arXiv subjects

Sergey Dolgov

Publications and source records attributed to Sergey Dolgov.

At least 37 records · Page 2Linked to original sources

arXiv2021

Deep composition of tensor-trains using squared inverse Rosenblatt transports

Characterising intractable high-dimensional random variables is one of the fundamental challenges in stochastic computation. The recent surge of transport maps offers a mathematical foundation and new insights for tackling this challenge by coupling intractable random variables with tractable reference random variables. This paper generalises the functional tensor-train approximation of the inverse Rosenblatt transport recently developed by Dolgov et al. (Stat Comput 30:603--625, 2020) to a wide class of high-dimensional non-negative functions, such as unnormalised probability density functions. First, we extend the inverse Rosenblatt transform to enable the transport to general reference measures other than the uniform measure. We develop an efficient procedure to compute this transport from a squared tensor-train decomposition which preserves the monotonicity. More crucially, we integrate the proposed order-preserving functional tensor-train transport into a nested variable transformation framework inspired by the layered structure of deep neural networks. The resulting deep inverse Rosenblatt transport significantly expands the capability of tensor approximations and transport maps to random variables with complicated nonlinear interactions and concentrated density functions. We demonstrate the efficiency of the proposed approach on a range of applications in statistical learning and uncertainty quantification, including parameter estimation for dynamical systems and inverse problems constrained by partial differential equations.

stat.ML↗

arXiv2021

Efficient Structure-preserving Support Tensor Train Machine

An increasing amount of collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever-present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data motivates the use of tailored low-rank tensor classification methods. In the presence of small amounts of training data, kernel methods offer an attractive choice as they provide the possibility for a nonlinear decision boundary. We develop the Tensor Train Multi-way Multi-level Kernel (TT-MMK), which combines the simplicity of the Canonical Polyadic decomposition, the classification power of the Dual Structure-preserving Support Vector Machine, and the reliability of the Tensor Train (TT) approximation. We show by experiments that the TT-MMK method is usually more reliable computationally, less sensitive to tuning parameters, and gives higher prediction accuracy in the SVM classification when benchmarked against other state-of-the-art techniques.

cs.LG↗

arXiv2021

Tensor Decomposition Methods for High-dimensional Hamilton-Jacobi-Bellman Equations

A tensor decomposition approach for the solution of high-dimensional, fully nonlinear Hamilton-Jacobi-Bellman equations arising in optimal feedback control of nonlinear dynamics is presented. The method combines a tensor train approximation for the value function together with a Newton-like iterative method for the solution of the resulting nonlinear system. The tensor approximation leads to a polynomial scaling with respect to the dimension, partially circumventing the curse of dimensionality. A convergence analysis for the linear-quadratic case is presented. For nonlinear dynamics, the effectiveness of the high-dimensional control synthesis method is assessed in the optimal feedback stabilization of the Allen-Cahn and Fokker-Planck equations with a hundred of variables.

math.OC↗

arXiv2020

Rank Bounds for Approximating Gaussian Densities in the Tensor-Train Format

Low-rank tensor approximations have shown great potential for uncertainty quantification in high dimensions, for example, to build surrogate models that can be used to speed up large-scale inference problems (Eigel et al., Inverse Problems 34, 2018; Dolgov et al., Statistics & Computing 30, 2020). The feasibility and efficiency of such approaches depends critically on the rank that is necessary to represent or approximate the underlying distribution. In this paper, a-priori rank bounds for approximations in the functional tensor-train representation for the case of Gaussian models are developed. It is shown that under suitable conditions on the precision matrix, the Gaussian density can be approximated to high accuracy without suffering from an exponential growth of complexity as the dimension increases. These results provide a rigorous justification of the suitability and the limitations of low-rank tensor methods in a simple but important model case. Numerical experiments confirm that the rank bounds capture the qualitative behavior of the rank structure when varying the parameters of the precision matrix and the accuracy of the approximation. Finally, the practical relevance of the theoretical results is demonstrated in the context of a Bayesian filtering problem.

math.NA↗

arXiv2020

Parallel time-dependent variational principle algorithm for matrix product states

Combining the time-dependent variational principle (TDVP) algorithm with the parallelization scheme introduced by Stoudenmire and White for the density matrix renormalization group (DMRG), we present the first parallel matrix product state (MPS) algorithm capable of time evolving one-dimensional (1D) quantum lattice systems with long-range interactions. We benchmark the accuracy and performance of the algorithm by simulating quenches in the long-range Ising and XY models. We show that our code scales well up to 32 processes, with parallel efficiencies as high as 86%. Finally, we calculate the dynamical correlation function of a 201-site Heisenberg XXX spin chain with $1/r^2$ interactions, which is challenging to compute sequentially. These results pave the way for the application of tensor networks to increasingly complex many-body systems.

