Condition Numbers Of Matrices Research Articles

Learning quantum many-body systems from a few copies

Estimating physical properties of quantum states from measurements is one of the most fundamental tasks in quantum science. In this work, we identify conditions on states under which it is possible to infer the expectation values of all quasi-local observables of a state from a number of copies that scales polylogarithmically with the system's size and polynomially on the locality of the target observables. We show that this constitutes a provable exponential improvement in the number of copies over state-of-the-art tomography protocols. We achieve our results by combining the maximum entropy method with tools from the emerging fields of classical shadows and quantum optimal transport. The latter allows us to fine-tune the error made in estimating the expectation value of an observable in terms of how local it is and how well we approximate the expectation value of a fixed set of few-body observables. We conjecture that our condition holds for all states exhibiting some form of decay of correlations and establish it for several subsets thereof. These include widely studied classes of states such as one-dimensional thermal and high-temperature Gibbs states of local commuting Hamiltonians on arbitrary hypergraphs or outputs of shallow circuits. Moreover, we show improvements of the maximum entropy method beyond the sample complexity that are of independent interest. These include identifying regimes in which it is possible to perform the postprocessing efficiently as well as novel bounds on the condition number of covariance matrices of many-body states.

Conditioning of random Fourier feature matrices: double descent and generalization error

Abstract We provide high-probability bounds on the condition number of random feature matrices. In particular, we show that if the complexity ratio $N/m$, where $N$ is the number of neurons and $m$ is the number of data samples, scales like $\log ^{-1}(N)$ or $\log (m)$, then the random feature matrix is well-conditioned. This result holds without the need of regularization and relies on establishing various concentration bounds between dependent components of the random feature matrix. Additionally, we derive bounds on the restricted isometry constant of the random feature matrix. We also derive an upper bound for the risk associated with regression problems using a random feature matrix. This upper bound exhibits the double descent phenomenon and indicates that this is an effect of the double descent behaviour of the condition number. The risk bounds include the underparameterized setting using the least squares problem and the overparameterized setting where using either the minimum norm interpolation problem or a sparse regression problem. For the noiseless least squares or sparse regression cases, we show that the risk decreases as $m$ and $N$ increase. The risk bound matches the optimal scaling in the literature and the constants in our results are explicit and independent of the dimension of the data.

A space–time Trefftz DG scheme for the time-dependent Maxwell equations in anisotropic media

In this paper we are concerned with Trefftz discretizations of the time-dependent Maxwell equations in anisotropic media in three-dimensional domains. We propose a class of space–time Trefftz DG methods, which include the Trefftz variational formulation from Egger et al. (2015). We prove the error estimates of the approximate solutions with respect to the meshwidth and the condition number of the coefficient matrices. Furthermore, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous Maxwell equations in anisotropic media. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

Brief review of the development of theories of robustness, roughness and bifurcations of dynamic systems

The development issues of theories of robustness, roughness and bifurcations of dynamic systems are considered. In the modern theory of dynamic systems and automatic control systems, researches of the properties of roughness and robustness of systems are becoming more and more important. The work considers methods of research and ensuring robust stability of interval dynamic systems of both algebraic and frequency directions of robust stability. The main results of the original algebraic method of robust stability for continuous and discrete time are given. In the frequency direction of robust stability, the issues of a frequency-robust method to the analysis and synthesis of robust multidimensional control systems based on the use of the frequency condition number of the transfer matrix of the “input-output” ratio are considered. The main provisions of the theory and method of topological roughness of dynamic systems based on the concept of roughness according to Andronov-Pontryagin are presented with the introduction of a measure of roughness of systems in the form of a condition number of matrices of reduction to a diagonal (quasi-diagonal) basis at special points of phase space. Criteria for dynamic systems bifurcations are formulated. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as Lorenz and Rössler attractors, Belousov-Jabotinsky, Chua systems, “predator-prey” and “predator-prey-food”, Hopf bifurcation, Schumpeter and Caldor economic systems, Henon mapping, and others. For research of weakly formalized and non-formalized systems, the use of the approach of analogies of theoretical-multiple topology and the abstract method to such systems is proposed. Further research suggests the development of roughness and bifurcation theories for complex nonlinear dynamical systems.

