张量空间与数值张量计算详解

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《张量空间与数值张量计算》是Springer系列计算机数学中的第42卷，由Wolfgang Hackbusch编著，隶属于Springer出版社的Computational Mathematics系列。这本书主要探讨了在现代科学与工程中广泛应用的张量理论及其在数值计算中的关键角色。本书的焦点在于张量空间，这是一种数学工具，用于描述多线性结构，特别适用于处理多变量的高阶问题，如在物理学、工程学、机器学习和数据科学等领域中的流体力学、电磁学、量子力学等问题。作者从基础概念出发，详细介绍了张量的定义、性质以及它们在向量和矩阵扩展后的表现形式，包括秩、协变和contra-variant张量的概念，以及如何在坐标变换下保持不变或变化。数值张量计算部分着重于将张量理论应用于实际数值方法，如数值微分、积分和方程求解。它涵盖了张量分解（如SVD、TT和CP分解）等技术，这些技术对于大型数据的高效存储和处理至关重要。书中还会探讨如何使用张量方法解决偏微分方程（PDEs）的离散化和求解，特别是在高维和多尺度问题中的应用。本书还涉及了数学分类，按照2010年AMS数学主题分类，包括15A69（张量代数）、35C99（泛函微分方程）、44A35（特殊函数的数值计算）、46A32（泛函分析中的核和算子）、46B28（函数空间上的运算）、47A80（算子理论中的近似和最佳逼近）、65F99（一般数值方法）、65N99（数值微分方程）、65L99（数值线性代数）、65N22（数值微分）、65N99（一般数值方法）以及65Z05（计算复杂度）等多个类别。版权信息方面，本书遵循德国1965年的版权法，强调所有权利保留，未经许可，任何形式的复制、重印、广播、存档或数据库存储都需要符合法律的规定。该书的国际标准书号(ISBN)和电子书号(e-ISBN)以及DOI均给出，方便读者查找和获取。《张量空间与数值张量计算》是一本深入研究张量理论在数值计算领域的专业著作，对于从事相关领域的研究人员、工程师以及研究生来说，具有很高的参考价值。

