Low-Rank Space-Time Decoupled Isogeometric Analysis for Parabolic Problems with Varying Coefficients

  • 1 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, A-4040, Linz, Austria
  • 2 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, A-4040, Linz, Austria
Angelos MantzaflarisORCID iD: https://orcid.org/0000-0001-7135-1084, Felix ScholzORCID iD: https://orcid.org/0000-0003-3339-0079 and Ioannis Toulopoulos
  • Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenberger Straße 69, A-4040, Linz, Austria
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Abstract

In this paper we present a space-time isogeometric analysis scheme for the discretization of parabolic evolution equations with diffusion coefficients depending on both time and space variables. The problem is considered in a space-time cylinder in d+1, with d=2,3, and is discretized using higher-order and highly-smooth spline spaces. This makes the matrix formation task very challenging from a computational point of view. We overcome this problem by introducing a low-rank decoupling of the operator into space and time components. Numerical experiments demonstrate the efficiency of this approach.

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 d+1, which becomes practically infeasible, even for a small or moderate number of degrees of freedom. In order to treat this problem efficiently, we use the low-rank decoupling techniques proposed in [17]. This provides us with an efficient Kronecker decomposition of the system matrix into space and time components, therefore reducing the dimension of the problem as well as the overall computational effort.

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 d, with some integer d1. From a practical point of view, we are interested in the cases d=1,2, or 3. Furthermore, 𝜶=(α1,,αd) denotes a multi-index of non-negative integers α1,,αd with degree |𝜶|=j=1dαj. For any 𝜶, we define the differential operator x𝜶=x1α1xdαd, with xj=xj, j=1,,d. As usual, L2(Ω) denotes the Lebesgue space for which Ω|v(x)|2𝑑x<, endowed with the norm

vL2(Ω)=(Ω|v(x)|2𝑑x)12.

For an integer 0, we define the standard Sobolev space

H(Ω)={vL2(Ω):x𝜶vL2(Ω) for all |𝜶|},

endowed with the norm

vH(Ω)=(|𝜶|x𝜶vL2(Ω)2)12,

and also we introduce the subspace H01(Ω)={vH1(Ω):v=0 on Ω}. Let J¯=[0,T] be the time interval for some final time T>0. For later use, we define the space-time cylinder Q=Ω×(0,T) and its boundary parts Σ=Ω×(0,T), ΣT=Ω×{T} and Σ0=Ω×{0} such that Q=ΣΣ¯0Σ¯T, see an illustration in Figure 1.

Figure 1
Figure 1

The space type cylinder Q and the IgA B-spline parameterization 𝚽:Q^Q.

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

According to the definition of x𝜶, we now define the spatial gradient xv=(x1v,,xdv). Let and m be positive integers. For functions defined in the space-time cylinder Q, we define the Sobolev spaces

H,m(Q)={vL2(Q):x𝜶vL2(Q), 0|𝜶| and tivL2(Q),i=1,,m},

where t=t, and, in particular, the subspaces

H01,0(Q)={vL2(Q):xv[L2(Q)]d,v=0 on Σ},
H0,0¯1,1(Q)={vL2(Q):xv[L2(Q)]d,tvL2(Q),v=0 on Σ,v=0 on ΣT},
H0,0¯1,1(Q)={vL2(Q):xv[L2(Q)]d,tvL2(Q),v=0 on Σ,v=0 on Σ0}.

We equip the above spaces with the norms and seminorms

vH,m(Q)=(|𝜶|x𝜶vL2(Q)2+m0=0mtm0vL2(Q)2)12,
|v|H,m(Q)=(|𝜶|=x𝜶vL2(Q)2+tmvL2(Q)2)12.

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 ab meaning that cabCa with generic positive constants c and C.

2.2 The Model Parabolic Problem

In Q¯=Ω¯×[0,T], we consider the initial boundary value problem

tu-divx(ρ(x,t)u)=fin Q,
u(x,t)=0on Σ,
u(,0)=u0on Σ0,

as model problem, where f:Q, with fL2(Q), and u0:Ω, with u0L2(Ω) are given functions, the diffusion coefficient ρmaxρ(x,t)ρmin>0 is a given smooth function, and u:Q¯ is the unknown.