Observability Analysis for Orbit Determination Using Spaceborne Gradiometer

The application of a spaceborne gradiometer to satellite orbit determination has been proposed in recent years. The advantages lie in its full autonomy and its immunity against spoofing attacks. The objective of this paper is to study the observability of the spaceborne gradiometer-based orbit determination (SGOD) system. To address the problem of unknown biases in spaceborne gradiometer measurements, the joint estimation of satellite orbital states and gradiometer biases is adopted. Then, the system observability matrices are formulated based on the theory of nonlinear observability. The local weak observability is assessed numerically by singular values and condition numbers of the observability matrices. A special unobservable case is demonstrated, which is the circular orbit in the Keplerian two-body model. Different orbital parameters, attitude control modes, and gravity models are furtherly employed to analyze their effects on system observability. The observability analysis shows that low altitudes, large eccentricities, large inclinations, attitude maneuverings, and the J2 perturbation can improve the system observability. Furthermore, a numerical simulation using the extended Kalman filter (EKF) is developed to obtain accuracies and convergence rates of the SGOD system. Effects of orbital parameters, attitude control modes, and the J2 perturbation on orbit determination results are also studied. It is shown that the simulation results agree with the observability analysis. Using a spaceborne gradiometer with 0.1 E (1 E=10−9 s−2) accuracy, position accuracy of tens of meters is achieved for a low Earth orbit.

Global space-time Trefftz DG schemes for the time-dependent linear wave equation

In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains $\Omega \subset \mathbb{R}^d~ (d\in \mathbb{N})$. We propose two variants of the Trefftz DG method, define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations, and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth $h$ and the condition number of the coefficient matrices, respectively. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous wave equation in anisotropic media. In particular, the error analysis holds for the (nonhomogeneous) Dirichlet, Neumann, and mixed boundary conditions from the original PDEs. Furthermore, a strategy to discretize the model in heterogeneous media is proposed. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy.

Fast solution methods for Riesz space fractional diffusion equations with non-separable coefficients

In this paper, efficient preconditioned iterative methods for solving the Riesz space fractional diffusion equations with variable diffusion coefficients are considered. Crank–Nicolson and weighted and shifted Grünwald difference scheme are applied to discretize the problem. For one-dimensional problems, we transform the asymmetric discretized linear system into a symmetric one and solve the resulted linear system by preconditioned conjugate gradient method. The preconditioned conjugate gradient method with a symmetric Toeplitz preconditioner and a sine transform based preconditioner are employed to solve the resulting linear system. Theoretically, we respectively prove that the condition numbers of preconditioned matrices have uniform upper bounds, which is independent of discretization step-sizes and fractional order. For two-dimensional problems, we propose an optional and symmetric splitting iteration method, which can be embedded by conjugate gradient method although the diffusion coefficients are non-separable. We prove that the optional and symmetric splitting iteration method is unconditionally convergent. Numerical results are reported to illustrate the efficiency of the proposed methods.

The Simple Solution for Nonlinear State Estimation of Ill-Conditioned Systems: The Normalized Unscented Kalman Filter

An easy-to-implement method for nonlinear state estimation for ill-conditioned systems is proposed. By propagating standard deviations and correlations instead of the covariance in the unscented Kalman filter (UKF), the condition numbers of relevant matrices are reduced. The reduction in the condition number is related to the scaling of the problem. Hence, what we propose is a normalization method that acts as an “auto-scaler”. Compared to other methods in state estimation for ill-conditioned systems, our proposed method factors the covariance matrix into physically meaningful statistics which can be used to check for filter divergence online. The method is compared to a standard UKF in a case study and shows a significant reduction in the condition number.

