arXiv subjects

Mattia Tani

Publications and source records attributed to Mattia Tani.

18 recordsLinked to original sources

arXiv2025

The Isogeometric Fast Fourier-based Diagonalization method

The construction of robust solvers for linear systems obtained from the discretization of partial differential equations using Isogeometric Analysis is challenging since the condition number of the system matrix not only grows with the reciprocal square of the grid size (for second order problems), but also exponentially with the spline degree. The Fast Diagonalization method allows the construction of a preconditioner that is robust both in grid size and spline degree. Although this method is efficient in practice, its computational complexity is superlinear in the number of degrees of freedom. In this work, we construct a variant of the Fast Diagonalization method that can exploit the Fast Fourier Transformation. Note that, because of boundary effects, a Fourier Transformation cannot diagonalize the overall problem. We circumvent this issue using a stable splitting of the spline space. The resulting preconditioner is still robust with respect to the grid size and spline degree and can be realized with a computational complexity that grows almost linearly with the number of degrees of freedom.

math.NA↗

arXiv2025

Factorizations and fast diagonalization for the heat equation

This work investigates diagonalization-based methods for efficiently solving linear evolution problems, with a particular focus on the heat equation. The plain diagonalization of the differential operator, though effective for elliptic problems where fast diagonalization can be used, exhibits instability when applied to the heat equation. To address this difficulty, we examine three alternative approaches, based on LU factorization, a suitable arrowhead factorization, and a low-rank modification. These methods introduce more robust factorizations of the time derivative, ensuring both computational efficiency and stability.

math.NA↗

arXiv2024

A low-rank solver for conforming multipatch Isogeometric Analysis

In this paper, we propose an innovative isogeometric low-rank solver for the linear elasticity model problem, specifically designed to allow multipatch domains. Our approach splits the domain into subdomains, each formed by the union of neighboring patches. Within each subdomain, we employ Tucker low-rank matrices and vectors to approximate the system matrices and right-hand side vectors, respectively. This enables the construction of local approximate fast solvers. These local solvers are then combined into an overlapping Schwarz preconditioner, which is utilized in a truncated preconditioned conjugate gradient method. Numerical experiments demonstrate the significant memory storage benefits and a uniformly bounded number of iterations with respect to both mesh size and spline degree.

math.NA↗

arXiv2023

Space-time least squares approximation for Schrödinger equation and efficient solver

In this work we present a space-time least squares isogeometric discretization of the Schrödinger equation and propose a preconditioner for the arising linear system in the parametric domain. Exploiting the tensor product structure of the basis functions, the preconditioner is written as the sum of Kronecker products of matrices. Thanks to an extension to the classical Fast Diagonalization method, the application of the preconditioner is efficient and robust w.r.t. the polynomial degree of the spline space. The time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution.

math.NA↗

arXiv2023

A low-rank isogeometric solver based on Tucker tensors

We propose an isogeometric solver for Poisson problems that combines i)low-rank tensor techniques to approximate the unknown solution and the system matrix, as a sum of a few terms having Kronecker product structure, ii) a Truncated Preconditioned Conjugate Gradient solver to keep the rank of the iterates low, and iii) a novel low-rank preconditioner, based on the Fast Diagonalization method where the eigenvector multiplication is approximated by the Fast Fourier Transform. Although the proposed strategy is written in arbitrary dimension, we focus on the three-dimensional case and adopt the Tucker format for low-rank tensor representation, which is well suited in low dimension. We show by numerical tests that this choice guarantees significant memory saving compared to the full tensor representation. We also extend and test the proposed strategy to linear elasticity problems.

math.NA↗

arXiv2022

A IETI-DP method for discontinuous Galerkin discretizations in Isogeometric Analysis with inexact local solvers

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines.

math.NA↗

arXiv2021

Weighted quadrature for hierarchical B-splines

We present weighted quadrature for hierarchical B-splines to address the fast formation of system matrices arising from adaptive isogeometric Galerkin methods with suitably graded hierarchical meshes. By exploiting a local tensor-product structure, we extend the construction of weighted rules from the tensor-product to the hierarchical spline setting. The proposed algorithm has a computational cost proportional to the number of degrees of freedom and advantageous properties with increasing spline degree. To illustrate the performance of the method and confirm the theoretical estimates, a selection of 2D and 3D numerical tests is provided.

math.NA↗

arXiv2020

Easy and Efficient preconditioning of the Isogeometric Mass Matrix

This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix-vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evidence of its good behaviour with respect to the spline degree and the (possibly singular) geometry parametrization. We also extend the preconditioner to the multipatch case through an Additive Schwarz method.

math.NA↗

arXiv2020

An efficient solver for space-time isogeometric Galerkin methods for parabolic problems

In this work we focus on the preconditioning of a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By incorporating some information on the geometry parametrization and on the equation coefficients, we keep high efficiency with non-trivial domains and variable thermal conductivity and heat capacity coefficients.

math.NA↗

arXiv2020

Compressive Isogeometric Analysis

This work is motivated by the difficulty in assembling the Galerkin matrix when solving Partial Differential Equations (PDEs) with Isogeometric Analysis (IGA) using B-splines of moderate-to-high polynomial degree. To mitigate this problem, we propose a novel methodology named CossIGA (COmpreSSive IsoGeometric Analysis), which combines the IGA principle with CORSING, a recently introduced sparse recovery approach for PDEs based on compressive sensing. CossIGA assembles only a small portion of a suitable IGA Petrov-Galerkin discretization and is effective whenever the PDE solution is sufficiently sparse or compressible, i.e., when most of its coefficients are zero or negligible. The sparsity of the solution is promoted by employing a multilevel dictionary of B-splines as opposed to a basis. Thanks to sparsity and the fact that only a fraction of the full discretization matrix is assembled, the proposed technique has the potential to lead to significant computational savings. We show the effectiveness of CossIGA for the solution of the 2D and 3D Poisson equation over nontrivial geometries by means of an extensive numerical investigation.