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 uH01,0(Q) such that

a(u,v)=l(v)for all vH0,0¯1,1(Q)

with the bilinear form

a(u,v)=-Qu(x,t)tv(x,t)𝑑x𝑑t+Qρ(x,t)xu(x,t)xv(x,t)𝑑x𝑑t

and the linear form

l(v)=Qf(x,t)v(x,t)𝑑x𝑑t+Ωu0(x)v(x,0)𝑑x,

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 u0=0. However, the analysis presented in the sequel can easily be generalized to other constellations of boundary conditions. The space-time variational formulation (2.2) has a unique solution, see, e.g., [10, 11]. In these monographs, besides existence and uniqueness results, one can also find useful a priori estimates and regularity results.

For our analysis, we make the following convenient assumption.

Assumption 2.1.

We assume that the solution u of (2.2) belongs to V=H0,0¯1,1(Q)H,m(Q) with some 2 and m1.

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 n1 denote the number of basis functions. Consider a partition

𝒵={0=z1<z2<<zM=1}

of the unit interval I¯=[0,1] with I¯j=[zj,zj+1], j=1,,M-1, being the intervals of the partition. Based on 𝒵, we consider a knot-vector Ξ={0=ξ1,ξ2,,ξn1+p+1=1} and its associated vector of knot multiplicities ={m1,,mM} with m1=mM=p+1, i.e.,

ξ1+m1++mj-1==ξm1++mj=zjfor j=1,,M.

We assume that mjp for all internal knots. The B-spline basis functions are defined by the Cox–de Boor formula, see, e.g., [2] and [3],

B^i,p=x-ξiξi+p-ξiB^i,p-1(x)+ξi+p+1-xξi+p+1-ξi+1B^i+1,p-1(x)with B^i,0(x)={1,if ξixξi+1,0,otherwise.

Now, let us consider the unit cube Q^=(0,1)d+1d+1, which we will refer to as the parametric domain. We extend the univariate B-spline concept to multiple dimensions with the use of tensor products. Let the integers p and nk denote the given B-spline degree (same for all directions) and the number of basis functions of the B-spline space that will be constructed in xk-direction with k=1,,d+1. We introduce the (d+1)-dimensional vector of knots 𝚵d+1=(Ξ1,,Ξk,,Ξd+1), k=1,,d+1, with the particular components given by Ξk={0=ξ1k,ξ2k,,ξnk+p+1k=1}. For all the internal knots, we assume that mjkp, with mjk to be the associated multiplicities. The basis functions of the multivariate B-spline space 𝔹^𝚵d+1,p are defined by the tensor-product of the corresponding univariate B-spline basis functions of 𝔹^Ξk,p spaces, that is,

𝔹^𝚵d+1,p=k=1d+1𝔹^Ξk,p=span{B^j(x^)}j=1nB,

where nB=n1nd+1 and each B^j(x^) has the form

B^j(x^)=B^j1(x^1)B^jk(x^k)B^jd+1(x^d+1)with B^jk(x^k)𝔹^Ξk,p.

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 𝐂jd+1, the domain Q is parameterized by the mapping

𝚽:Q^Q,x=𝚽(x^)=j=1nB𝐂jB^j(x^)Q,

where x^=𝚽-1(x). cf. [2]. The components Ξk of 𝚵d+1 form a mesh Th^,Q^={E^m}m=1M in Q^, where E^m are the elements and h^ is the mesh-size. We construct a mesh Th,Q={Em}m=1M in Q, where the elements Em are the images of E^mTh^,Q^ under 𝚽. We define the isogeometric discretization space on Q as

VΦ,𝚵d+1,p:=span{B^j𝚽-1:B^j𝔹^𝚵d+1,p,j=1,,nB}.

To keep notation simple, we denote the above space by Vh, that is, we omit to write the domain parameterization, the spline degree and knot vectors. Furthermore, we introduce the space

V0h:=VhH0,0¯1,1={vhVh:vh|ΣΣ0=0}.
Assumption 2.2.