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xvi Contents

13.6 Hadamard Product of Tensors . . . .............................406

13.7 ConvolutionofTensors .....................................407

13.8 Matrix-Matrix Multiplication . . . .............................408

13.9 Matrix-Vector Multiplication . . . .............................408

13.9.1 IdenticalFormats ....................................409

13.9.2 SeparableForm(13.25a) .............................409

13.9.3 Elementary Kronecker Tensor (13.25b) . . . ..............410

13.9.4 Matrix in p-TermFormat(13.25c)......................411

13.10FunctionsofTensors,FixedPointIterations ....................412

13.11Example:OperationsforQuantumChemistryApplications .......414

14 Tensorisation ..................................................417

14.1 Basics ...................................................417

14.1.1 Notations, Choice of T

..............................417

14.1.2 Format H

tens

.......................................419

14.1.3 OperationswithTensorisedVectors ....................420

14.1.4 ApplicationtoRepresentationsbyOtherFormats .........421

14.1.5 Matricisation .......................................423

14.1.6 GeneralisationtoMatrices ............................423

14.2 ApproximationofGridFunctions ............................425

14.2.1 GridFunctions ......................................425

14.2.2 Exponential Sums . ..................................425

14.2.3 Polynomial Approximations for Asymptotically Smooth

Functions ..........................................426

14.2.4 Multiscale Feature and Conclusion . ...................427

14.2.5 LocalGridReﬁnement ...............................428

14.3 Convolution...............................................429

14.3.1 Notations ..........................................429

14.3.2 PreviewandMotivation ..............................430

14.3.3 Tensor Algebra A(

) ................................432

14.3.4 Algorithm ..........................................438

14.4 FastFourierTransform .....................................441

14.4.1 FFT for C

Vectors ..................................441

14.4.2 FFTforTensorisedVectors ...........................442

14.5 TensorisationofFunctions ..................................444

14.5.1 Isomorphism Φ

....................................444

14.5.2 Scalar Products . . . ..................................445

14.5.3 Convolution ........................................446

14.5.4 Continuous Functions . . . .............................446

15 Generalised Cross Approximation ...............................447

15.1 ApproximationofGeneralTensors ...........................447

15.1.1 ApproximationofMultivariateFunctions................448

15.1.2 Multiparametric Boundary Value Problems and PDE with

StochasticCoefﬁcients ...............................449

15.1.3 FunctionofaTensor .................................451

Contents xvii

15.2 Notations .................................................451

15.3 PropertiesintheMatrixCase ................................453

15.4 Case d ≥ 3 ...............................................455

15.4.1 Matricisation .......................................456

15.4.2 Nestedness .........................................457

15.4.3 Algorithm ..........................................459

16 Applications to Elliptic Partial Differential Equations ..............463

16.1 GeneralDiscretisationStrategy ..............................463

16.2 Solution of Elliptic Boundary Value Problems ..................464

16.2.1 SeparableDifferentialOperator ........................465

16.2.2 Discretisation .......................................465

16.2.3 SolutionoftheLinearSystem .........................467

16.3 Solution of Elliptic Eigenvalue Problems . . . ...................469

16.3.1 RegularityofEigensolutions ..........................469

16.3.2 IterativeComputation ................................471

16.3.3 AlternativeApproaches ..............................472

16.4 OnOtherTypesofPDEs ....................................472

17 Miscellaneous Topics ...........................................473

17.1 Minimisation Problems on V ................................473

17.1.1 Algorithm ..........................................473

17.1.2 Convergence .......................................474

17.2 SolutionofOptimisationProblemsinvolvingTensorFormats .....475

17.2.1 FormulationoftheProblem ...........................476

17.2.2 Reformulation,Derivatives,andIterativeTreatment .......477

17.3 OrdinaryDifferentialEquations ..............................478

17.3.1 TangentSpace ......................................478

17.3.2 Dirac-FrenkelDiscretisation ..........................478

17.3.3 Tensor Subspace Format T

...........................479

17.3.4 Hierarchical Format H

..............................481

17.4 ANOVA ..................................................483

17.4.1 Deﬁnitions .........................................483

17.4.2 Properties ..........................................484

17.4.3 CombinationwithTensorRepresentations ...............486

17.4.4 SymmetricTensors ..................................486

References ........................................................487

Index .............................................................495

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张量空间与数值张量计算详解

张量分析教科书（Tensor Algebra And Tensor Analysis For Engineers ）

RuntimeError: Input type (torch.cuda.LongTensor) and weight type (torch.cuda.FloatTensor) should be the same

:Dictionary inputs to traced functions must have consistent type. Found Optional[Tensor] and List[Tensor]

The size of tensor a (10) must match the size of tensor b (3) at non-singleton dimension 1

AssertionError: size of input tensor and input format are different. tensor shape: (1, 8, 32, 56, 56), input_format: CHW

tensor 和longtensor

can't convert Sparse layout tensor to numpy. Use Tensor.dense() first.

Create a 2D tensor with shape (2, 3) and values [[1, 2, 3], [4, 5, 6]]. Transpose the tensor.

sparsetensor转tensor

tensor_input =torch.cat([tensor1, tensor2, tensor3], dim=0) RuntimeError: zero-dimensional tensor (at position 0) cannot be concatenated

test_tensor = load_and_preprocess_image(test_img) test_tensor = tf.expand_dims(test_tensor, axis=0)

is_keras_tensor

将tensor变成tensor元组

The size of tensor a (32) must match the size of tensor b (512) at non-singleton dimension 1

tensor变量和longtensor变量

如何判断tensor是空tensor

tensor数组转tensor

TypeError: unsupported operand type(s) for +: 'Tensor' and 'function'、

两个tensor合成一个tensor

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