quant-ph↗

arXiv2020

Guaranteed a posteriori error bounds for low rank tensor approximate solutions

We propose a guaranteed and fully computable upper bound on the energy norm of the error in low-rank Tensor Train (TT) approximate solutions of (possibly) high dimensional reaction-diffusion problems. The error bound is obtained from Euler-Lagrange equations for a complementary flux reconstruction problem, which are solved in the low-rank TT representation using the block Alternating Linear Scheme. This bound is guaranteed to be above the energy norm of the total error, including the discretization error, the tensor approximation error, and the error in the solver of linear algebraic equations, although quadrature errors, in general, can pollute its evaluation. Numerical examples with the Poisson equation and the Schroedinger equation with the Henon-Heiles potential in up to 40 dimensions are presented to illustrate the efficiency of this approach.

math.NA↗

arXiv2019

Solving differential Riccati equations: A nonlinear space-time method using tensor trains

Differential algebraic Riccati equations are at the heart of many applications in control theory. They are time-depent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton-Kleinman iteration. Approximating the space-time problem in low-rank form requires fewer applications of the discretized differential operator and gives a low-rank approximation to the overall solution.

math.NA↗

arXiv2019

Greedy low-rank algorithm for spatial connectome regression

Recovering brain connectivity from tract tracing data is an important computational problem in the neurosciences. Mesoscopic connectome reconstruction was previously formulated as a structured matrix regression problem (Harris et al., 2016), but existing techniques do not scale to the whole-brain setting. The corresponding matrix equation is challenging to solve due to large scale, ill-conditioning, and a general form that lacks a convergent splitting. We propose a greedy low-rank algorithm for connectome reconstruction problem in very high dimensions. The algorithm approximates the solution by a sequence of rank-one updates which exploit the sparse and positive definite problem structure. This algorithm was described previously (Kressner and Sirković, 2015) but never implemented for this connectome problem, leading to a number of challenges. We have had to design judicious stopping criteria and employ efficient solvers for the three main sub-problems of the algorithm, including an efficient GPU implementation that alleviates the main bottleneck for large datasets. The performance of the method is evaluated on three examples: an artificial "toy" dataset and two whole-cortex instances using data from the Allen Mouse Brain Connectivity Atlas. We find that the method is significantly faster than previous methods and that moderate ranks offer good approximation. This speedup allows for the estimation of increasingly large-scale connectomes across taxa as these data become available from tracing experiments. The data and code are available online.

math.NA↗

arXiv2019

Approximation and sampling of multivariate probability distributions in the tensor train decomposition

General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions that is based on low-rank surrogates in the tensor-train format. We construct a tensor-train approximation to the target probability density function using the cross interpolation, which requires a small number of function evaluations. For sufficiently smooth distributions the storage required for the TT approximation is moderate, scaling linearly with dimension. The structure of the tensor-train surrogate allows efficient sampling by the conditional distribution method. Unbiased estimates may be calculated by correcting the transformed random seeds using a Metropolis--Hastings accept/reject step. Moreover, one can use a more efficient quasi-Monte Carlo quadrature that may be corrected either by a control-variate strategy, or by importance weighting. We show that the error in the tensor-train approximation propagates linearly into the Metropolis--Hastings rejection rate and the integrated autocorrelation time of the resulting Markov chain. These methods are demonstrated in three computed examples: fitting failure time of shock absorbers; a PDE-constrained inverse diffusion problem; and sampling from the Rosenbrock distribution. The delayed rejection adaptive Metropolis (DRAM) algorithm is used as a benchmark. We find that the importance-weight corrected quasi-Monte Carlo quadrature performs best in all computed examples, and is orders-of-magnitude more efficient than DRAM across a wide range of approximation accuracies and sample sizes. Indeed, all the methods developed here significantly outperform DRAM in all computed examples.

math.NA↗

arXiv2019

Kriging in Tensor Train data format

Combination of low-tensor rank techniques and the Fast Fourier transform (FFT) based methods had turned out to be prominent in accelerating various statistical operations such as Kriging, computing conditional covariance, geostatistical optimal design, and others. However, the approximation of a full tensor by its low-rank format can be computationally formidable. In this work, we incorporate the robust Tensor Train (TT) approximation of covariance matrices and the efficient TT-Cross algorithm into the FFT-based Kriging. It is shown that here the computational complexity of Kriging is reduced to $\mathcal{O}(d r^3 n)$, where $n$ is the mode size of the estimation grid, $d$ is the number of variables (the dimension), and $r$ is the rank of the TT approximation of the covariance matrix. For many popular covariance functions the TT rank $r$ remains stable for increasing $n$ and $d$. The advantages of this approach against those using plain FFT are demonstrated in synthetic and real data examples.

