 and the scaling of computational effort for tensor contractions in post-HF methods can be reduced to roughly N 4 while the de...

Tensor truncation techniques are based on singular value decompositions. Therefore, the direct error control is restricted to $$\ell ^{2}$$ or $$L^{2}$$ norms. On the other hand, one wants to approximate multivariate (grid) functions in appropriate tensor formats in order to perform cheap pointwise evaluations, which require $$\ell ^{\infty }$$ or $$L^{\infty }$$ error estimates. Due to the huge dimensions of the tensor spaces, a direct estimate of $$\left\| \cdot \right\| _{\infty }$$ by $$\left\| \cdot \right\| _{2}$$ is hopeless. In the paper we prove that, nevertheless, in cases where the function to be approximated is smooth, reasonable error estimates with respect to $$\left\| \cdot \right\| _{\infty }$$ can be derived from the Gagliardo---Nirenberg inequality because of the special nature of the singular value decomposition truncation.

In this article we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log-normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)