The mesh Th^,Q^ is uniform, i.e., for every E^Th^,Q^, there exists a number γ>0 such that γh^rE^, where rE^ is the radius of the inscribed circle of E^.

Remark 1.

Since the parameterization 𝚽 is fixed, under Assumption 2.2, we have hh^.

The parameterization 𝚽 can be considered to be bi-Lipschitz homeomorphisms, see [1]. For simplifying the analysis, we consider the following regularity properties on 𝚽.

Assumption 2.3.

We assume that 𝚽 and 𝚽-1 are sufficiently smooth (i.e., C1 diffeomorphisms) and that there exist constants 0<c<C such that c|detJ𝚽|C, where J𝚽 is the Jacobian matrix of 𝚽.

2.4 Discrete Variational Forms

We denote by 𝐧=(n1,,nd,nd+1)=(nx,nt) the normal on Σ. Let vhV0h and whn=vh+θhtvh, where θ is a positive parameter. We multiply (2.1) by wh and we integrate over Q; by applying integration by parts we obtain

aQ(u,vh)=Qtu(vh+θhtvh)+ρ(x,t)(xuxvh+θhxuxtvh)dxdt+Qnxρ(x,t)xu(vh+θhtvh)𝑑s=Qf(vh+θhtvh)𝑑x𝑑t.

Since wh=0 on Σ, and nx=0 on Σ0ΣT, we can obtain

aQ(u,vh)=Qtu(vh+θhtvh)+ρ(x,t)(xuxvh+θhxuxtvh)dxdt=Qf(vh+θhtvh)𝑑x𝑑t.

The space-time IgA method for (2.1) can be formulated as follows: Find uhV0h such that

ah(uh,vh)=Lh(vh)for all vhV0h,
ah(uh,vh)=aQ(uh,vh)andLh(vh)=Qf(vh+θhtvh)𝑑x𝑑t.

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 ρ(x,t) is constant, i.e., we have ρ(x,t)=ρ>0. The analysis can be extended easily to the general problem given in (2.1). Motivated by (2.7b), we define the norm on V0h as follows:

vh=(ρ12xvL2(Q)2+θhtvL2(Q)2+12vL2(ΣT)2)12.

Let Assumption 2.1 hold and let V be the space defined there. We define the space V0h,*=V+V0h endowed with the norm

vh,*=(vh2+(θh)-1vL2(Q)2)12.
Lemma 2.

The discrete bilinear form ah(,), defined in (2.7a), is V0h-elliptic, i.e.,

ah(vh,vh)Cevhh2for vhV0h,

where Ce=1.

Proof.

Using Green’s formula

Qtvhvh+vhtvhdxdt=Qntvh2𝑑s,

the fact that nt=0 on Σ and vh(x,0)=0, we obtain the identity

Qtvhvhdxdt=12Qtvh2dxdt=12ΣTvh2𝑑s.

The definition of ah(,) and identity (2.8) yield

ah(vh,vh)=Q12tvh2+θh(tvh)2+ρ|xvh|2+ρθh2t|xvh|2dxdt
=12vhL2(ΣT)2+θhtvhL2(Q)2+ρ12xvhL2(Q)2+θh2Qρ|xvh|2nt𝑑s
=12vhL2(ΣT)2+θhtvhL2(Q)2+ρ12xvhL2(Q)2+θh2(ρ12xvhL2(ΣT)2-ρ12xvhL2(Σ0)2).

Now, since it holds that vh(x,0)=0, we also get xvh(x,0)=0 for all xΣ0. Using this in (2.9), we can arrive at

ah(vh,vh)=12vhL2(ΣT)2+θhtvhL2(Q)2+ρ12xvhL2(Q)2+θh2ρ12xvhL2(ΣT)2vhh2.
Lemma 3.

The discrete bilinear form ah(,), is uniformly bounded on V0h,*×V0h, i.e.,

|ah(u,vh)|μbuh,*vhhfor uV0h,* and vhV0h,

where the constant μb>0 depends on the constants that appear in the inverse inequalities.

Proof.

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

Lemma 4.