Distribution of the Scaled Condition Number of Single-Spiked Complex Wishart Matrices

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}\in \mathbb {C}^{n\times m}$ </tex-math></inline-formula> ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m\geq n$ </tex-math></inline-formula> ) be a random matrix with independent columns each distributed as complex multivariate Gaussian with zero mean and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single-spiked</i> covariance matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {I}_{n}+ \eta \mathbf {u}\mathbf {u}^{*}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {I}_{n}$ </tex-math></inline-formula> is the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\times n$ </tex-math></inline-formula> identity matrix, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {u}\in \mathbb {C}^{n\times 1}$ </tex-math></inline-formula> is an arbitrary vector with unit Euclidean norm, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta \geq 0$ </tex-math></inline-formula> is a non-random parameter, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(\cdot)^{*}$ </tex-math></inline-formula> represents the conjugate-transpose. This paper investigates the distribution of the random quantity <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})=\sum _{k=1}^{n} \lambda _{k}/\lambda _{1}$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0\le \lambda _{1}\le \lambda _{2}\le \ldots \leq \lambda _{n} &lt; \infty $ </tex-math></inline-formula> are the ordered eigenvalues of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}\mathbf {X}^{*}$ </tex-math></inline-formula> (i.e., single-spiked Wishart matrix). This random quantity is intimately related to the so called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">scaled condition number</i> or the Demmel condition number (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}(\mathbf {X})$ </tex-math></inline-formula> ) and the minimum eigenvalue of the fixed trace Wishart-Laguerre ensemble (i.e., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{-2}(\mathbf {X})$ </tex-math></inline-formula> ). In particular, we use an orthogonal polynomial approach to derive an exact expression for the probability density function of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> which is amenable to asymptotic analysis as matrix dimensions grow large. Our asymptotic results reveal that, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n\to \infty $ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m-n$ </tex-math></inline-formula> is fixed and when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\eta $ </tex-math></inline-formula> scales on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1/n$ </tex-math></inline-formula> , <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> scales on the order of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n^{3}$ </tex-math></inline-formula> . In this respect we establish simple closed-form expressions for the limiting distributions. It turns out that, as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m,n\to \infty $ </tex-math></inline-formula> such that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n/m\to c\in (0,1)$ </tex-math></inline-formula> , properly centered <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\kappa _{\text {SC}}^{2}(\mathbf {X})$ </tex-math></inline-formula> fluctuates on the scale <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$m^{\frac {1}{3}}$ </tex-math></inline-formula> .

On the Condition Number of the Shifted Real Ginibre Ensemble

We derive an accurate lower tail estimate on the lowest singular value $\sigma_1(X-z)$ of a real Gaussian (Ginibre) random matrix $X$ shifted by a complex parameter $z$. Such shift effectively changes the upper tail behaviour of the condition number $\kappa(X-z)$ from the slower $\mathbf{P}(\kappa(X-z)\ge t)\lesssim 1/t$ decay typical for real Ginibre matrices to the faster $1/t^2$ decay seen for complex Ginibre matrices as long as $z$ is away from the real axis. This sharpens and resolves a recent conjecture in [arXiv:2005.08930] on the regularizing effect of the real Ginibre ensemble with a genuinely complex shift. As a consequence we obtain an improved upper bound on the eigenvalue condition numbers (known also as the eigenvector overlaps) for real Ginibre matrices. The main technical tool is a rigorous supersymmetric analysis from our earlier work [arXiv:1908.01653].

Block symmetric-triangular preconditioners for generalized saddle point linear systems from piezoelectric equations

Based on the symmetric-triangular (ST) decomposition technique, a class of block ST (BST) preconditioners are proposed for generalized saddle point linear systems arising from the meshfree discretization of piezoelectric equations. By applying the BST preconditioners, we first transform the generalized saddle point linear systems into symmetric positive definite ones, which then can be solved in a fast and efficient way by the classical conjugate gradient (CG) or the preconditioned CG (PCG) iteration methods. Two practical BST preconditioners are presented and analyzed in detail. Eigen-properties and upper bounds of the condition numbers of the preconditioned matrices are proved. Implementation aspects are discussed. Finally, two numerical experiments arising from the piezoelectric strip shear deformation problem and the piezoelectric strip bending problem are presented. Numerical results show that the iteration steps of the PCG methods are independent of the number of degrees of freedom and the proposed BST preconditioners perform much better than some existing preconditioning techniques.

