## 1 Introduction

In this work we apply an efficient Isogeometric Analysis (IgA) scheme [2, 8] to parabolic initial-boundary value problems, which are frequently used to describe time evolution phenomena in physics, medicine, and so on.

The standard approach for these problems is to discretize separately
in space and in time. For high dimensions, there are works which
propose tensor methods in order to tackle the, so-called, *curse of
dimensionality*, notably using the tensor-train format for a global space-time
approximation [4, 5]. Recently, it was proposed
in [13] to use IgA to
discretize a pure heat conduction problem simultaneously in both space
and time. In particular, the time variable was regarded as an extra
spatial variable, and the problem was lifted in one dimension
higher. Consequently, the space-time cylinder was parameterized by a
NURBS volume and high-order and highly smooth splines were used to
discretize the problem.

One issue in the efficiency of IgA is the increased cost of computations already for three-dimensional problems. When adding an extra dimension, the cost related to computing the discretized operator increases significantly [15]. Indeed, the dependence of the computational complexity with respect to the dimension is exponential [14]. In the recent work [17] a partial low-rank tensor decomposition was proposed for decoupling the integrals arising in isogeometric schemes, thereby accelerating their computation.

The present work combines [13] and [17]. In particular, the scheme used in [13] is applied to a general parabolic problem with varying diffusion coefficient.
We revisit the analysis in [13] for the case in question, and provide the corresponding discretization error estimate in the appropriate norm.
Moreover, the fully varying coefficient requires highly accurate numerical integration in

The IgA space-time scheme in [13] is based on the space-time variational formulation presented in the books [10, 11]. In these works, the authors proved the uniqueness of the corresponding weak solution. Working in a different direction, in [7] a time discontinuous Galerkin space-time IgA scheme has been analyzed for solving simple parabolic problems (i.e., without varying coefficients).

The rest of the paper is organized as follows. We start by introducing our model problem, deriving the discrete variational form and the error estimates for spline discretizations in Section 2. In Section 3 we focus on the efficient computation of the matrix expressing the discrete operator and we bound the computational complexity in terms of the number of degrees of freedom and the polynomial degree of the discretization. We provide numerical results and computation times in Section 4. We conclude the paper and provide some future research directions in Section 5.

## 2 The Model Problem

### 2.1 Preliminaries

Let Ω be a bounded Lipschitz domain in

For an integer

endowed with the norm

and also we introduce the subspace

According to the definition of *m* be positive integers.
For functions defined in the space-time cylinder *Q*, we define the Sobolev spaces

where

We equip the above spaces with the norms and seminorms

respectively.
In what follows, positive constants *c* and *C* appearing in inequalities are
generic constants which do not depend on the mesh-size *h*.
We write *c* and *C*.

### 2.2 The Model Parabolic Problem

In

as model problem, where

Using the standard procedure and integration by parts with respect to both *x* and *t*,
we can easily derive the following space-time variational formulation
of (2.1): Find

with the bilinear form

and the linear form

where note that the last integral in (2.3) is related to the initial conditions in (2.1).
For simplicity, we only consider homogeneous
Dirichlet boundary conditions on Σ. Also, in the rest of the paper, we will consider that

For our analysis, we make the following convenient assumption.

We assume that the solution *u* of (2.2) belongs to

### 2.3 B-Spline Spaces

In this subsection, we briefly present the B-spline spaces and the form of the B-spline parameterizations for the physical space-time patches (called also space-time subdomains). We refer to [2, 3], for a more detailed presentation.

We start by presenting the B-spline space for the univariate case. Let the integer *p* denote the
B-spline degree and the integer

of the unit interval

We assume that

Now, let us consider the unit cube *p* and

where

In the frame of IgA, the representation of any volumetric domain is defined by a B-spline basis, see (2.4), and the associated control points, see
[2].
Given the associated control points
*Q* is parameterized
by the mapping

where *Q*,
where the elements *Q* as

To keep notation simple, we denote the above space by

The mesh

Since the parameterization

The parameterization

We assume that

### 2.4 Discrete Variational Forms

We denote by *Q*; by applying integration by parts we obtain

Since

The space-time IgA method for (2.1) can be formulated as follows: Find

Note that, under Assumption 2.1 and the derivation of (2.6), we can conclude that
the solution *u* of (2.2) satisfies (2.7a).