Let s2 be a positive integer and let a function vV=Hs(Q)H0,0¯1,1(Q). Then there exists a tensor-product quasi-interpolant Πh,p:VV0h such that

u-Πh,puhCintp,1hmin(p+1,s)-1uHs(Q),u-Πh,puh,*Cintp,2hmin(p+1,s)-1uHs(Q),

where the constants Cintp,i, i=1,2 depend on the B-spline parameterization Φ, but not on u and h.

Proof.

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 h.

Theorem 5.

Let uV=Hs(Q)H0,0¯1,1(Q), s2, solve (2.2) and let uh solve (2.7a). Under Assumption 2.2, there exists a constant c>0, independent of u and h such that

u-uhhchruHs(Ω)with r=min(p+1,s)-1.
Proof.

Using the properties of bilinear form ah(,), i.e., Lemma 2 and Lemma 3, as well as the consistency of u, we obtain

uh-Πhuh2Ceah(uh-Πh,pu,uh-Πh,pu)=ah(u-Πh,pu,uh-Πh,pu)μbu-Πhuh,*uh-Πhuh,

where we immediately get

uh-ΠhuhμbCeu-Πhuh,*.

Hence, applying the triangle inequality u-uhhu-Πhuh,*+uh-Πhuh, and using the interpolation estimates (2.10), we can derive the desired estimate (2.11). ∎

3 Matrix Assembly and Decoupling

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

F:(0,1)dΩ.

We obtain a parameterization 𝚽:Q^Q of the space-time cylinder Q, see (2.5), by lifting F linearly, i.e.,

𝚽(x^1,,x^d,t^)=(F(x^1,,x^d),t^T),

where instead of x^d+1 we use t^ to denote the time variable.

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 ah(,), see (2.7). For the general case of varying coefficient ρ, we apply the decoupling technique presented in [17] in order to fully decouple the assembly. Note that in the case of a constant diffusion coefficient, i.e., ρ1, the decoupling of the integration in time and in space follows naturally from the form of (2.1) and (3.1). In the sequel we present first the case ρ1.

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 ah(,), on the basis functions of the discrete space. Using the parameterization (3.1), we can transform each appearing integral on the (d+1)-dimensional space-time domain Q to the parametric domain Q^=(0,1)d+1. As a consequence of the B-spline basis’ tensor–product structure, the resulting (d+1)-variate integrals split into a product of a d-variate integral over the spatial parametric domain and a univariate integral over the unit interval.

Let i=(i1,,id,id+1) and j=(j1,,jd,jd+1) be two multi-indices. We consider each of the four terms of ah(Bi,Bj) separately.

For the first term, the transformation of the integral yields

QtBiBjdxdt=Q^|detJF|t^B^iB^jdx^dt^=(0,1)d|detJF|B^i1idB^j1jddx^01t^B^id+1B^jd+1dt^,

where JF is the Jacobian of the spatial geometric mapping F. For the second term we arrive at

θhQtBitBjdxdt=θhTQ^|detJF|t^B^it^B^jdx^dt^
=θhT(0,1)d|detJF|B^i1idB^j1jddx^01t^B^id+1t^B^jd+1dt^.

Transforming the third term leads to

QxBixBjdxdt=TQ^|detJF|x^B^iJF-1JF-x^B^jdx^dt^
=T(0,1)d|detJF|x^B^i1idJF-1JF-x^B^jdx^01B^id+1B^jd+1dt^.

Finally, the fourth integral becomes

θhQxBixtBjdxdt=θhQ^|detJF|x^B^iJF-1JF-x^B^jdx^dt^
=θh(0,1)d|detJF|x^B^i1idJF-1JF-x^B^j1jddx^01B^id+1t^B^jd+1dt^.