Variational quantum linear solver with a dynamic ansatz

Variational quantum algorithms have found success in the noisy intermediate-scale quantum era owing to their hybrid quantum-classical approach which mitigate the problems of noise in quantum computers. In our paper, we introduce the dynamic ansatz in the variational quantum linear solver for a system of linear algebraic equations. In this improved algorithm, the number of layers in the hardware efficient ansatz circuit is evolved, starting from a small and gradually increasing until convergence of the solution is reached. We demonstrate the algorithm advantage in comparison to the standard static ansatz by utilizing fewer quantum resources and with a smaller quantum depth on average in the presence and absence of quantum noise and in cases when the number of qubits or condition number of the system matrices are increased. The numbers of iterations and layers can be altered by a switching parameter. The performance of the algorithm in using quantum resources is quantified by a newly defined metric.

離散deRham系列を満たす高次多角形要素を用いた静磁場問題に対する領域分割法

A Balancing Domain Decomposition (BDD) method is considered as the preconditioner of an iterative Domain Decomposition Method (DDM) for perturbed magnetostatic problems, where the magnetic vector potential is regarded as an unknown function approximated by the Ned´ elec curl conforming finite element. To reduce the number of the Degrees´ Of Freedom (DOF) of coarse spaces in BDD methods, Polynomial Element Methods (PEMs) are introduced. Owing to the introduction of PEMs and the result of BDD method originally proposed by Mandel, the condition number of coefficient matrices derived from the iterative DDM is evaluated. Therefore, the number of iterations of the iterative DDM can be kept even when the number of the subdomains becomes larger. Moreover, the approximate coarse space by PEMs can admit that each subdomain becomes more general polyhedron. The results in case of higher order PEMs are also shown.

Efficient block preconditioners for integral constrained elliptic optimal control problems with finite element approximations

In this paper, focusing on a distributed optimal control problem for the elliptic equations with integral control constraint, we propose efficient block diagonal preconditioners to solve the corresponding linearized algebraic system with finite element methods. We derive the first order necessary and sufficient optimality conditions for an integral constraint on the control, and divide the discretized optimal conditions into three matrix forms. Then block-diagonal preconditioners for the corresponding linear algebraic systems are constructed. With respect to both the mesh size and the regularization parameter, the robustness of our preconditioners is proved in detail. In fact, the condition numbers of the preconditioned matrices are a constant for different parameters. Meanwhile, based on the equivalent matrix forms, an algorithm is proposed for this kind of constrained optimal control problems. Numerical experiments are given to depict the efficiency of our proposed preconditioners.

Another Hilbert inequality and critically separated interpolation nodes

AbstractAbstract: Estimates on the condition number of Vandermonde matrices have implications on several algorithms ranging from polynomial interpolation to sparse super resolution in fluorescence microscopy. Classically, the situation is studied for monomials on real intervals, the complex unit disk, and the complex unit circle. Except for roots of unity and well separated nodes on the unit circle, the condition number grows strongly with increasing polynomial degree. Here, we show that the condition number of the Vandermonde matrix for a particular instance of critically separated nodes on the complex unit circle grows logarithmically with the polynomial degree. The proof is based on a variant of Hilbert's inequality with remainder term.

Large-deviation asymptotics of condition numbers of random matrices

AbstractLet $\mathbf{X}$ be a $p\times n$ random matrix whose entries are independent and identically distributed real random variables with zero mean and unit variance. We study the limiting behaviors of the 2-normal condition number k(p,n) of $\mathbf{X}$ in terms of large deviations for large n, with p being fixed or $p=p(n)\rightarrow\infty$ with $p(n)=o(n)$ . We propose two main ingredients: (i) to relate the large-deviation probabilities of k(p,n) to those involving n independent and identically distributed random variables, which enables us to consider a quite general distribution of the entries (namely the sub-Gaussian distribution), and (ii) to control, for standard normal entries, the upper tail of k(p,n) using the upper tails of ratios of two independent $\chi^2$ random variables, which enables us to establish an application in statistical inference.