stat.CO↗

arXiv2019

Parallel cross interpolation for high-precision calculation of high-dimensional integrals

We propose a parallel version of the cross interpolation algorithm and apply it to calculate high-dimensional integrals motivated by Ising model in quantum physics. In contrast to mainstream approaches, such as Monte Carlo and quasi Monte Carlo, the samples calculated by our algorithm are neither random nor form a regular lattice. Instead we calculate the given function along individual dimensions (modes) and use this data to reconstruct its behaviour in the whole domain. The positions of the calculated univariate fibers are chosen adaptively for the given function. The required evaluations can be executed in parallel both along each mode (variable) and over all modes. To demonstrate the efficiency of the proposed method, we apply it to compute high-dimensional Ising susceptibility integrals, arising from asymptotic expansions for the spontaneous magnetisation in two-dimensional Ising model of ferromagnetism. We observe strong superlinear convergence of the proposed method, while the MC and qMC algorithms converge sublinearly. Using multiple precision arithmetic, we also observed exponential convergence of the proposed algorithm. Combining high-order convergence, almost perfect scalability up to hundreds of processes, and the same flexibility as MC and qMC, the proposed algorithm can be a new method of choice for problems involving high-dimensional integration, e.g. in statistics, probability, and quantum physics.

math.NA↗

arXiv2019

Tensor product approach to quantum control

In this proof-of-concept paper we show that tensor product approach is efficient for control of large quantum systems, such as Heisenberg spin wires, which are essential for emerging quantum computing technologies. We compute optimal control sequences using GRAPE method, applying the recently developed tAMEn algorithm to calculate evolution of quantum states represented in the tensor train format to reduce storage. Using tensor product algorithms we can overcome the curse of dimensionality and compute the optimal control pulse for a 41 spin system on a single workstation with fully controlled accuracy and huge savings of computational time and memory. The use of tensor product algorithms opens new approaches for development of quantum computers with 50 to 100 qubits.

math.NA↗

arXiv2018

A low-rank tensor method for PDE-constrained optimization with isogeometric analysis

Isogeometric analysis (IGA) has become one of the most popular methods for the discretization of partial differential equations motivated by the use of NURBS for geometric representations in industry and science. A crucial challenge lies in the solution of the discretized equations, which we discuss in this talk with a particular focus on PDE-constrained optimization discretized using IGA. The discretization results in a system of large mass and stiffness matrices, which are typically very costly to assemble. To reduce the computation time and storage requirements, low-rank tensor methods have become a promising tool. We present a framework for the assembly of these matrices in low-rank form as the sum of a small number of Kronecker products. For assembly of the smaller matrices only univariate integration is required. The resulting low rank Kronecker product structure of the mass and stiffness matrices can be used to solve a PDE-constrained optimization problem without assembling the actual system matrices. We present a framework which preserves and exploits the low-rank Kronecker product format for both the matrices and the solution. We use the block AMEn method to efficiently solve the corresponding KKT system of the optimization problem. We show several numerical experiments with 3D geometries to demonstrate that the low-rank assembly and solution drastically reduces the memory demands and computing times, depending on the approximation ranks of the domain.

math.NA↗

arXiv2018

A hybrid Alternating Least Squares -- TT Cross algorithm for parametric PDEs

We consider the approximate solution of parametric PDEs using the low-rank Tensor Train (TT) decomposition. Such parametric PDEs arise for example in uncertainty quantification problems in engineering applications. We propose an algorithm that is a hybrid of the alternating least squares and the TT cross methods. It computes a TT approximation of the whole solution, which is beneficial when multiple quantities of interest are sought. This might be needed, for example, for the computation of the probability density function (PDF) via the maximum entropy method [Kavehrad and Joseph, IEEE Trans. Comm., 1986]. The new algorithm exploits and preserves the block diagonal structure of the discretized operator in stochastic collocation schemes. This disentangles computations of the spatial and parametric degrees of freedom in the TT representation. In particular, it only requires solving independent PDEs at a few parameter values, thus allowing the use of existing high performance PDE solvers. In our numerical experiments, we apply the new algorithm to the stochastic diffusion equation and compare it with preconditioned steepest descent in the TT format, as well as with (multilevel) quasi-Monte Carlo and dimension-adaptive sparse grids methods. For sufficiently smooth random fields the new approach is orders of magnitude faster.