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 (Kij)=(ah(Bj,Bi)) as the sum of Kronecker products

K=X1(Y1+Y2)+X2(Y3+Y1),

where Xk are n1nd×n1nd-matrices containing the d-variate integrals given in (3.2)–(3.5) and Yk are nd+1×nd+1-matrices containing the corresponding univariate integrals. This representation (3.6) is also called the 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 ρ^(x^,t^)=(ρ𝚽)(x^,t^) into 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

ρ^(x^,t^)r=1R𝒰r(x^)𝒱r(t^),

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

This decomposition of ρ^ leads to a decomposition of the (d+1)-variate integrals. In particular, the third term and the fourth term become, respectively,

Qρ(x,t)xBixBjdxdtTr=1R(0,1)d|detJF|𝒰r(x^)x^B^i1idJF-1JF-x^B^jdx^01𝒱r(t^)B^id+1B^jd+1dt^,
θhQρ(x,t)xBixtBjdxdtθhr=1R(0,1)d|detJF|𝒰r(x^)x^B^i1idJF-1JF-x^B^j1jddx^
   ×01𝒱r(t^)B^id+1t^B^jd+1dt^.

We arrive at a Kronecker format representation of rank R+1 of the system matrix:

KX1(Y1+Y2)+r=1RUr(V1r+V2r),

where the matrices X1,Y1,Y2 are defined as in (3.6) and Ur,V1r,V2r are the resulting matrices in (3.7).

If the diffusion coefficient is matrix-valued, let us say ρpq, then the same method can be applied by decomposing each of the components of the matrix. In this case the rank 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 x^k, k=1,,d+1, direction are the same, i.e., n=n1==nd+1 and p=p1==pd+1. The complexity of assembling the system matrix is bounded from below by the number of its non-zeros, which is O(nd+1pd+1). The classical assembly method using element-wise Gauss quadrature rules has complexity O(nd+1p3(d+1)).

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 X1,Ur is thus O(ndp3d) while the complexity of computing the matrices Y1,Y2,Vr is clearly dominated by this. Thus, the complexity of the quadrature step is O(Rndp3d). Generating the global matrix K by computing the Kronecker product (3.8) then costs O(Rnd+1pd+1). Depending on the dimension, the overall complexity is either dominated by the d-variate quadrature or by the sum of Kronecker products. Since usually np, for d=2 the complexity of the sum of Kronecker products (3.8) is dominating. For d=3, the complexity of the quadrature step dominates. For details on the complexity, see [17].

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 ρ=1. The method produces a block-bidiagonal system matrix which can be solved sequentially.

Example 1: Q2+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

u(x1,x2,t)=(cos(2π(x-y))-cos(2π(x+y)))sin(2πt)

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.

Figure 2
Figure 2Figure 2

The space-time domain for d=2 and a time slice of the exact solution at t=0.7. On the left, the spatial domain is an annulus and the time direction is parallel to the z-axis.

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

For solving the linear system, a parallel GMRES solver from Trilinos [6] is used with tolerance set to 10-8 and Krylov subspace dimension 200. We used a parallel domain decomposition preconditioner (AZ_dom_decomp) and a direct solver (incomplete LU) in each processor/sub-problem.

Table 1 shows the error convergence as well as the computation times for assembling the matrix in the case of a constant diffusion coefficient ρ1. We observe that the convergence rates are in agreement with the theoretical predicted rates in Theorem 5. We also remark that the assembly times given in last column are significantly low compared to the size of the system.

Table 1

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

h#DOFErrorError rateAssembly time
p=30.253438.309490.0061 s
0.1251,3315.099440.700.0052 s
0.06256,8590.4920963.370.0259 s
0.0312542,8750.0345563.830.0926 s
0.01562300,7630.003926123.140.374241 s
p=40.255127.977060.0111 s
0.1251,7285.789540.460.0613 s
0.06258,0000.23354.630.4567 s
0.0312546,6560.006014855.281.9032 s
0.01562314,4320.0002860934.3911.2130 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

ρ(x1,x2,t)=(x1-4)2(x2-4)2(t-4)2+(x1-4)4(x2-4)4(t-4)4.

As tolerance for the projection and truncation error in the low-rank approximation we chose ϵ=10-6 which results in an approximation of rank 2. See Table 2.