#### 2.4.1 Discretization Error Analysis

For simplifying the presentation, we derive the analysis for the case where

Let Assumption 2.1 hold and let *V* be the space defined there.
We define the space

*The discrete bilinear form *

*where
*

Using Green’s formula

the fact that

The definition of

Now, since it holds that

*The discrete bilinear form *

*where the constant *

The proof can be given following the same steps as in [13, Lemma 4]. ∎

*Let *

*where the constants *

*u*and

*h*.

The proof can be given using the classical inverse and trace inequalities and using the quasi-interpolation estimates on B-spline spaces presented in [1]. See also [13, Lemma 6]. ∎

Now, we can give the main discretization error estimate in terms of the discrete norm

*Let *

*u*and

*h*such that

Using the properties of bilinear form *u*, we obtain

where we immediately get

Hence, applying the triangle inequality

## 3 Matrix Assembly and Decoupling

Let us assume that the spatial domain Ω is described as the image of a regular B-spline parameterization

We obtain a parameterization *Q*, see (2.5), by lifting *F* linearly, i.e.,

where instead of

In this section, we exploit the tensor product structure of the spline space to
vastly improve the computational complexity of computing the discrete bilinear form

### 3.1 Fast Assembly for Constant Diffusion Coefficients

In this case, we can exploit the tensor product structure of the B-spline basis and the
corresponding structure of the parameterization (3.1) directly,
in order to separate the integration in space and time.
For assembling the system matrix produced in (2.7), we need to compute the bilinear form *Q* to the parametric domain *d*-variate integral over the spatial parametric domain and a univariate integral over the unit interval.

Let

For the first term, the transformation of the integral yields

where *F*.
For the second term we arrive at

Transforming the third term leads to

Finally, the fourth integral becomes

We observe that these representations only consist of entries of the stiffness and mass matrices for space and time, as well as those of the matrix containing the mixed time derivatives.

The decomposition of the integrals implies that we can write the system matrix

where *d*-variate integrals given in
(3.2)–(3.5) and *Kronecker format*.
We define the Kronecker rank of a matrix to be the number of summands in the Kronecker format, that is,
in our case the Kronecker rank of *K* is 2.

### 3.2 Fast Assembly for Space-Time Dependent Diffusion

Next we consider the case where the diffusion coefficient ρ is a smooth function depending on
both *x* and *t*. Now,
the terms (3.4) and (3.5) no longer decompose directly as in the previous case. In order to decouple integration in space and time in this case, we use the partial tensor decomposition method presented in [17] to decompose the parametric diffusion coefficient *d*-variate and univariate functions by projecting into a spline space and computing the singular value decomposition of the coefficient tensor. This results in an approximation

where *d*-variate and univariate spline functions respectively and *R* is the smallest rank such that a given error tolerance is satisfied.

This decomposition of

We arrive at a Kronecker format representation of rank

where the matrices

If the diffusion coefficient is matrix-valued, let us say *R* in the Kronecker format is the total rank, i.e., the sum of the ranks of all components.

### 3.3 Computational Complexity

In the following complexity analysis, we assume that the degrees of freedom and polynomial degrees in each

In the proposed method we compute *d*-variate and univariate integrals by element-wise Gauss quadrature exploiting the decomposition of the integrals. The complexity of computing each matrix *K* by computing the Kronecker product (3.8) then costs *d*-variate quadrature or by the sum of Kronecker products. Since usually

## 4 Numerical Examples

In this section we perform our experiments on a single patch.
For a multi-patch domain one can apply continuous or discontinuous discretization techniques and ultimately treat the problem patch-wise. Hence the methods developed in this work can be applied for handling the resulting local problems.
Typically, the global system matrix for a multi-patch discretization has sparse block structure. Each block has either tensor-product structure similar to (3.8), or is a very sparse block coming from interface coupling, which can be done in different ways.
For instance, the isogeometric tearing and interconnecting (IETI) method from [9] can be applied if the multi-patch discretization is continuous.
Otherwise a discontinuous Galerkin approach can be used as is described in [7] for constant coefficient

### Example 1: $Q\subset {\mathbb{R}}^{2+1}$ .

For our first numerical example, the space-time domain *Q* is the
quarter annulus in space prolongated into the unit interval in time (see Figure 2).
We consider the function

to be the exact solution of the problem.
The right-hand-side *f*, the boundary data and the initial data are computed accordingly.
Besides the system matrix *K*, we also compute the load vector using a partial low-rank approximation
of the right-hand-side as presented in [17].
The parameter θ is set to 1.
The method was implemented using the G+Smo C++ library [12, 16].
The assembly was performed on a single 2GHz processor, using B-spline basis functions of degrees 3 and 4.