Numerical solution using radial basis functions for multidimensional fractional partial differential equations of type Black–Scholes

In this paper, as far as the authors know, for the first time, a one-dimensional partial differential model is generalized using fractional differential operators and the same principle that provides the dimensional invariance of the radial basis functions methodology, resulting in a multidimensional fractional partial differential model that can be solved using a numerical scheme of radial basis functions. A radial basis functions scheme is proposed to solve numerically, on different node configurations, multidimensional fractional partial differential equations, both in space and in time. Using the QR factorization, a way to reduce the condition number of the interpolation matrices of the proposed scheme is presented, the resulting scheme is used to numerically solve the diffusion equation that may be obtained from the Black–Scholes model, as well as some generalizations of this diffusion model with fractional differential operators and multiple dimensions. The Caputo fractional derivative is discretized with an order error $${\mathcal {O}}(\mathrm{{d}}t^{n-\alpha +1})$$ , with $$(n-1)<\alpha \le n$$ . The examples of fractional partial differential equations that are presented involve the Caputo fractional operator in the temporal part due to the memory phenomenon, and the Riemann–Liouville fractional operator in the spatial part due to the property of nonlocality.

A generalized plane wave discontinuous Galerkin method for three-dimensional anisotropic Helmholtz equations with variable wave numbers

In this paper we are concerned with the numerical method for three-dimensional anisotropic Helmholtz equations with variable wave numbers, where positive definite matrices define anisotropic media. We define novel generalized plane wave basis functions based on rigorous choice of the coordinate transformation. Then we derive the desired error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under an assumption on the shape regularity of polyhedral meshes. Numerical results verify the validity of the theoretical results, and indicate that the approximate solutions generated by the proposed method possess high accuracies.

Improving convergence by optimizing the condition number of the stiffness matrices arising from least-squares finite element methods

The least-squares finite element method (LSFEM) has drawn much attention with desirable properties, such as always symmetric positive-definite stiffness matrix and approximation of primal and non-primal variables with equivalent accuracy. Despite a condition number comparable to that of Galerkin FEM, of O(h−2), it is sometimes found that LSFEM with low-order elements does not show a satisfactory accuracy of solution. In this report, we propose an easy-to-implement technique that optimizes the condition number of the stiffness matrix. The least-squares functional is applied to the first-order governing equations and a linear combination of the weak form domain and boundary equations, with normalization (i.e., weighting) parameters is constructed. The impact of these parameters on the condition number of the resulting global stiffness matrix is proved to be independent of mesh size or shape function order. By use of a global optimization scheme that tunes these parameters, we observe notable decrease in the condition number compared to cases where the least-squares principle is directly applied to the governing equations, and hence convergence is significantly improved. It is further shown that the technique can be applied to the elemental stiffness matrix for a computationally efficient means of determining normalization parameters. To show the general applicability of the method, we also apply this technique to solve for a two-dimensional (2D) lid-driven flow problem and to calculate the electromagnetic field distribution in a 2D rectangular resonant cavity. In both problems, the LSFEM solution accuracy is poor prior to optimization of weighting parameters, and accuracy does not increase much by mesh refinement alone. Moreover, the process can be generalized to Galerkin FEM and other methods to improve convergence properties.

Plane wave discontinuous Galerkin methods for the Helmholtz equation and Maxwell equations in anisotropic media

In this paper we are concerned with plane wave discontinuous Galerkin (PWDG) methods for Helmholtz equation and time-harmonic Maxwell equations in three-dimensional anisotropic media, for which the coefficients of the equations are matrices instead of numbers. We first define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations. Then we derive the error estimates of the resulting approximate solutions with respect to the condition number of the coefficient matrices, under a new assumption on the shape regularity of polyhedral meshes. Numerical results verify the validity of the theoretical results, and indicate that the approximate solutions generated by the proposed PWDG method possess high accuracy.