Table 2

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

h#DOFErrorError rateAssembly time
p=30.253438.402220.0425 s
0.1251,3315.117430.720.0250 s
0.06256,8590.4922693.380.1312 s
0.0312542,8750.0345593.830.9555 s
0.01562300,7630.003662863.246.2758 s
p=40.255128.038950.01667 s
0.1251,7285.811140.470.1474 s
0.06258,0000.2335694.640.2993 s
0.0312546,6560.006014935.281.9955 s
0.01562314,4320.0002934.3616.8282 s

For the cases of p5, we only show the computation times of the assembly, since the resulting systems are not only non-symmetric but also quite large and dense, therefore rather hard to work with. Figure 3 shows the dependence of the computation times on the number of degrees of freedom for the partial tensor decomposition method as well as for the classical element-wise Gauss quadrature. It can be seen in the presented computation times that the assembly method using partial low-rank tensor approximation is very efficient. In particular, it outperforms a classical element-wise Gauss rule approach by far. We refer to [17] for further experimental comparisons between the classical element-wise Gauss quadrature and the low-rank partial tensor decomposition method, for the stiffness matrix.

For the cases of p5, we only show the computation times of the assembly, since the resulting systems are not only non-symmetric but also quite large and dense, therefore very hard to solve. Figure 3 shows the dependence of the computation times on the number of degrees of freedom for the three-dimensional problem. As it can be seen in the presented computation times, the assembly method using partial low-rank tensor approximation is very efficient. In particular, it outperforms a classical element-wise Gauss rule approach by far. We refer to [17] for an experimental comparison between the classical element-wise Gauss quadrature and the low-rank partial tensor decomposition method, for the stiffness matrix.

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 p=3.

Figure 3
Figure 3

Computation times for assembling the global system matrix on the example domain (Figure 2) for the space-timedependent diffusion coefficient (4.2).

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

Figure 4
Figure 4

Computation times for assembling the Kronecker format on the example domain (Figure 2) for the space-timedependent diffusion coefficient (4.2).

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

Example 2: Q3+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

ρ(x1,x2,x3,t)=(x1-4)2(x2-4)2(x3-4)2(t-4)2+(x1-4)4(x2-4)4(x3-4)4(t-4)4.

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 ρ(x1,x2,x3,t) given in (4.3).

Figure 5
Figure 5

The volumetric shell shape of Example 2.

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

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 6
Figure 6

Computation times for assembling the global system matrix on the four-dimensional space-time domain(with spatial part as in Figure 5) for the space-time dependent diffusion coefficient (4.3).

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

Figure 7
Figure 7

Computation times for assembling the Kronecker format on the four-dimensional space-time domain(with spatial part as in Figure 5) for the space-time dependent diffusion coefficient (4.3).

Citation: Computational Methods in Applied Mathematics 19, 1; 10.1515/cmam-2018-0024

Figure 7 shows the computation times for the system matrix in Kronecker format. They are dominated by the trivariate spatial integrals. For p=2 we assembled for a maximum number of about 19 million degrees of freedom. Finally, we observe one more time that the partial tensor decomposition method leads to a large speed-up of the computation compared to the classical element-wise Gauss quadrature.

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

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

    • Crossref
    • Export Citation
  • [2]

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

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

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    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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [10]

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

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    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.

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    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.

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    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.

    • Crossref
    • Export Citation
  • [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.

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    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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

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

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
  • [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.

    • Crossref
    • Export Citation
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  • View in gallery

    The space type cylinder Q and the IgA B-spline parameterization 𝚽:Q^Q.

  • View in gallery View in gallery

    The space-time domain for d=2 and a time slice of the exact solution at t=0.7. On the left, the spatial domain is an annulus and the time direction is parallel to the z-axis.

  • View in gallery

    Computation times for assembling the global system matrix on the example domain (Figure 2) for the space-timedependent diffusion coefficient (4.2).

  • View in gallery

    Computation times for assembling the Kronecker format on the example domain (Figure 2) for the space-timedependent diffusion coefficient (4.2).

  • View in gallery

    The volumetric shell shape of Example 2.

  • View in gallery

    Computation times for assembling the global system matrix on the four-dimensional space-time domain(with spatial part as in Figure 5) for the space-time dependent diffusion coefficient (4.3).

  • View in gallery

    Computation times for assembling the Kronecker format on the four-dimensional space-time domain(with spatial part as in Figure 5) for the space-time dependent diffusion coefficient (4.3).