math.NA↗

arXiv2020

A domain decomposition method for Isogeometric multi-patch problems with inexact local solvers

In Isogeometric Analysis, the computational domain is often described as multi-patch, where each patch is given by a tensor product spline/NURBS parametrization. In this work we propose a FETI-like solver where local inexact solvers exploit the tensor product structure at the patch level. To this purpose, we extend to the isogeometric framework the so-called All-Floating variant of FETI, that allows us to use the Fast Diagonalization method at the patch level. We construct then a preconditioner for the whole system and prove its robustness with respect to the local mesh-size $h$ and patch-size $H$ (i.e., we have scalability). Our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree $p$.

math.NA↗

arXiv2019

Space-time least-squares isogeometric method and efficient solver for parabolic problems

In this paper, we propose a space-time least-squares isogeometric method to solve parabolic evolution problems, well suited for high-degree smooth splines in the space-time domain. We focus on the linear solver and its computational efficiency: thanks to the proposed formulation and to the tensor-product construction of space-time splines, we can design a preconditioner whose application requires the solution of a Sylvester-like equation, which is performed efficiently by the fast diagonalization method. The preconditioner is robust w.r.t. spline degree and mesh size. The computational time required for its application, for a serial execution, is almost proportional to the number of degrees-of-freedom and independent of the polynomial degree. The proposed approach is also well-suited for parallelization.

math.NA↗

arXiv2017

Matrix-free weighted quadrature for a computationally efficient isogeometric $k$-method

The $k$-method is the isogeometric method based on splines (or NURBS, etc.) with maximum regularity. When implemented following the paradigms of classical finite element methods, the computational resources required by the $k-$method are prohibitive even for moderate degree. In order to address this issue, we propose a matrix-free strategy combined with weighted quadrature, which is an ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more important, greatly reduces memory consumption. Our strategy also requires an efficient preconditioner for the linear system iterative solver. In this work we deal with an elliptic model problem, and adopt a preconditioner based on the Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our numerical tests show that the isogeometric solver based on MF-WQ is faster than standard approaches (where the main cost is the matrix formation by standard Gaussian quadrature) even for low degree. But the main achievement is that, with MF-WQ, the $k$-method gets orders of magnitude faster by increasing the degree, given a target accuracy. Therefore, we are able to show the superiority, in terms of computational efficiency, of the high-degree $k$-method with respect to low-degree isogeometric discretizations. What we present here is applicable to more complex and realistic differential problems, but its effectiveness will depend on the preconditioner stage, which is as always problem-dependent. This situation is typical of modern high-order methods: the overall performance is mainly related to the quality of the preconditioner.

math.NA↗

arXiv2017

Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.

math.NA↗

arXiv2017

A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis

In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the recently-developed weighted quadrature for the Galerkin discretization. In this paper, we are interested in the solution of the linear systems arising from the discretization of the Poisson problem using one of these approaches. In [SIAM J. Sci. Comput. 38(6) (2016) pp. A3644--A3671], a well-established direct solver for linear systems with tensor structure was used as a preconditioner in the context of Galerkin isogeometric analysis, yielding promising results. In particular, this preconditioner is robust with respect to the mesh size $h$ and the spline degree $p$. In the present work, we discuss how a similar approach can applied to the considered nonsymmetric linear systems. The efficiency of the proposed preconditioning strategy is assessed with numerical experiments on two-dimensional and three-dimensional problems.

math.NA↗

arXiv2016

Fast formation of isogeometric Galerkin matrices by weighted quadrature

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for $p$ large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree $k$-refinement in isogeometric analysis.

math.NA↗

arXiv2016

Isogeometric preconditioners based on fast solvers for the Sylvester equation

We consider large linear systems arising from the isogeometric discretization of the Poisson problem on a single-patch domain. The numerical solution of such systems is considered a challenging task, particularly when the degree of the splines employed as basis functions is high. We consider a preconditioning strategy which is based on the solution of a Sylvester-like equation at each step of an iterative solver. We show that this strategy, which fully exploits the tensor structure that underlies isogeometric problems, is robust with respect to both mesh size and spline degree, although it may suffer from the presence of complicated geometry or coefficients. We consider two popular solvers for the Sylvester equation, a direct one and an iterative one, and we discuss in detail their implementation and efficiency for 2D and 3D problems on single-patch or conforming multi-patch NURBS geometries. Numerical experiments for problems with different domain geometries are presented, which demonstrate the potential of this approach.

math.NA↗

arXiv2014

Preconditioning of Active-Set Newton Methods for PDE-constrained Optimal Control Problems

We address the problem of preconditioning a sequence of saddle point linear systems arising in the solution of PDE-constrained optimal control problems via active-set Newton methods, with control and (regularized) state constraints. We present two new preconditioners based on a full block matrix factorization of the Schur complement of the Jacobian matrices, where the active-set blocks are merged into the constraint blocks. We discuss the robustness of the new preconditioners with respect to the parameters of the continuous and discrete problems. Numerical experiments on 3D problems are presented, including comparisons with existing approaches based on preconditioned conjugate gradients in a nonstandard inner product.

math.NA↗