For solving the linear system, a parallel GMRES solver from Trilinos [6] is used
with tolerance set to

Table 1 shows the error convergence as well as the computation times for assembling the matrix in the case of a constant diffusion coefficient

Three-dimensional experiment on the example space-time domain (see Figure 2)with a constant diffusion coefficient.

h | #DOF | Error | Error rate | Assembly time | |

0.25 | 343 | 8.30949 | 0.0061 s | ||

0.125 | 1,331 | 5.09944 | 0.70 | 0.0052 s | |

0.0625 | 6,859 | 0.492096 | 3.37 | 0.0259 s | |

0.03125 | 42,875 | 0.034556 | 3.83 | 0.0926 s | |

0.01562 | 300,763 | 0.00392612 | 3.14 | 0.374241 s | |

0.25 | 512 | 7.97706 | 0.0111 s | ||

0.125 | 1,728 | 5.78954 | 0.46 | 0.0613 s | |

0.0625 | 8,000 | 0.2335 | 4.63 | 0.4567 s | |

0.03125 | 46,656 | 0.00601485 | 5.28 | 1.9032 s | |

0.01562 | 314,432 | 0.000286093 | 4.39 | 11.2130 s |

Next, we study the behavior of the method when we have a space-time dependent diffusion coefficient. We consider again the same exact solution given in (4.1), but the diffusion coefficient is defined as

As tolerance for the projection and truncation error in the low-rank approximation we chose

Three-dimensional experiment on the example space-time domain (see Figure 2)with a space-time dependent diffusion coefficient (4.2).

h | #DOF | Error | Error rate | Assembly time | |

0.25 | 343 | 8.40222 | 0.0425 s | ||

0.125 | 1,331 | 5.11743 | 0.72 | 0.0250 s | |

0.0625 | 6,859 | 0.492269 | 3.38 | 0.1312 s | |

0.03125 | 42,875 | 0.034559 | 3.83 | 0.9555 s | |

0.01562 | 300,763 | 0.00366286 | 3.24 | 6.2758 s | |

0.25 | 512 | 8.03895 | 0.01667 s | ||

0.125 | 1,728 | 5.81114 | 0.47 | 0.1474 s | |

0.0625 | 8,000 | 0.233569 | 4.64 | 0.2993 s | |

0.03125 | 46,656 | 0.00601493 | 5.28 | 1.9955 s | |

0.01562 | 314,432 | 0.000293 | 4.36 | 16.8282 s |

For the cases of

For the cases of

In some cases it can be beneficial to avoid evaluating the sum of Kronecker products in (3.8) by using
the *Kronecker format* representation directly. Since matrix–vector multiplication can be implemented
easily for a matrix in this format, it can be used for solving the system iteratively. The main advantage is the reduction of the needed memory for storing the system matrix which allows us to assemble up to a very large number of degrees of freedom. The assembly is also very fast, since we only have to compute bivariate and univariate integrals.
Figure 4 shows the computation times for the bivariate and univariate integrals in the matrices of the right-hand side in (3.8).
The maximum number of degrees of freedom that were computed in this experiment are over 136 million for

### Example 2: $Q\subset {\mathbb{R}}^{3+1}$ .

In this example the space-time cylinder *Q* is
the product of a volumetric shell shape (see Figure 5) in space and the unit interval in time.
The diffusion coefficient is chosen to be

We focus on the computation times for matrix assembly.
Figure 6 shows the dependence of the assembly time on the number of degrees of freedom for
the

We note that in the four-dimensional case the overall complexity is no longer dominated by the sum of Kronecker products (3.8) but by the trivariate quadrature. For this reason, the computation of the matrix in Kronecker format is no longer significantly faster than the computation of the global matrix by performing the sum of Kronecker products in (3.8). However, the advantages stemming from the reduction in memory still apply to the four-dimensional case and we can assemble up to many more degrees of freedom using the same amount of memory.