math.NA↗

arXiv2018

Preconditioners and Tensor Product Solvers for Optimal Control Problems from Chemotaxis

In this paper, we consider the fast numerical solution of an optimal control formulation of the Keller--Segel model for bacterial chemotaxis. Upon discretization, this problem requires the solution of huge-scale saddle point systems to guarantee accurate solutions. We consider the derivation of effective preconditioners for these matrix systems, which may be embedded within suitable iterative methods to accelerate their convergence. We also construct low-rank tensor-train techniques which enable us to present efficient and feasible algorithms for problems that are finely discretized in the space and time variables. Numerical results demonstrate that the number of preconditioned GMRES iterations depends mildly on the model parameters. Moreover, the low-rank solver makes the computing time and memory costs sublinear in the original problem size.

math.NA↗

arXiv2017

Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods

Many problems in computational science and engineering are simultaneously characterized by the following challenging issues: uncertainty, nonlinearity, nonstationarity and high dimensionality. Existing numerical techniques for such models would typically require considerable computational and storage resources. This is the case, for instance, for an optimization problem governed by time-dependent Navier-Stokes equations with uncertain inputs. In particular, the stochastic Galerkin finite element method often leads to a prohibitively high dimensional saddle-point system with tensor product structure. In this paper, we approximate the solution by the low-rank Tensor Train decomposition, and present a numerically efficient algorithm to solve the optimality equations directly in the low-rank representation. We show that the solution of the vorticity minimization problem with a distributed control admits a representation with ranks that depend modestly on model and discretization parameters even for high Reynolds numbers. For lower Reynolds numbers this is also the case for a boundary control. This opens the way for a reduced-order modeling of the stochastic optimal flow control with a moderate cost at all stages.

math.NA↗

arXiv2016

Fast iterative solution of the Bethe-Salpeter eigenvalue problem using low-rank and QTT tensor approximation

In this paper, we study and implement the structural iterative eigensolvers for the large-scale eigenvalue problem in the Bethe-Salpeter equation (BSE) based on the reduced basis approach via low-rank factorizations in generating matrices, introduced in the previous paper. The approach reduces numerical costs down to $\mathcal{O}(N_b^2)$ in the size of atomic orbitals basis set, $N_b$, instead of practically intractable $\mathcal{O}(N_b^6)$ complexity scaling for the direct diagonalization of the BSE matrix. As an alternative to rank approximation of the static screen interaction part of the BSE matrix, we propose to restrict it to a small active sub-block, with a size balancing the storage for rank-structured representations of other matrix blocks. We demonstrate that the enhanced reduced-block approximation exhibits higher precision within the controlled numerical cost, providing as well a distinct two-sided error estimate for the BSE eigenvalues. It is shown that further reduction of the asymptotic computational cost is possible due to ALS-type iteration in block tensor train (TT) format applied to the quantized-TT (QTT) tensor representation of both long eigenvectors and rank-structured matrix blocks. The QTT-rank of these entities possesses almost the same magnitude as the number of occupied orbitals in the molecular systems, $N_o$, hence the overall asymptotic complexity for solving the BSE problem can be estimated by $\mathcal{O}(\log(N_o) N_o^{2})$. We confirm numerically a considerable decrease in computational time for the presented iterative approach applied to various compact and chain-type molecules, while supporting sufficient accuracy.

math.NA↗

arXiv2015

Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format

We apply the Tensor Train (TT) decomposition to construct the tensor product Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic elliptic diffusion PDE with the stochastic Galerkin discretization, and to compute some quantities of interest (mean, variance, exceedance probabilities). We assume that the random diffusion coefficient is given as a smooth transformation of a Gaussian random field. In this case, the PCE is delivered by a complicated formula, which lacks an analytic TT representation. To construct its TT approximation numerically, we develop the new block TT cross algorithm, a method that computes the whole TT decomposition from a few evaluations of the PCE formula. The new method is conceptually similar to the adaptive cross approximation in the TT format, but is more efficient when several tensors must be stored in the same TT representation, which is the case for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin matrix and to compute the solution of the elliptic equation and its post-processing, staying in the TT format. We compare our technique with the traditional sparse polynomial chaos and the Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial degree is bounded for each random variable independently. This provides higher accuracy than the sparse polynomial set or the Monte Carlo method, but the cardinality of the tensor product set grows exponentially with the number of random variables. However, when the PCE coefficients are implicitly approximated in the TT format, the computations with the full tensor product polynomial set become possible. In the numerical experiments, we confirm that the new methodology is competitive in a wide range of parameters, especially where high accuracy and high polynomial degrees are required.

math.NA↗