Figure 7 shows the computation times for the
system matrix in Kronecker format. They are dominated by the
trivariate spatial integrals. For

## 5 Conclusions

We presented and analyzed a space-time IgA scheme for linear parabolic problems with varying coefficients.
We employed low-rank approximation techniques to speed-up the computation of the discrete operator.
In this way we were able to compute with several millions of degrees of freedom. However, a standing challenge is
the solution of the linear system. Indeed, this system is non-symmetric and with a rapidly increasing bandwidth with respect to the polynomial degree. The GMRES solver that we used, combined with domain decomposition and ILU preconditioner
performed reasonably good for a moderate number of degrees of freedom.
Another challenging task is the extension of this approach
to problems with discontinuous diffusion coefficients. In this case, the
space-time cylinder *Q* could be described as a multi-patch space-time domain (cf. [12]) compatible with the discontinuities of
the diffusion coefficient. The low-rank approximation would be applied patch-wise and appropriate discontinuous Galerkin techniques, i.e., numerical fluxes, would be needed for coupling the local patch-wise problems.

## References

- [1]↑
L. Beirão da Veiga, A. Buffa, G. Sangalli and R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numer. 23 (2014), 157–287.

- [2]↑
J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis. Toward integration of CAD and FEA, John Wiley & Sons, Chichester, 2009.

- [3]↑
C. de Boor, A Practical Guide to Splines, revised ed., Applied Math. Sci. 27, Springer, New York, 2001.

- [4]↑
S. Dolgov and B. Khoromskij, Simultaneous state-time approximation of the chemical master equation using tensor product formats, Numer. Linear Algebra Appl. 22 (2015), no. 2, 197–219.

- [5]↑
S. V. Dolgov, B. N. Khoromskij and I. V. Oseledets, Fast solution of parabolic problems in the tensor train/quantized tensor train format with initial application to the Fokker–Planck equation, SIAM J. Sci. Comput. 34 (2012), no. 6, A3016–A3038.

- [6]↑
M. Heroux, An overview of trilinos, Technical Report SAND2003-2927, Sandia National Laboratories, 2003.

- [7]↑
C. Hofer, U. Langer, M. Neumüller and I. Toulopoulos, Time-multipatch discontinuous Galerkin space-time isogeometric analysis of parabolic evolution problems, Electron. Trans. Numer. Anal. 49 (2018), 126–150.

- [8]↑
T. J. R. Hughes, J. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods Appl. Mech. Engrg. 194 (2005), no. 39–41, 4135–4195.

- [9]↑
S. K. Kleiss, C. Pechstein, B. Jüttler and S. Tomar, IETI—isogeometric tearing and interconnecting, Comput. Methods Appl. Mech. Engrg. 247/248 (2012), 201–215.

- [10]↑
O. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, Appl. Math. Sci. 49, Springer, New York, 1985.

- [11]↑
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monogr. 23, American Mathematical Society, Providence, 1968.

- [12]↑
U. Langer, A. Mantzaflaris, S. E. Moore and I. Toulopoulos, Multipatch discontinuous Galerkin isogeometric analysis, Isogeometric Analysis and Applications—IGAA 2014, Lect. Notes Comput. Sci. Eng. 107, Springer, Cham (2015), 1–32.

- [13]↑
U. Langer, S. E. Moore and M. Neumüller, Space-time isogeometric analysis of parabolic evolution problems, Comput. Methods Appl. Mech. Engrg. 306 (2016), 342–363.

- [14]↑
A. Mantzaflaris, B. Jüttler, B. N. Khoromskij and U. Langer, Matrix generation in isogeometric analysis by low rank tensor approximation, Curves and Surfaces, Lecture Notes in Comput. Sci. 9213, Springer, Cham (2015), 321–340.

- [15]↑
A. Mantzaflaris, B. Jüttler, B. N. Khoromskij and U. Langer, Low rank tensor methods in Galerkin-based isogeometric analysis, Comput. Methods Appl. Mech. Engrg. 316 (2017), 1062–1085.

- [16]↑
A. Mantzaflaris and F. Scholz, G+smo (Geometry plus Simulation modules) v0.8.1, (2017), http://gs.jku.at/gismo.

- [17]↑
F. Scholz, A. Mantzaflaris and B. Jüttler, Partial tensor decomposition for decoupling isogeometric Galerkin discretizations, Comput. Methods Appl. Mech. Engrg. 336 (2018), 485–506.